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Physics 304

Thermodynamics Notes

An Introduction to Thermal Physics , D. V. Schroeder, Addison Wesley Longman, 2000

© J Kiefer 2009

I. First “Law” of Thermodynamics

• 1 A. Temperature Sec 1.1..........................................................................
  • 1.1 1. Energy....................................................................................
  • 1.2 2. Thermal Equilibrium..............................................................
  • 1.3 3. Thermometers........................................................................
• 2 B. Work...................................................................................................
  • 2.1 1. First “Law” of Thermodynamics Sec 1.4.............................
  • 2.2 2. Compressive Work Sec 1.5...................................................
  • 2.3 3. Other Works...........................................................................
• 3 C. Heat Capacity Sec 1.6.......................................................................
  • 3.1 1. Changing Temperature...........................................................
  • 3.2 2. Heat Capacity and Degrees of Freedom................................
• 1 II. Second “Law” of Thermodynamics...................................................
  • 1.1 A. Combinatorics....................................................................................
    • 1.1.1 1. Two State Systems Sec 2.1..................................................
    • 1.1.2 2. Einstein Solid Sec 2.2, 2.3...................................................
  • 1.2 B. Entropy
    • 1.2.1 1. Large Systems Sec 2.4.........................................................
    • 1.2.2 2. Second “Law” Sec 2.6.........................................................
  • 1.3 C. Creating Entropy................................................................................
    • 1.3.1 1. Temperature Sec 3.1, 3.2......................................................
    • 1.3.2 2. Pressure Sec 3.4...................................................................
    • 1.3.3 3. Chemical Potential Sec 3.5..................................................
    • 1.3.4 4. Expanding & Mixing Sec 2.6
• 2 III. Processes............................................................................................
  • 2.1 A. Cyclic Processes................................................................................
    • 2.1.1 1. Heat Engines & Heat Pumps Sec 4.1, 4.2.............................
    • 2.1.2 2. Otto, Diesel, & Rankine Sec 4.3...........................................
  • 2.2 B. Non-cyclic Processes or Thermodynamic Potentials.........................
    • 2.2.1 1. Thermodynamic Potentials Sec 5.1......................................
    • 2.2.2 2. Toward Equilibrium Sec 5.2.................................................
    • 2.2.3 3. Phase transformations Sec 5.3..............................................
• 3 IV. Statistical Mechanics.........................................................................
  • 3.1 A. Partition Function..............................................................................
    • 3.1.1 1. Boltzmann Sec 6.1...............................................................
    • 3.1.2 2. Probability Sec 6.2................................................................
    • 3.1.3 3. A Couple of Applications.......................................................
  • 3.2 B. Adding up the States Sec 1.2, 1.7, 2.5, 6.6, 6.7.......................................
    • 3.2.1 1. Two-State Paramagnet Sec 6.6............................................
    • 3.2.2 2. Ideal Gas Sec 6.7.................................................................
    • 3.2.3 3. Thermodynamic Properties of the Ideal Monatomic Gas......
    • 3.2.4 4. Solids Sec 2.2, 3.3, 7.5..........................................................
    • 3.2.5 5. Photons Sec 7.4.....................................................................
    • 3.2.6 6. Specific Heat (Heat Capacity) of Solids Sec. 7.5.................
• A. Temperature Sec 1.

b. Heat The word heat refers to energy that is transferred, or energy that flows, spontaneously by virtue of a difference in temperature. We often say heat flows into a system or out of a system, as for instance heat flowed from block A to block B above. It is incorrect to say

that heat resides in a system, or that a system contains a certain amount of heat.

There are three mechanisms of energy transfer: conduction, convection, and radiation. Two objects, or two systems, are said to be in contact if energy can flow from one to the other. The most obvious example is two aluminum blocks sitting side by side, literally touching. However, another example is the Sun and the Earth, exchanging energy by radiation. The Sun has the higher temperature, so there is a net flow of energy from the

Sun to the Earth. The Sun and the Earth are in contact.

c. Zeroth “Law” of Thermodynamics Two systems in thermal equilibrium with each other have the same temperature. Clearly, if we consider three systems, A, B, & C, if A & B are in thermal equilibrium, and A & C are in thermal equilibrium, then B & C are also in thermal equilibrium, and all three have the same temperature.

3. Thermometers

a. Temperature scales What matter are temperature differences. We can feel that one object is hotter than another, but we would like to have a quantitative measure of temperature. A number of temperature scales have been devised, based on the temperature difference between two

easily recognized conditions, such as the freezing and boiling of water. Beyond that, the definition of a degree of temperature is more or less arbitrary. The Fahrenheit scale has 180 degrees between the freezing and boiling points, while the Celsius scale has 100. Naturally, we find 100 more convenient than 180. On the other hand, it turns out that the freezing and boiling points of water are affected by other variables, particularly air pressure. Perhaps some form of absolute scale would be more useful. Such a scale is the Kelvin scale, called also the absolute temperature scale. The temperature at which the pressure of a dilute gas at fixed volume would go to zero is called the absolute zero temperature. Kelvin temperatures are measured up from that lowest limit. The unit of absolute temperature is the kelvin (K), equal in size to a degree Celsius. It turns out that 0 K = -273.15 oC. [The text continues to label non-absolute temperatures with the degree

symbol: oC, etc., as does the introductory^ University Physics^ textbook. The latter also claims that temperature intervals are labeled with the degree symbol following the letter, as C o. That’s silly.]

b. Devices Devices to measure temperature take advantage of a thermal property of matter— material substances expand or contract with changes in temperature. The electrical conductivity of numerous materials changes with temperature. In each case, the thermometer must itself be brought into thermal equilibrium with the system, so that the system and the thermometer are at the same temperature. We read a number from the thermometer scale, and impute that value to the temperature of the system. There are bulb thermometers, and bi-metallic strip thermometers, and gas thermometers, and

thermometers that detect the radiation emitted by a surface. All these must be calibrated, and all have limitations on their accuracies and reliabilities and consistencies. B. Work

1. First “Law” of Thermodynamics Sec 1. a. Work Heat is defined as the spontaneous flow of energy into or out of a system caused by a difference in temperature between the system and its surroundings, or between two objects whose temperatures are different. Any other transfer of energy into or out of a system is called work. Work takes many forms, moving a piston, or stirring, or running an electrical current through a resistance. Work is the non-spontaneous transfer of energy. Question: is lighting a Bunsen burner under a beaker of water work? The hot gasses of the flame are in contact with the beaker, so that’s heat. But, the gases are made hot by combustion, so that’s work.

b. Internal energy There are two ways, then, that the total energy inside a system may change—heat and/or work. We use the term internal energy for the total energy inside a system, and the symbol U. Q and W will stand for heat and work, respectively. Energy conservation gives us the First “Law” of Thermodynamics:.

Now, we have to be careful with the algebraic signs. In this case, Q is positive as the heat entering the system, and W is positive as the work done on the system. So a positive Q and a positive W both cause an increase of internal energy, U.

2. Compressive Work Sec 1.

a. PV diagrams Consider a system enclosed in a cylinder oriented along the x-axis, with a moveable piston at one end. The piston has a cross sectional area A in contact with the system. We may as well imagine the system is a volume of gas, though it may be liquid or solid. A force applied to the piston from right to left (-x direction) applies a pressure on the gas of. If the piston is displaced a distance , then the work done by the force is If the displacement is slow enough, the system can adjust so that the pressure is uniform over the area of the piston. In that case, called quasistatic , the work becomes.

Now it is quite possible, even likely, that the pressure will change as the volume changes. So we imagine the compression (or expansion) occurring in infinitesimal steps, in which case the work becomes an integral:

Naturally, to carry out the integral, we need to have a specific functional form for P ( V ). On a PV diagram, then, the work is the area under the P ( V ) curve. In addition, the P ( V ) curve is traversed in a particular direction—compression or expansion, so the work will be positive or negative accordingly. Notice over a closed path on a PV diagram the work is not necessarily zero.

b. Internal energy of the Ideal Gas Sec 1.

isotherm is parabolic, since ; that’s a special case. A curve along which Q = 0 is called an adiabat.

3. Other Works In our discussion of energy conservation, we spoke of work as being any energy flow into or out of the system that was not heat. We spoke of compressive work (sometimes called piston work ) and “all other forms of work.” The all other forms of work included stirring (called shaft work ) and combustion and electrical currents and friction. It would also include any work done by external forces beyond the compressive work, particularly work done by the force of gravity. We also have been assuming that the center of mass of the system is not moving, so there was no kinetic energy associated with translation of the entire system. A general form of the First “Law” of Thermodynamics ought to include all the energy of the system, not only its internal energy. Thus for instance, the total energy of a system might be.

a. Steady flow process We might consider a situation in which a fluid is flowing steadily without friction, but with heat flow into the fluid and a change in elevation and changes in volume and pressure and some stirring.

In engineering real devices, all the various sources of work have to be taken into account. In any specific device, some works can be neglected and other works not.

b. The turbine In a turbine, a fluid flows through a pipe or tube so quickly that Q = 0, and normally the entry and exit heights are virtually the same. In the case of an electrical generator, the moving fluid turns a fan, so that the shaft work is negative. The energy balance equation for a volume element of the fluid having a mass, M , would look something like this:

An equation like this tells us how to design our turbine to maximize the shaft work.

c. Bernoulli’s Equation Suppose both Q and W (^) shaft are zero.

If the fluid is incompressible, then the volume is constant, and we can divide through by

V to obtain Bernoulli’s Equation. The internal energy is also constant because Q = 0 and no compressive work is done. [ is the mass density of the fluid.]

C. Heat Capacity Sec 1.

1. Changing Temperature a. Definition

By definition, the heat capacity of an object is. The specific heat capacity is the heat

capacity per unit mass,. This definition is not specific enough, however, since. A heat capacity could be computed for any combination of conditions—constant V , constant P , constant P & V , etc.

b. Constant pressure heat capacity If pressure is constant, then . The second term on the right is the energy expended to expand the system, rather than increase the temperature.

c. Constant volume heat capacity , since.

2. Heat Capacity and Degrees of Freedom

a. Degrees of freedom A degree of freedom is essentially a variable whose value may change. In the case of a physical system, the positions of the particles that comprise the system are degrees of freedom. For a single particle in 3-dimensional space, there are three degrees of freedom. Three coordinates are required to specify its location. We are particularly interested in variables that determine the energy of the system—the velocities determine the kinetic energy, the positions determine the potential energy, etc. In other words, we expect to associate some kinetic energy and some potential energy with each degree of freedom.

In effect, this text treats the kinetic and potential energies as degrees of freedom. An isolated single particle, having no internal structure, but able to move in three- dimensional space, has three degrees of freedom which may have energy associated with them: the three components of its velocity. Since the particle is not interacting with any other particle, we do not count its position coordinates as degrees of freedom. On the other hand, a three-dimensional harmonic oscillator has potential energy as well as kinetic energy, so it has 6 degrees of freedom. Molecules in a gas have more degrees of freedom than simple spherical particles. A molecule can rotate as well as translate and its constituent parts can vibrate. A water molecule is comprised of three atoms, arranged in the shape of a triangle. The molecule can translate in three dimensions, and rotate around three different axes. That’s 3+3 = 6 degrees of freedom for an isolated water molecule. Within the molecules, the atoms can vibrate relative to the center of mass in three distinct ways, or modes. That’s another 2x3 = 6 degrees of freedom. Now, finally if the water molecule is interacting with other water molecules, then there is interaction between the molecules, and the degrees of freedom are 2x3+2x3+2x3 = 18. Notice that if we regard the molecule as three interacting atoms, not as a rigid shape, there are 3x2x3 = 18 degrees of freedom.

A system of N particles, such as a solid made of N harmonic oscillators, has 6 degrees of freedom per particle for a total of 6 N degrees of freedom.

The idea is that each degree of freedom, as it were, contains some energy. The total internal energy of a system is the sum of all the energies of all the degrees of freedom.

II. Second “Law” of Thermodynamics

A. Combinatorics

1. Two State Systems Sec 2.

a. Micro- and macro-states

Consider a system of three coins, as described in the text. The macrostate of this system is described by the number of heads facing up. There are four such macrostates, labeled 0, 1, 2, & 3. We might even call these energy levels 0, 1, 2, & 3.

Specifying the orientation of each individual coin defines a microstate. We can list the microstates, using H for heads and T for tails: TTT, HTT, THT, TTH, HHT, HTH, THH, HHH.

Now, we sort the microstates into the macrostate energy levels.

energy level microstates multiplicity, 0 TTT 1 1 HTT, THT, TTH 3 2 HHT, HTH, THH 3 3 HHH 1

The multiplicity is the number of distinct ways that a specified macrostate can be realized. The total multiplicity of the system is the total of all the possible microstates. For these three coins, that’s.

b. Two-state paramagnet Consider a large number of non-interacting magnetic dipole moments, let’s say^ N^ of them. These dipoles may point in one of only two ways: up or down. If an external uniform magnetic field is applied, say in the up direction, each dipole will experience a torque tending to rotate it to the up direction also. That is to say, parallel alignment with the external field is a lower energy state than is anti-parallel alignment.

The energy of the system is characterized by the number of dipoles aligned with the external field, q. But, we don’t care which q dipoles of the N total are in the up state. Having q dipoles up specifies the energy macrostate, which may be realized by the selection of any q dipoles out of N to be up. The number of microstates for each macrostate is just the number of combinations, the number of ways of choosing q objects from a collection of N objects.

Now, what are the odds of observing this paramagnet to be in a particular energy macrostate? Assuming every microstate is equally likely, then we have . Notice that the total multiplicity is because each dipole has only two possible states.

Here is a microstate for a system of N = 10 dipoles, with q = 6 (6 dipoles point up).

The probability function, P ( q ), for this system looks like this:

[I did the calculation of using the COMBIN function in Excel.]

As we increase the numbers, the s become very large very quickly, as illustrated by the text example on pages 58 & 59. B. Entropy

1. Large Systems Sec 2. a. Very large numbers Macroscopic systems contain multiples of Avogadro’s number, , perhaps many, many multiples. The factorials of such large numbers are even larger—very large numbers. We’ll use Stirling’s Approximation to evaluate the factorials: . Ultimately, we will want the logarithm of N !:.

b. Multiplicity function Consider an Einstein solid with a large number of oscillators, N , and energy units, q.

The multiplicity function is. Take the logarithm, using Stirling’s formula.

Now further assume that q >> N. In that case,. The becomes , whence

c. Interacting systems The multiplicity function for a pair of interacting Einstein solids is the product of their separate multiplicity functions. Let’s say and. Then. If we were to graph this function vs. q (^) A , what would it look like? Firstly, we

expect a peak at with a height of. That’s a very large number. How about the width of the curve? In the text, the author shows that the curve is a Gaussian: , where. The

origin has been shifted to the location of. The point at which occurs when. Now, this is a large number, but compared to the scale of the horizontal axis , that peak is very narrow, since N is a large number in itself. That is, the half width of the peak is of the whole range of the independent variable.

The upshot is that as N and q become large, the multiplicity function peak becomes narrower and narrower. The most probable macrostate becomes more and more probable

relative to the other possible macrostates. Put another way, fluctuations from the most probable macrostate are very small in large systems.

2. Second “Law” Sec 2.

a. Definition of entropy We define the entropy of a system to be. The units of entropy are the units of the Boltzmann Constant, J/K.

The total entropy of two interacting systems, such as the two Einstein solids above, is.

The Second “Law” of Thermodynamics says: Systems tend to evolve in the direction of increasing multiplicity. That is, entropy tends to increase. This is simply because the

macrostate of maximum multiplicity is the most probable to be observed by the time the

system has reached thermal equilibrium.

b. Heat capacities

We cannot measure entropy directly, but we can measure changes in entropy indirectly, through the heat capacity. For instance, if no work is being done on the system,

Of course, we need to know C (^) V as a function of T. This is obtained by measuring Q or U

vs. T. In general, the heat capacity decreases with decreasing temperature. At higher temperatures, the heat capacity approaches the constant (Dulong-Petit). For instance, the C (^) V vs. T for a monatomic substance would look like this:

The Third “Law” of Thermodynamics says that as , or alternatively, that S = 0 when T = 0K. In reality, there remains residual entropy in a system at T = 0K—near absolute zero, the relaxation time for the system to settle into its very lowest energy state is very, very long.

Now notice, if indeed as , then absolute zero can not be attained in a finite number of steps, since as. It’s like the famous example of approaching a wall in a series of steps, each one half the previous step.

For example, let us say that we wish to cool an ideal gas to absolute zero. We’d have to “get rid” of the entropy in the gas in a series of steps.

i) isothermal compression—heat and entropy is transferred to a reservoir ii) adiabatic expansion—temperature decreases, entropy is constant, Q = 0 repeat

Now if we were to graph these S ( T ) points we have generated we would see two curves. But the curves are not parallel; they appear to converge at T = 0. As a result, the gets smaller for each successive two-stage step, the closer we get to T = 0.

In practice, a real gas would condense at some point. The text describes three real-life high-tech coolers. In any case, there will be a series of ever smaller steps downward between converging curves on the S ( T ) graph, toward absolute zero.

2. Pressure Sec 3.

a. Mechanical equilibrium Consider two systems whose volumes can change as they interact. An example might be two gases separated by a moveable membrane. The total energy and volume of the two systems are fixed, but the systems may exchange energy and volume. Therefore, the entropy is a function of the volumes as well as the internal energies. However, we will be keeping the numbers of particles in each system fixed.

At the equilibrium point, and.

As we did with temperature, we can identify the pressure with the derivative of entropy with volume, thusly:.

b. Thermodynamic identity

Now if we envision a system whose internal energy and volume are changing, we would write the change in entropy (a function of both U and of V ) as follows:

c. Creating entropy with mechanical work Remember that compressive work () is just one form of work. If the compression is slow, and no other form of work is done on the system, then the volume change is quasistatic, and. In such a case, we are allowed to combine the First “Law” with the thermodynamic identity to obtain . But, if the work done on the system is greater than , then. In other words, the amount of entropy created in the system is more than that accounted for by the heat flow into the system. This might happen, for instance, with a compression that occurs faster than the pressure can equalize throughout the volume of the system. It will happen if other forms of work are being done, such as mixing, or stirring. In a similar vein, if a gas is allowed to expand freely into a vacuum, no work is done by the gas, and no heat flows into or out of the gas. Yet the gas is occupying a larger volume, so its entropy is increased.

3. Chemical Potential Sec 3.

Now consider a case in which the systems can exchange particles as well as energy and volume.

a. Diffusive equilibrium

Define the chemical potential as. Evidently, the minus sign is attached so that particles will tend to diffuse from higher toward lower chemical potential.

b. Generalized thermodynamic identity For infinitesimal changes in the system,

This equation contains within it all three of the partial-derivative formulas for T , P and for. For instance, assume that entropy and volume are fixed. Then the thermodynamic identity says , whence we can write. To apply the partial-derivative formulae to a particular case, we need specific expressions for the interdependence of the variables, i.e., U as a function of N.

4. Expanding & Mixing Sec 2. a. Free expansion Imagine a container of volume 2 V , isolated from its surroundings, and with a partition that divides the container in half. An ideal gas is confined to one side of the container. The gas is in equilibrium, with temperature T and Pressure P. Now, imagine removing the partition. Over time, the gas molecules will diffuse to fill the larger volume.

However, in expanding the gas does no work, hence the phrase free expansion. Because the container is isolated, no heat flows into or out of the gas, nor does the number of molecules, N , change..

However, the entropy increases..

III. Processes A. Cyclic Processes

1. Heat Engines & Heat Pumps Sec 4.1, 4. A heat engine is a device that absorbs heat from a reservoir and converts part of it to work. The engine carries a working substance through a PVT cycle, returning to the state at which it starts. It expels “waste” heat into a cold reservoir, or into its environment. It must do this in order that the entropy of the engine itself does not increase with every cycle.

a. Efficiency The efficiency of the heat engine is defined as the ratio of work done by the engine to the heat absorbed by the engine.

We’d like to express e in terms of the temperatures of the hot and cold reservoirs. The First “Law” says that. The Second “Law” says that. Putting these together, we obtain. Firstly, notice that e cannot be greater than one. Secondly, e cannot be one unless Tc = 0 K, which cannot be achieved. Thirdly, is the greatest e^ can be—in practice,^ e^ is less than the theoretical limit, since always.

b. Carnot cycle Can a cycle be devised for which? That’s the Carnot cycle, which uses a gas as the working substance.

i) the gas absorbs heat from the hot reservoir. To minimize dS , we need ; the gas is allowed to expand isothermally in order to maintain the.

ii) the gas expands adiabatically, doing work, and cools from T (^) h to T (^) c.

iii) the gas is compressed isothermally, during which step heat is transferred to the cold reservoir.

iv) the gas is compressed adiabatically, and warms from Tc to T (^) h.

Now, for the total change in entropy to be very small, the temperature differences between the gas and the reservoirs must be very small. But that means that the heat transfers are very slooow. Therefore, the Carnot cycle is not very useful in producing useful work. [Empirically, the rate at which heat flows is proportional to the temperature difference — .]

c. Heat pump The purpose of a heat pump is to transport energy from a cold reservoir to a hot one by doing work on the working substance. The work is necessary because the temperature of the working fluid must be raised above that of the hot reservoir in order for heat to flow in the desired direction. Likewise, at the other side of the cycle the working fluid must be made colder than the cold reservoir.

Rather than efficiency, the corresponding parameter for a heat pump is the coefficient of performance ,

The First “Law” says. The Second “Law” says. Putting these together, we obtain. A Carnot cycle running in reverse will give the maximum COP.

2. Otto, Diesel, & Rankine Sec 4. Real heat engines need to produce work at a more rapid rate than a Carnot engine. Consequently, their efficiencies are lower than that of a Carnot engine. Of course, real engines do not achieve even their theoretical efficiencies due to friction and conductive heat loss through the cylinder walls and the like.

a. Otto cycle The Otto cycle is the basis for the ordinary 4-stroke gasoline engine.

i) air-fuel mixture is compressed adiabatically from V 1 to^ V 2 ; pressure rises from^ P 1 to^ P 2. ii) air-fuel mixture is ignited, the pressure rises isochorically from P 2 to^ P 3. iii) combustion products expand adiabatically from V 2 to V 1 ; pressure falls from P 3 to P 4. iv) pressure falls isochorically from P 4 to^ P 1.

The temperatures also change from step to step. The efficiency is given by

The quotient is the compression ratio. The greater the compression ratio, the greater is the efficiency of the engine. However, so is T (^) 3 greater. If T 3 is too great, the air-fuel

mixture will ignite prematurely, before the piston reaches the top of its stroke. This reduces power, and damages the piston and cylinder. Up to a point, chemical additives to the fuel can alleviate the premature detonation.

Notice that there is no hot reservoir per se ; rather the heat source is the chemical energy released by the combustion of the fuel.

b. Diesel cycle The Diesel cycle differs from the Otto cycle in that the air is first compressed adiabatically in the cylinder, then the fuel is injected into the hot air and ignited spontaneously, without need of a spark. The fuel injection takes place as the piston has begun to move downward, so that constant pressure is maintained during the fuel injection. Since the fuel is not in the cylinder during the compression, much higher compression ratios can be used, leading to greater efficiencies.

c. Rankine cycle In some ways the steam engine is a more nearly exact example of a heat engine than is the Otto engine. No chemical reaction or combustion takes place within the working fluid, and at least in principle the working fluid is not replaced at the beginning of each cycle. i) water is pumped to a high pressure into a boiler. ii) the water is heated at constant pressure and changes to steam (water vapour).