PHYS 7.3: Internal energy, heat and energy transfer	PPLATO @
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1 Opening items

1.1 Module introduction

Consider the following observations which you might make if you went on a cycle tour starting at sea level and heading into the mountains.

1

At the start of your tour, after pumping up your bike tyres, you notice that the end of the pump has become quite warm.

2

You set off; as you cycle higher and higher, you observe that the air is getting cooler even though the Sun is still out.

3

The Sun goes behind a cloud and you immediately feel colder.

4

You put on a jacket and feel warmer.

All of these observations are commonplace and you would not be surprised by any of them but can you explain the physics of what is happening? By the end of this module you should be able to do so, but first you will need to spend some time learning about heat, work and internal energy.

The study of these concepts forms the basis of thermodynamics (from the Greek thermos: warm and dynamis: power). Although it is apparently concerned with simple ideas, thermodynamics is an area of physics which requires precise thinking. For this reason the terminology needs to be carefully defined. Some of the more important definitions are introduced in Section 2.

The relationship between heat, work and internal energy is expressed in the first law of thermodynamics which is described in Section 3. It is this law which can be used to explain observations (1) and (2) above. Section 4 deals with the transfer of heat by conduction, convection and radiation. An understanding of these processes helps to explain observations (3) and (4).

Study comment Having read the introduction you may feel that you are already familiar with the material covered by this module and that you do not need to study it. If so, try the following Fast track questions. If not, proceed directly to the Subsection 1.3Ready to study? Subsection.

1.2 Fast track questions

Study comment Can you answer the following Fast track questions? If you answer the questions successfully you need only glance through the module before looking at the Subsection 5.1Module summary and the Subsection 5.2Achievements. If you are sure that you can meet each of these achievements, try the Subsection 5.3Exit test. If you have difficulty with only one or two of the questions you should follow the guidance given in the answers and read the relevant parts of the module. However, if you have difficulty with more than two of the Exit questions you are strongly advised to study the whole module.

1.3 Ready to study?

2 Some basic ideas of thermodynamics

2.1 Systems, states and equilibrium

Thermodynamics is the branch of science that deals with the macroscopic (i.e. large scale) aspects of temperature, energy and the transfer of energy. It is not concerned with microscopic (atomic scale) behaviour, and the whole subject can be developed logically without mentioning atoms or molecules, though we shall not adopt any such puritanical stance in this module.

The subject of any particular thermodynamic investigation is generally referred to as the system. The rest of the universe is then referred to as its environment. A system may be almost anything, a car engine, the air in a bicycle pump, the Earth’s atmosphere or a human being. The environment sounds rather grand, but in practice we usually need only take account of that portion of the universe which has a significant effect on the system, particularly with regard to energy transfer. In the case of a car engine, this ‘effective’ environment might be the petrol tank, the drive shaft and the surrounding air; in the case of the air in a bicycle pump it might be the pump itself and the air in the tyre to which the pump is attached.

The systems that are studied in thermodynamics are generally ones that can exist in several different states, each of which is specified by the values of various macroscopic properties of the system. For example, a fixed quantity of air trapped in a bicycle pump might be at a high temperature or a low one, its volume might be relatively large or relatively small, and its pressure might be quite high or quite low. By quoting values for these macroscopic properties (temperature, volume and pressure), or at least for as many of them as are independent, we can specify uniquely the state of the air in the pump and thereby enable any other investigator to reproduce the conditions we are describing. Macroscopic properties that can play this role have to be carefully chosen, and are sometimes referred to as thermodynamic coordinates.

In this module we will be concerned mainly with systems in, or close to, thermodynamic equilibrium. When a system is in thermodynamic equilibrium neither its macroscopic properties, nor the state they define, change with time; any such unchanging state is called an equilibrium state. By saying that a system is ‘close’ to equilibrium we imply that its bulk properties (and hence its state) may change with time, but only so slowly that the instantaneous state of the system can be approximated by an equilibrium state. A process in which the state of a system changes but is always very close to an equilibrium state is said to be a quasistatic process. Clearly, not all processes are quasistatic but a surprisingly large number are, and we can learn a lot about the real world even if we restrict our considerations to equilibrium states and quasistatic processes.

2.2 The ideal gas – an illustrative example

Thermodynamics is a powerful discipline, based on laws and concepts of wide applicability. However, when meeting thermodynamic concepts for the first time it is usually a good idea to keep a particular example in mind; otherwise the generality of the concepts can make them seem abstract and remote. The simple system traditionally used to introduce thermodynamic ideas is one consisting of a fixed quantity of ideal gas, at pressure P and temperature T, enclosed in a cylinder of variable volume V (see Figure 2a). If the cylinder contains n moles of gas, then, provided the system is in equilibrium, the pressure, volume and temperature of the gas will be related by the equation of state of an ideal gas:

PV = nRT(1)

where R = 8.314 J K⁻¹ mol⁻¹ is the molar gas constant. Note that the temperature in Equation 1 must be expressed in kelvin (K) for the equation to be valid. The behaviour of many real gases can be investigated, at least approximately, by treating them as ideal gases.

The relationship between P, V and T given by Equation 1 can be shown graphically by using a three–dimensional graph of the sort shown in Figure 2b. For a given value of n (i.e. for a fixed quantity of gas) each set of values for P, V and T that satisfies Equation 1 specifies a single point on the graph and corresponds to a unique equilibrium state of the system. The set of all such points (i.e. the set of all equilibrium states of the fixed quantity of gas) constitutes a continuous two–dimensional surface in the three–dimensional PVT space of the graph. Such a surface is called the PVT surface or equilibrium surface of the gas. The surface shown in Figure 2b has been truncated for ease of display, but ignoring that, every possible equilibrium state of the system is represented by a point on the equilibrium surface and, conversely, any point not on the equilibrium surface does not represent a possible equilibrium state of the system.

Looking at Figure 2b, or at the equation of state it represents, it is easy to see that for a given value of n (i.e. for a given equilibrium surface) a unique equilibrium state can be specified by assigning values to any two of the thermodynamic coordinates P, V and T. Once two of the coordinates have been fixed the other is automatically determined by the equation of state (Equation 1).

Question T2 shows that once the equilibrium state of a system is specified, certain other quantities (density and temperature in that particular case) are automatically determined by that state. Such quantities are said to be functions of state. In Question T2 the temperature and density were evaluated without worrying about how the gas came to be in the particular equilibrium state that it occupied. This is always possible for functions of state: their values depend only on the equilibrium state and not the manner in which the system reached that state.

2.3 Other thermodynamic systems

Although we shall pay a good deal of attention to ideal gases in this module, it must be remembered that there are countless other thermodynamic systems that can be studied in much the same way. Most obviously, there are the real gases, such as hydrogen and helium. Real gases at low pressure behave like ideal gases, but the behaviour of real gases is invariably more complicated when the pressure increases.

If you compress a sample of ideal gas all you get is a denser sample of ideal gas. However if you compress a real gas, particularly one that has been cooled to a relatively low temperature, it may well turn into a liquid, and if you continue the compression it may even become a solid. These different forms of matter – gas, liquid and solid – are said to be different phases of matter, and the processes that allow one phase to become another – such as the condensation of a gas, or the freezing of a liquid – are said to be changes of phase or phase transitions. The fact that a typical substance can exist as a solid, a liquid or a gas inevitably makes its equilibrium (PVT) surface more complicated than that of an ideal gas.

The equilibrium surface of a fixed quantity of a typical substance is shown in Figure 3. As you can see, when the temperature and volume are large the substance is a gas, but when they are low it is a solid. The liquid phase occurs under intermediate conditions and is distinctly separated from the solid phase, but not from the gas phase, so the distinction between a gas and a liquid is not always clear. Changes of phase, such as freezing and condensation, occur when the temperature, pressure and/or volume of the system are altered in such a way that the state of the system crosses one of the regions in which two different phases coexist.

Even the wide variety of real substances, typical or atypical, does not exhaust the range of systems that can be studied using thermodynamics. Working physicists are almost as likely to use it to study the behaviour of a cavity filled with light (as you will see in Subsection 4.3) as that of a cylinder filled with gas.

3 Heat, work and internal energy

In this section we will define and explore the relationship between heat, work and internal energy. This will enable us to write down the first law of thermodynamics.

3.1 An elementary problem

Imagine using a bicycle pump to pump up a tyre. After pumping for a few minutes, the end of the pump where gas is being compressed will become quite warm. The temperature of the air in that part of the pump is raised each time the pump is operated and the thermal contact that exists between the warm gas and the metal of the pump allows the pump itself to become warm as a result of the pumping action.

No prizes for getting the answer right – it was a very straightforward question. Here is something equally easy, but with wide ranging implications.

This point being made here is really the same point that was made at the end of Subsection 2.2. Temperature is a function of state, so its value depends only on the equilibrium state of the system, and not on how that state was attained. The system has no ‘memory’ of how it arrived at any particular equilibrium state. It may appear that a lot of fuss is being made over a rather simple point, but the equivalence (in the above sense) of directly heating a system or doing mechanical work on a system lies at the heart of the first law of thermodynamics.

3.2 Internal energy

We can gain more insight into the ‘curious’ behaviour of the bicycle pump by thinking in terms of energy rather than temperature.

The pump as a whole can have potential energy by virtue of its gravitational interaction with the Earth, and kinetic energy by virtue of its motion. However, in addition to these two contributions, its total energy will also contain a contribution arising from the atoms of which the pump is composed. Even when the pump as a whole is stationary, these atoms have their own microscopic motions which give the pump an internal kinetic energy that is quite distinct from the kinetic energy of its overall motion. Similarly, the electrical interactions between the atoms of the pump also give it an internal potential energy that can be distinguished from the gravitational potential energy associated with its overall position. The sum of these internal kinetic and potential energies, and any other internal energies that might exist, comprises the total internal energy of the pump. i

Internal energy plays an important part in thermodynamics and is related to temperature in an interesting way. As you might expect, when the temperature of a system is raised its internal energy increases. In the case of a fixed quantity of ideal gas confined in a cylinder this relationship is easy to understand; from a microscopic point of view the gas consists of tiny randomly moving particles that are incessantly colliding with one another and with the walls of their container. Raising the temperature of the gas simply corresponds to increasing the average kinetic energy of the particles.

So in this case, the temperature of the gas is a measure of the average internal kinetic energy associated with the random motion of the molecules that make up the gas. In fact, it can be shown that for an ideal gas of simple point like particles the internal energy U is related to the absolute temperature T by the equation:

$U = \dfrac3nRT$ i

For other systems, such as real gases or the metal of the pump, the situation is broadly similar, but the precise relationship between temperature and internal energy is more complicated. Although it remains true that an increase in temperature corresponds to an increase in internal energy, it is not necessarily the case that an increase in internal energy corresponds to an increase in temperature. For instance, if you take a bucket of ice at a temperature of −10°C (≈ 263 K) and directly heat it on a stove, you will raise its temperature and increase its internal energy. At least, that is what you will do until the temperature reaches 0°C and the ice starts to melt. Once melting commences, continued heating will still increase the internal energy of the contents of the bucket, but it won’t raise the temperature – it will just cause more of the ice to melt. Eventually, when all of the ice has changed its phase to become water (at 0°C) further increases of internal energy due to heating will once again cause the temperature to rise.

The complicated relationship between temperature and internal energy for a fixed quantity of a typical substance is shown in Figure 5.

The important point about internal energy is that it too is a function of state.i In other words, whenever the system is in an equilibrium state, its internal energy is determined by that equilibrium state and not by the processes that produced it. This explains why the cause of the raised temperature of the pump is bound to be ambiguous, unless we happen to know the detailed history of the pump. The raised pump temperature we observe simply reflects the raised internal energy of the pump. The fact that the internal energy is a function of state means that, in the absence of any additional information, there is no way of telling whether the internal energy was raised by direct heating (e.g. by immersing the pump in hot water) or by doing work (e.g. by pumping up a tyre).

3.3 Heat and work

In the preceding discussion, the terms ‘direct heating’ and ‘doing work’ were used in their everyday sense to indicate two different ways of increasing the internal energy of a system. In this subsection we shall give general definitions of heat and work, both of which are important concepts in thermodynamics. Let’s deal with heat first.

If two bodies at different temperatures are brought into thermal contact, the warmer one will be cooled and the cooler one warmed. If the two bodies are thermally isolated from the rest of the universe, they will end up at the same temperature. Everyone would agree that this process involves the flow of heat from one body to the other, but what exactly is heat, and how does it influence temperature?

Two hundred years ago, some scientists would have said that heat was a substance, sometimes called caloric. They would have explained the behaviour of bodies in thermal contact like this:

‘All bodies contain a fluid called caloric. Caloric flows from high temperature to low temperature and if you increase the caloric in a body its temperature will rise.’

Not all would have agreed. Experiments performed by Benjamin Thompson (1753–1814) (who later became Count Rumford of Bavaria) indicated that the caloric theory was incorrect. While working in Bavaria, Rumford studied the production of cannons (Figure 6) which were cooled by water while they were being bored. Rumford observed that more water was boiled away when the boring tool was blunt – a fact which could not be explained by caloric being squeezed out of the cannon into the water. Rumford’s alternative explanation was that the heating effect was caused by the motion of the borer itself. Later, the British scientist James Prescott Joule (1818–1889) refined these experiments and showed that there was an equivalence between heating and mechanical work. A given amount of work, performed in the course of stirring a liquid or rubbing a body or in some other appropriate way, had a fixed heating effect. This discovery paved the way for the eventual realization, by William Thomson (1824–1907) (who later became Lord Kelvin) and others, that direct heating and the performance of mechanical work were alternative methods of transferring energy to a system.

Armed with this insight into the importance of energy we would now describe the flow of heat in the following way;

‘All bodies contain internal energy. When two bodies with different temperatures are placed in thermal contact, energy is transferred from the body with the higher temperature to the body with the lower temperature. This net transfer of energy continues, altering the internal energies of the two bodies, until they attain the same temperature.’

What exactly is heat itself? It is notoriously difficult to clear away the clutter of history that surrounds this topic, but the conventional modern answer to the question is the following:

There are several points that should be noted about this highly restrictive definition of heat. First, you should notice that heating is not necessarily due to heat. We often say that a body has been ‘heated’ whenever we find that its temperature has been raised; but as you have seen, this might be due to energy that has been transferred from a hotter body (i.e. heat), or it might be the result of work being done e.g. stirring, rubbing etc.

Second, notice that the energy which is transferred as heat is not identified as some particular form of energy, it is simply energy. This means that once some heat has been absorbed into the internal energy of a body there is no way of distinguishing it from any of the other energy that makes up the total internal energy of the body. It is therefore not generally meaningful to speak about the ‘heat content’ of a body. Although physicists are well aware of this, it is still common practice to refer to the internal kinetic energy of a body as its ‘heat energy.’ Even more confusingly, when bodies collide it is not uncommon for them to do work on each other, with the consequence that the internal energy of each is increased and its temperature rises. In such circumstances it is quite normal to hear a physicist say that some of the kinetic energy involved in the collision has been dissipated as heat, even though the bodies may not have been at different temperatures throughout the process. These departures from modern terminology are not usually of much concern, but if you want to know whether heat (in the modern sense) has really been transferred in any particular process, just ask yourself if bodies with different temperatures were in thermal contact during the process.

If the only transfer of energy between a system and its environment is that due to temperature difference, then the increase in internal energy of the system ∆U must be equal to the heat Q transferred to it so,

∆U = Q(2)

The sign convention adopted in Equation 2 implies that heat flowing to a system is positive. Heat flowing from a system will therefore be negative and will decrease the internal energy of the system. i

Having defined heat, we can now lump together all other forms of energy transfer and give them the name of ‘work’.

Work is energy transferred between a system and its environment by any process in which a temperature difference is not directly involved.

Note that this definition of work is much more general than the definition of work used in mechanics, where it is essentially the (scalar) product of a force and a displacement in the direction of the force. However, the mechanical definition of work is fully consistent with the thermodynamic definition, and can be used to quantify the work done in certain situations as we shall now show.

Consider the system shown in Figure 7; a cylinder of internal length x and cross–sectional area A, containing a fixed quantity of ideal gas at pressure P and temperature T. One of the end walls of the cylinder is a piston which will be subject to a force F_x = PA due to the pressure of the gas.

Initially we will assume the system is in equilibrium with pressure P_a and volume V_a. This means that there must be some external agency (some part of the environment) that is balancing the force F_x exerted by the gas and preventing the piston from moving. We shall not be concerned with the details of that external force, but it should be kept it in mind since it is vital for the equilibrium of the system. Since the quantity of gas is fixed, the pressure and volume are sufficient to specify the equilibrium state of the gas as described in Section 2. The temperature, density and other bulk properties can be easily calculated.

Now suppose that the external force preventing the piston from moving is relaxed a little, so that the gas expands quasistatically until it arrives at another equilibrium state with pressure P_b and volume V_b. The force that the gas exerts on the piston will have done some mechanical work during this expansion and will have transferred some energy to the environment; we call this the work done by the gas on the environment. How much work will be done by the gas in the given expansion?

A constant force F_x acting in the x–direction and moving an object (such as the piston) through a small distance ∆x does a small amount of work given by:

∆W = F_x∆x

In the case of the gas filled cylinder we know that F_x = PA, so we can write ∆W = PA∆x, but we also know that the product A∆x is equal to the change in volume of the gas ∆V so:

∆W = PA∆x = P∆V

As the gas expands its pressure may fall, so the force F_x = PA that concerns us may not be constant. Nonetheless, we can still use the expression ∆W = P∆V to find the approximate amount of work done over a sufficiently small expansion. This means we can find an approximate expression for the total work done by the gas as its volume increases from V _a to V _b by dividing the total change of volume into many small volume increments and summing up the work done in each of those small increments (See Figure 8). Adopting this approach we see that the total work done by the thermally isolated gas is approximately:

W ≈ P(V₁)∆V₁ + P(V₂)∆V₂ + P(V₃)∆V₃ + ... + P(V_n)∆V_n

i.e.$\displaystyle W \approx \sum_{i=1}^n P(V_i)\Delta V_i$

where V₁ = V_a, V_{n + 1} = V_b, and P(V_i) represents the pressure in the gas at the start of the i^th small volume increase.

We can get a more accurate expression for the work done by reducing the size of the largest step, and letting the number of steps correspondingly increase. In fact, in the limit as the largest step becomes vanishingly small this sum yields the exact value of W, so we may write:

$\displaystyle W = \lim_{\Delta V\to 0}\left(\sum_{i=1}^n P(V_i)\Delta V_i\right)$(3)

A limit of a sum of this kind is generally referred to as a definite integral, and is written as follows:

$\displaystyle W = \int_{V_{\rm a}}^{V_{\rm b}}P(V)\,dV$(4)

where P(V) now represents the pressure of the gas when its volume is V. The steps leading to Equation 4 are explained in greater detail in the block of modules devoted to integration in the maths strand of FLAP, but if you are not already familiar with the techniques of integration you will probably find it easier to think of the process in graphical terms. In this case the smooth curve of Figure 8 shows the variation of the pressure P with the volume V for the process we are considering.

The work done in any small part of the expansion from V_i to V_i+1 is approximately represented by the area of the corresponding rectangular strip, P(V_i)∆V_i. The total amount of work done in expanding from V_a to V_b is approximately represented by the sum of the areas of all the rectangles. As the number of rectangles increases and they become narrower, their total area approaches the area under the smooth curve between V_a and V_b. The area under the curve between V_a and V_b, measured in the scale units appropriate to the graph, may therefore be said to equal the definite integral of P(V) from V_a to V_b.

In order to use Equation 4, or the equivalent graph, to evaluate the work done in any particular process we need to know the precise relationship between P and V at every stage of that process. Remember, any quasistatic process corresponds to a particular pathway on the PVT surface and therefore implies a particular relationship between P and V.

For the sake of definiteness, let us suppose that the cylinder in Figure 7 is thermally isolated from its environment so that no heat can flow in or out of the cylinder during the process. A process of this kind is called an adiabatic process. We shall not pause to prove it here, but it can be shown that during an adiabatic process a fixed quantity of ideal gas satisfies an equation of the form:

P = BV^−γ

where γ is a constant determined by the nature of the gas, and B is a constant that characterizes the isothermal process. i It should be noted that the gas satisfies this adiabatic condition as well as the ideal gas equation of state (PV = nRT). Given a value for γ, the initial state of the process will determine the value of B, and the adiabatic condition will then determine those equilibrium states which can be reached by an adiabatic process starting from the given initial state.

Using the adiabatic relation between P and V, we see from Equation 4 that the work done in an adiabatic process in which the volume changes from V_a to V_b is

$\displaystyle W = \int_{V_{\rm a}}^{V_{\rm b}}BV)^{-\gamma}\,dV = \left[\dfrac{BV^{1-\gamma}}{1-\gamma}\right]_{V_{\rm a}}^{V_{\rm b}} = \dfrac{B\left(V_b^{1-\gamma}-V_a^{1-\gamma}\right)}{1-\gamma}$ i

Since no heat is exchanged between a system and its environment during an adiabatic process any change in the internal energy of the system must be entirely due to work done by or on the gas. If, as above, we use W to represent the work done by the system during a process, it follows that the change in the internal energy of the system ∆U during an adiabatic process is given by:

∆U = −W(5) i

The sign convention adopted in Equation 5 implies that work done by a system is positive. Work done on a system will therefore be negative and will increase the internal energy of the system. Note that Equation 5 is true of any adiabatic process, not just those involving ideal gases.

The example we have been discussing involved mechanical work – but it is worth remembering that our definition of work includes all processes of energy transfer that do not directly involve a temperature difference. For example connecting a wire (the system) between the terminals of a battery (part of the environment) will cause the wire’s temperature (and hence its internal energy) to rise. If the wire is thermally insulated then Equation 5 still applies but this time W is the (negative) chemical work done on the wire by the battery.

3.4 The first law of thermodynamics

Subsection 3.3 described two methods of increasing the internal energy of a system: transferring heat to it and doing work on it. We deliberately considered processes in which either one method of energy transfer occurred or the other. However, many processes involve both methods.

In the presence of both methods of energy transfer we might expect to find that ∆U = Q − W. As we shall shortly demonstrate, the values of Q and W involved in any particular process depend on the details of that process, but U is a function of state, so the value of ∆U will depend only on the initial and final equilibrium states that mark the beginning and end of the process.

The formal statement that this is so is known as the first law of thermodynamics: i

Since heat and work have been defined in such a way that they include all possible methods of transferring energy between a system and its environment, the first law amounts to the assertion that energy is conserved.

The internal energy of a system is only changed when energy is transferred to or from that system. In the context of the nineteenth century, the first law highlighted the general importance of energy, an importance that was only beginning to be appreciated around 1850. It also plays an important part in the logical structure of thermodynamics since it implies the existence of a function of state U without any recourse to microscopic arguments about the energies of individual atoms or molecules of the sort we permitted ourselves in Subsection 3.1.

Let’s now use the first law to show that Q is generally not a function of state, thereby demonstrating that the idea that bodies have an identifiable ‘heat content’ must be wrong. Once again we wish to consider a cylinder containing a fixed quantity of ideal gas, and a process in which the equilibrium state of the gas changes from one in which P = P_a and V = V_a to one in which P = P_b and V = V _b. However, this time, instead of thermally isolating the cylinder, we will allow heat exchange with the environment by arranging for part of the cylinder to be in contact with a thermal reservoir (that is a large body to or from which heat can be transferred without changing its temperature T_res, see Figure 9).

We needn’t worry about how a thermal reservoir might be constructed, but we will suppose that we can adjust its temperature T_res to be any value we desire. That means we can control the flow of heat into or out of the system by setting T_res to be greater or less than the temperature of the gas.

Using this apparatus we can investigate a number of different processes leading from state 1 (defined by P_a and V_a) to state 4 (defined by P_b, V_b). Two of these processes are shown in Figure 10, each is a two–step process; the first will be called 1–2–4 and leads from the initial state to the final state via state 2(P_b, V_a); the second will be called 1–3–4 and goes from the initial state to the final state via state 3(P_a, V_b). These processes are described in more detail below; neither of them is adiabatic.

Table 1 Description of the processes shown in Figure 10.

Process 1–2–4	Process 1–3–4
Step 1 Holding the piston fixed, so that V remains constant at Va, decrease the value of Tres so that heat flows out of the system, its temperature falls and its pressure decreases until it reaches state 2.	Step 1 By suitable adjustments of Tres and movement of the piston, increase the volume while keeping the pressure constant at Pb until the gas reaches state 3.
Step 2 By suitable adjustments of Tres and movement of the piston, increase the volume while keeping the pressure at Pb constant until the gas reaches state 4.	Step 2 Holding the piston fixed, so that V remains constant at Vb, decrease the value of Tres so that heat flows out of the system, its temperature falls and its pressure decreases until it reaches state 4.

The answer to Question T6 shows that the work done by the system in moving between the initial and final equilibrium states depends on the path taken.

As noted earlier, the fact that internal energy and temperature are functions of state while work and heat are not is the reason why it is impossible to tell whether a temperature rise in a bike pump has been caused by direct heating or by work done on the gas or by a combination of the two.

In Subsection 3.3 we considered an adiabatic process in which no heat flowed in or out of the system, and we saw that in such a process a fixed quantity of ideal gas obeys the condition PV^γ = constant. In this subsection we have considered constant volume processes (such as Step 1 of process 1–2–4) and constant pressure processes (such as Step 1 of process 1–3–4).

An example of each of these processes is shown in Figure 11, along with their projections onto the PV plane. Also shown is another kind of process, an isothermal process, in which the temperature remains constant.

All these simple processes play important roles in thermodynamics.

For a fixed quantity of ideal gas, it follows from the equation of state (PV = nRT) that in an isothermal process the gas must obey the condition

PV = D(7)

where D is a constant that characterizes the isothermal process. It should be noted that the gas satisfies this isothermal condition as well as the ideal gas equation of state (PV = nRT). The initial state of the process will determine the value of D, and the isothermal condition will then determine those equilibrium states that are accessible in an isothermal process that starts from the given initial state.

The work done by a fixed sample of ideal gas during an isothermal expansion from V_a to V_b is given by:

$\displaystyle W = \int_{V_{\rm a}}^{V_{\rm b}}P\,dV = \int_{V_{\rm a}}^{V_{\rm b}}\dfrac DV\,dV = D\loge\left(\dfrac{V_{\rm b}}{V_{\rm a}}\right)$ i

3.5 The first law and the Earth’s atmosphere

Subsection 1.1 mentioned the commonplace observation that atmospheric temperature decreases with height. In fact this holds good up to heights of about 10 km and the average reduction of temperature with height is about 6 K km⁻¹. This phenomenon can at least in part be explained by the first law of thermodynamics. First consider atmospheric pressure.

Now consider what happens if a body of air is forced to rise because of a mountain. As it rises it expands because of the reduced pressure from the surrounding air acting on it. In thermodynamic terms, it is doing a positive amount of work W on its environment. As air is a poor conductor of heat (see Section 4), we may assume that there is negligible heat exchange with the surrounding air, so the expansion is adiabatic. Consequently Q = 0 and, from the first law of thermodynamics,

∆U = −W

Since W is positive, ∆U is negative and thus the internal energy will decrease and the temperature will fall.

We have described the decrease in temperature with height for a single, imaginary block of air. In reality most of the air in the bottom 10 km of the atmosphere is continually rising and falling mainly due to convection (see Section 4) and the effects of mountain ranges. If the air was dry this would result in an overall temperature variation with height exactly corresponding to that of a rising, adiabatically expanding, block of air (about 10 K km⁻¹). In fact the actual value is about 6 K km⁻¹ due mainly to the effect of the condensation of water vapour. This is the kind of cooling which you might expect to observe on the cycle tour described in Subsection 1.1.

3.6 Summary of Section 3

1

Internal energy U is the sum of the potential and kinetic energies associated with the constituent molecules of a system.

2

Heat is energy transferred between a system and its environment as a result of the temperature difference between them. Heat always flows from high temperature to low temperature.

3

Work is the energy transferred between a system and its environment by any process in which a temperature difference is not directly involved.

4

Internal energy is a function of state. Heat and work are not functions of state.

5

The first law of thermodynamics states that, if a system undergoes a change from one equilibrium state to another, the difference between the heat Q supplied to the system and the work W done by the system will depend only on the initial and final states and not on the process by which this change is brought about. This provides a thermodynamic justification for introducing the internal energy as a function of state U such that:

∆U = Q − W

6

An adiabatic process is one in which there is no heat exchange between the system concerned and its environment. An isothermal process is one in which there is no change of temperature.

7

The reduction of temperature with height in the bottom few kilometres of the atmosphere can be at least partially explained as the result of adiabatic expansion.

4 Methods of heat transfer

In Section 3 heat was defined as energy transferred because of temperature difference. In this section we will explore three different mechanisms by which the transfer of heat is effected. They are conduction, convection and radiation.

4.1 Conduction

Perhaps the simplest mode of heat transfer is conduction_of_heatconduction. It can occur in all three phases of matter: solids, liquids and gases. To fix our ideas we think first of a thin slab of solid shown in Figure 13 with thickness ∆l, cross–sectional area A and a temperature difference between its flat faces of ∆T. We can assume that ∆l is so small that heat transfer out of the slab through the sides is negligible. If we imagine a quantity of heat ∆Q flows through the slab in the small time interval ∆t, then the rate of flow of heat can be written as $\dfrac{\Delta Q}{\Delta t}$. It is found experimentally that $\dfrac{\Delta Q}{\Delta t}$is proportional to the temperature gradient $\dfrac{\Delta T}{\Delta l}$and the cross–sectional area A.

Thus$\dfrac{\Delta Q}{\Delta t} \propto A\dfrac{\Delta T}{\Delta l}$

or$\dfrac{\Delta Q}{\Delta t} = -\kappa A\dfrac{\Delta T}{\Delta l}$

where κ (the Greek letter kappa) is a constant called the thermal conductivity coefficient i that is characteristic of the material of the slab. The negative sign indicates that the direction of heat flow is in the opposite direction to the temperature gradient. That is, it flows from hot to cold. In the limit as ∆l and ∆t approach zero, the equation can be written using derivatives, as

$\dfrac{dQ}{dt} = -\kappa A\dfrac{dT}{dl}$(8)

Equation 8 is known as Fourier’s law and it can be applied to practical problems such as heat flow through walls or along bars. i Often the calculations are complicated by the fact that κ varies with temperature and the heat flow is three-dimensional, but we will only consider simple examples where the flow is one-dimensional.

Such a situation is shown in Figure 14.

If in addition a steady state has been reached so that there is no change of temperature with time at any point in the bar, a simplified form of Equation 8 can be used:

$\dfrac Qt = -\kappa A\dfrac{T_2-T_1}{l}$    (in the absence of convection)(9)

where l is the length of the bar, and Q is the heat that flows in time t.

4.2 Convection

Consider a fluid initially in mechanical equilibrium so that is there is no bulk movement. Now imagine that a small source of heat is placed somewhere in the fluid. The temperature of the fluid next to the heat source will increase causing expansion and resulting in a decrease in density.

The weight of fluid occupying a given volume is now less but the force due to pressure difference which balanced the original weight of fluid (see Figure 12a) has not changed. Thus there is a net upward force on the less dense fluid. As it rises its place is taken by cooler, denser fluid which will in turn warm, expand and rise. If a mechanism exists whereby the warm, rising fluid transfers heat to some cooler body (a heat sink) then a cyclical process is established so that the fluid circulates and heat is continually transferred from heat source to heat sink. This method of heat transfer is called convection. i

In general convection is a complicated phenomenon, depending sensitively on:

• the shape of the containing vessel,
• whether the convection is forced or free. Free convection implies that the fluid movement is purely in response to the change of density described above. Forced convection implies that the fluid movement is mechanically assisted. Examples of forced convection include the use of a fan in an oven, stirring a cup of tea or a cool breeze across your cheek,
• the directions and magnitudes of the temperature gradient and the gravitational force,
• the viscosity (loosely, the ‘stickiness’) of the fluid.

 

There exist a few simple cases which can be analysed in detail. We will briefly mention one.

Consider natural convection between two horizontal plane surfaces of large area A and small separation ($l \ll \sqrt{A}$) as shown in Figure 15a. i

The surfaces above and below the fluid layer are held at temperatures T_a and T_b respectively. If T_a > T_b then no convection can occur and heat transfer will be entirely by conduction. We can thus measure the thermal conductivity κ if we know the rate of heat flow and the dimensions of the fluid.

This is given by rearranging Equation 9,

$\dfrac Qt = -\kappa A\dfrac{T_2-T_1}{l}$    (in the absence of convection)(Eqn 9)

so$\kappa = - \dfrac Qt\times\dfrac1A\times\dfrac{l}{\Delta T}$    (in the absence of convection)

where ∆T = T_a − T_b.

Plotting κ against different values of ∆T should show only small variations as in Figure 15b. If the experiment is repeated with T_b > T_a, so that the heat flow is upwards, something more complicated happens. For small ∆T, the value of −Ql/(tA∆T) is the same as before, but as ∆T increases beyond a certain critical value ∆T_c, the value of this quantity begins to increase (Figure 15b). The changing value of this quantity with ∆T shows that the flow of heat is no longer simply proportional to ∆T.

The critical temperature difference ∆T_c represents the temperature difference at which convection switches on and the results are then roughly consistent with $\dfrac{dQ}{dt}$ proportional to (∆T)^5/4 (rather than ∆T) for ∆T > ∆T_c.

The flow pattern in the fluid is also of interest; for this geometry the fluid organizes itself into a regular array of hexagonal cells with fluid moving in one direction along the axis of the hexagons and in the opposite direction along their boundaries. All this complexity from natural convection in a simple geometry! Just imagine how much more complicated it can get. Do not despair! Although it is very difficult to model the intricate details of convection in all but the most simple situations, useful results can often be obtained using a simple empirical formula:

where h is called the convective heat transfer coefficient. As you can see, this formula is a little like Fourier’s law in Equation 8,

$\dfrac{dQ}{dt} = -\kappa A\dfrac{dT}{dl}$(Eqn 8)

but it does not contain a length l.

The convective heat transfer coefficient is something which can be looked up in tables or calculated from standard formulae. It depends on shape and orientation, and may also depend on ∆T as in the example shown in Figure 15. From a practical point of view we cannot afford to overlook convection.

Empirical equations such as Equation 10 are useful in appropriate situations, but one should retain a sense of caution about depending too heavily on such calculations in view of the complexity of the phenomenon.

4.3 Electromagnetic radiation and the black–body spectrum

The third process of heat transfer we wish to discuss is commonly called radiation_electromagneticradiation. In the context of heat transfer, this is taken to mean the emission and absorption of electromagnetic radiation.

Electromagnetic radiation is a wave phenomenon, consisting of oscillating electric and magnetic fields and it is something you are in contact with every second of your life. i Light, radio waves, infra–red radiation, ultraviolet radiation, gamma rays and X–rays are all forms of electromagnetic radiation differing only in their wavelength (the distance between successive wave peaks).

These different kinds of radiation collectively form the electromagnetic spectrum, as illustrated in Figure 16.

As far as this module is concerned, the important thing about electromagnetic radiation is that it carries energy. All objects emit energy in the form of electromagnetic radiation. i You yourself are radiating energy at about the same rate as a domestic light bulb as you read this module!

The power emitted per unit area of emitting surface is usually carried away by electromagnetic radiation of various wavelengths, so it makes sense to describe the emission in terms of the power radiated per unit area per unit wavelength range. This quantity is called the spectral brightness R_λ i of a surface and generally depends on the temperature and nature of the emitting surface, as well as the wavelength being observed. The spectral brightness of your own body would be greatest at infra–red wavelengths, whereas that of the Sun is greatest at visible wavelengths.

Thermodynamic arguments beyond the scope of this module imply that at a given temperature there is an upper limit to the spectral brightness of a surface at any given wavelength. Any body which produces this maximum spectral brightness at a given wavelength would constitute a perfect emitter at that wavelength and should, according to thermodynamics, also be a perfect absorber of radiation at that wavelength. Consequently, any object which is a perfect emitter at all wavelengths will also be a perfect absorber at all wavelengths. For this reason a perfect emitter of radiation is rather surprisingly known as a black body, though the alternative term full radiator is sometimes used.

Exactly how much radiation is emitted by a black body at a given temperature? If we had such an object it would be easy to make measurements. Unfortunately, no real surface emits perfectly at all wavelengths. However a very good approximation can be achieved by forming a cavity with a small aperture in a solid object as shown in Figure 17.

Almost all radiation entering through the aperture will eventually be absorbed somewhere inside the cavity, so the cavity constitutes an (almost) perfect absorber and is therefore a good approximation to a black body. Some of the radiation emitted by the walls of the cavity will inevitably escape through the aperture, and this may be used to study black–body radiation. By adjusting the (uniform) temperature of the cavity’s walls we can investigate how the spectral brightness of a black body varies with temperature as well as wavelength. Such a device is called a cavity radiator. In this situation the hole will always appear brighter than the entire body. This is a direct result of the hole being nearer to a true black body than is the surface of the body. A similar effect is sometimes seen from a cavity within an open fire.

The variation of spectral brightness with wavelength for a black body at a given temperature defines the black–body spectrum at that temperature. A sequence of black–body spectra at different temperatures is shown graphically in Figure 18. As you can see, the shape of the spectrum is the same for all temperatures. As the temperature rises, the peak in the curve moves to shorter wavelengths and the total emitted power (represented by the area under the curve) increases. This is in line with everyday observation. A heated object will start to glow dull red as its brightness peak enters the visible region of the spectrum. As its temperature continues to rise it turns orange, yellow and then white as the peak moves to shorter and shorter wavelengths. A very hot star appears blue since its peak emission is in the ultraviolet and it is seen mainly by the light emitted in the blue end of the spectrum.

The theory of black–body radiation is now well understood. The relationship between R_λ, λ and T for a black body was formulated by Max Planck (1858–1947) in 1900 and is known as Planck’s function. It takes the form

$R_\lambda = \dfrac{c_1}{\lambda^5}\dfrac{1}{\left({\rm e}^{c_2/kT}-1\right)}$(11a) i

where c₁ and c₂ are constants.

Integrating the Planck function of Equation 11a to find the total power emitted per unit area by a black body across all wavelengths, looks a bit formidable but fortunately the final result can be expressed simply. If R is the total emitted power per unit area, then:

Equation (12a) is known as Stefan’s law and σ is Stefan’s constant. In SI units σ = 5.67 × 10⁻⁸ W m⁻² K⁻⁴ i

For real objects, the power emitted at a given wavelength is usually less than the black–body spectral brightness. Figure 19 compares the spectrum for tungsten at 2000 K with the black body curve.

Equations 11a and 12a can be adjusted to account for the behaviour of real objects by incorporating a factor known as the emissivity ε (the Greek letter epsilon). Thus Equation 11a becomes

$R_\lambda = \dfrac{c_1}{\lambda^5}\varepsilon_\lambda\dfrac{1}{\left({\rm e}^{c_2/kT}-1\right)}$(11b)

The subscript λ emphasizes the fact that the emissivity may vary with wavelength. ε_λ is always between 0 and 1.

Equation 12a becomes:

R = εσT⁴(12b)

where ε is now a representative value of ε_λ over the whole spectrum and therefore may vary with temperature.

4.4 Heat transfer by radiation

It is now just a small step to see the link between electromagnetic radiation and heat transfer. Any object is continually radiating energy into its environment and absorbing energy from its environment. If for the moment we ignore all other forms of energy transfer, then from the first law of thermodynamics any difference between the incoming and outgoing radiation must lead to a change in internal energy of the object:

∆U = E_in − E_out(13)

where E_in is the incoming radiation in a given time, and E_out is the outgoing radiation in the same time. For an object in thermal equilibrium with its environment, E_in = E_out so there is no change in ∆U and thus no change in temperature.

Question T22 proves that for this special case, the net rate of energy transfer by radiation between the two bodies is proportional to the difference between the fourth powers of their absolute temperatures. Fortunately, this result is also true whether or not the objects are black bodies and whatever their shape and separation. This is not difficult to prove but there is not room to do so in this module.

In general then, the net rate of heat transfer by electromagnetic radiation to a body at temperature T₁ from a body at temperature T₂ satisfies the relation:

$\dfrac{dQ}{dt} \propto -(T_1^4 - T_2^4)$

For a body of area A with temperature T₁ and emissivity ε₁ surrounded by an environment with emissivity ε₂ and temperature T₂:

The minus sign is for consistency with the previous convention that energy flow to the body is positive.

There is one more point to be made. Equations 9,

$\dfrac Qt = -\kappa A\dfrac{T_2-T_1}{l}$    (in the absence of convection)(Eqn 9)

and Equation 10 show that the rate of heat flow in both conduction and convection is proportional to the temperature difference. It would be very useful if Equation 14 could be written in the same form.

Fortunately it can, provided T₁ − T₂ is small compared with T₁ and T₂. Under these circumstances it can be shown (using the binomial expansion) that (T₁ⁿ − T₂ⁿ) ≈ nT⁽ⁿ⁻¹⁾(T₁ − T₂). Thus, for small temperature differences, heat transfer by conduction, convection and radiation are all proportional to the temperature difference. This makes life a lot easier in complex situations where all three processes are involved and explains the basis of an empirical result known as Newton’s law of cooling which states that the rate of heat transfer from a hot body to its environment is proportional to the difference in temperature between them.

4.5 Human beings and heat transfer

We are now in a position to look back at the rest of the observations made in the introductory example in Subsection 1.1 with regard to an imaginary cycling tour. i

The third observation was that you feel colder when the sun goes behind a cloud. Of course you do; you are no longer in a position to absorb the direct electromagnetic radiation. Having got this far in the module, we can perhaps be a little more quantitative.

The solar radiation received at the Earth’s surface with the Sun directly overhead in a clear sky is of the order of 1 kW m⁻². Obviously this will decrease if the atmosphere is not clear and the sun is not overhead. The power per unit area your skin receives will also depend on the orientation relative to the Sun’s rays but the point we wish to make is that the incoming solar radiation is of the same order of magnitude as the radiant energy loss for each unit area of exposed skin. Therefore it is not surprising that you notice whether or not you are receiving it.

The final observation was that you put on a jacket and feel warmer. Again we know this to be true – but is it consistent with the theory we have of heat transfer? As the Sun has gone behind a cloud, the solar radiation component is not so important. Radiative heat transfer between the jacket and the body is probably the same as between the rest of the environment and the body and so is unchanged. Your skin surface temperature is typically 32°C–35°C. This is warmer than the atmosphere in most circumstances and thus free convection will be initiated as the layer of air next to the skin is warmed, expands and rises. The efficiency of the convection depends very much on the clothing (or lack of it). Tight–fitting clothing or clothing designed to trap pockets of air next to the skin effectively prevents convection. The only way in which energy can now be transferred from the body is by conduction – but the air is a very poor conductor, so you feel warm. Loose–fitting clothing allows the air to circulate and feels cooler.

5 Closing items

5.1 Module summary

1

Thermodynamics is concerned with the macroscopic aspects of temperature, energy and the transfer of energy between a system and its environment. Amongst other things it deals with a system’s equilibrium states, which can usually be completely specified by the values of a small number of macroscopic properties, and quasistatic processes in which the changing state of a system is always very close to an equilibrium state. A typical example of a thermodynamic system is n moles of ideal gas, confined in a cylinder of volume V at temperature T and pressure P. Such a gas obeys the equation of state of an ideal gas PV = nRT, where R is the molar gas constant, and has a set of equilibrium states that can be represented by a continuous two–dimensional PVT surface.

2

Heat is energy which is transferred between a system and its environment as a direct result of temperature differences. The direction of heat flow is from high temperature to low temperature.

3

Work is energy transferred between a system and its environment by any other means.

4

An adiabatic process is one in which no heat enters or leaves a system. When a fixed quantity of ideal gas undergoes an adiabatic expansion from V_a to V_b, P = BV_−γ and the work done by the gas is:

$\displaystyle W = \int_{V_{\rm a}}^{V_{\rm b}}BV)^{-\gamma}\,dV = \left[\dfrac{BV^{1-\gamma}}{1-\gamma}\right]_{V_{\rm a}}^{V_{\rm b}} = \dfrac{B\left(V_b^{1-\gamma}-V_a^{1-\gamma}\right)}{1-\gamma}$

where B is a constant that characterizes the process.

An isothermal process is one in which the temperature of the system does not change. When a fixed quantity of ideal gas undergoes an isothermal expansion from V_a to V_b, P = D/V and the work done by the gas is:

$\displaystyle W = \int_{V_{\rm a}}^{V_{\rm b}}P\,dV = D\loge\left(\dfrac{V_{\rm b}}{V_{\rm a}}\right)$

where D is a constant that characterizes the process.

In a constant pressure process the work done by the gas is W = P(V_b − V_a), and in a constant volume process the gas does no work.

5

The first law of thermodynamics asserts that if a system undergoes a change from one equilibrium state to another, the difference between the heat Q supplied to the system and the work W done by the system will depend only on the initial and final equilibrium states and not on the process by which the change is brought about. This justifies the introduction of a function of state known as the internal energy U which changes by an amount

∆U = Q − W(Eqn 6)

in any quasistatic process. For many common systems U may be interpreted as the sum of the disordered microscopic energies of the atoms and molecules that make up the system, and may be related to the temperature of a system. Because Q and W are not functions of state their respective contributions to U cannot be disentangled, and it does not generally make sense to speak of the ‘heat content’ of a system.

6

The rate at which heat is transferred along a uniform bar of cross–sectional area A by conduction is related to the temperature gradient along the bar by Fourier’s law:

$\dfrac{dQ}{dt} = -\kappa A\dfrac{dT}{dl}$(Eqn 8)

where κ is the thermal conductivity coefficient, a characteristic of the material of the bar. Conduction does not involve any bulk motion of the conducting material.

7

Convection is a complex heat transfer process that can occur in a gravitational field when part of a fluid is heated, expands, and rises above denser parts. In some simple situations, the rate of heat transfer by convection between two parallel planes of area A that differ in temperature by an amount ∆T can be approximated by the equation:

$\dfrac{dQ}{dt} = -hA\Delta T$

where h is the convective heat transfer coefficient.

8

All objects emit and absorb electromagnetic radiation. The power emitted per unit area per unit wavelength is known as the spectral brightness. The spectral brightness of a surface varies with wavelength and generally depends on the temperature and nature of the emitting surface.

9

Surfaces which are strong/weak emitters at a given wavelength are also strong/weak absorbers at that wavelength. A black body or full radiator is a body which absorbs all incident radiation at all wavelengths. Such a body is also a perfect emitter at all wavelengths. The variation of spectral brightness with temperature T and wavelength λ for a black body is described by Planck’s function:

$R_\lambda = \dfrac{c_1}{\lambda^5}\dfrac{1}{\left({\rm e}^{c_2/kT}-1\right)}$(Eqn 11a)

10

The total power emitted per unit area of surface can be calculated from Stefan’s law:

R = εσT⁴(Eqn 12b)

Where ε is the (average) emissivity of the surface at the relevant temperature. For a black body ε = 1.

11

For a body of surface area A and emissivity ε₁ at temperature T₁ surrounded by an environment of emissivity ε₂ at temperature T₂, the rate of heat transfer to the body by radiation is:

$\dfrac{dQ}{dt} = -A\varepsilon_1\varepsilon_2\sigma(T_1^4 - T_2^4)$(Eqn 14)

If (T₁ − T₂) is small, then dQ ∝ −(T₁ − T₂) is a good approximation.

5.2 Achievements

Having completed this module, you should be able to:

A1

Define the terms that are emboldened and flagged in the margins of the module.

A2

Distinguish between internal energy, heat and work, and evaluate those quantities in various circumstances. (Particularly in the case of an ideal gas undergoing various simple processes.)

A3

State the first law of thermodynamics, explain its significance, and use it in simple calculations.

A4

Write down an equation relating heat flow by conduction to temperature gradient and do simple problems based on this equation.

A5

Describe the process of heat transfer by convection, and write down an empirical equation that may be used to estimate the rate of heat transfer by convection in simple situations.

A6

Explain what is meant by a black body, sketch a graph showing the variation of the spectral brightness of a black body with wavelength (Planck’s function), and describe how the main features of the graph vary with the temperature of the black body.

A7

Write down, interpret and use Stefan’s law for the total power emitted per unit area of a black body at temperature T.

A8

Describe the way in which absorption and emission of electromagnetic radiation give rise to heat transfer.

Study comment You may now wish to take the following Exit test for this module which tests these Achievements. If you prefer to study the module further before taking this test then return to the topModule contents to review some of the topics.

5.3 Exit test

Study comment Having completed this module, you should be able to answer the following questions each of which tests one or more of the Achievements.

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