CHAPTER 14 THE CLASSICAL STATISTICAL TREATMENT OF AN IDEAL GAS

This chapter is a continuation of the previous one. We show how a partition function is used, in this case with an ideal gas. We also obtain a formula for the distribution of speeds in a gas, a result that is often obtained from the kinetic theory of gases.

In the previous chapter we derived the following equations appropriate for a system fo r which the Maxwell-Boltzmann Distribution is applicable.

Thermodynamic properties from the partition function In this chapter we will concentrate on an ideal gas. The Maxwell-Boltzmann distribution is used. We develop expressions for various thermodynamic variables in terms of ln(Z), a process that we began in Chapter 13. Internal energy: We recall that Differentiating Z gives Therefore or more compactly

Gibbs Function The Gibbs thermodynamic potential has previously been written in terms of the chemical potential. For a single component system so Enthalpy: We have (see diagram on Fact Sheet) H=G+TS and using the expression for U Pressure: From the VFT-VUS diagram dF=-PdV-SdT Considering F=F(V,T)

This gives (reciprocity relationship) Hence, evaluating this partial derivative using the expression for F=F(lnZ), that is, F = -NkT[lnZ-lnN-1] results in

Summary (M-B distribution!) potentials

Partition function for a gas.( See text for different approach.) We first note that, instead of using in which the sum is over the energy levels, we can use in which the sum is over energy states. Consider an ideal monatomic gas in a cubical container of volume V. The energy states are given by The partition function is then a sum over the n’s, with the integers going from 1 to infinity. We temporarily set This permits us to write

For this system the partition function is then Unless T or V is extremely small the energy levels are very closely spaced and a good approximation is to replace sums by integrals. The integral is well-known (see Fact Sheet) Using the expression for a: or constant

Properties of a monatomic ideal gas. Once we have lnZ we can rapidly compute the other thermodynamic variables. Consider the pressure: But so PV=NkT (equation of state) Next let us calculate the internal energy. We have which quickly yields This again reminds us that the average kinetic energy per particle is (3/2)kT Hence, using a model of a gas and statistical concepts, we have derived the empirical relationships for P and U. These results can be checked by experiments.

Now we consider the entropy Consider 1 mole of an ideal monatomic gas: This is called the Sackur-Tetrode Equation. Before a discussion of statistical mechanics we obtained an expression for s which included a constant, , whose value was unknown. The equation was

Statistical mechanics provides us with an expression for this constant. This equation agrees well with experiment. The theoretical expression is very important in a study of the thermodynamics of chemical reactions. We should note that the expression for s is not valid as T goes to zero as, in this expression, s does not approach zero. This is not surprising as we have used Maxwell-Boltzmann statistics, which are not valid at low temperatures.

Distribution of speeds in an ideal monatomic gas. It is often necessary to compute the averages of various functions of the molecular speeds (in astrophysics, for example). To do so, we must know the probability distribution of molecular speeds, the so-called Maxwell-Boltzmann distribution of speeds. This was first derived by Maxwell on the basis of molecular collisions and later by Boltzmann using statistical methods. For a continuous distribution we calculated the number of states in an energy region . For bosons with s=0, the density of states is: We also have and Hence

In addition, for an ideal gas, Using (1) and (2) in is the number of molecules per unit energy interval. is then the number of molecules in an infinitesimal energy interval. Notice that Planck’s constant does not appear in the expression. The work of Planck came after the death of Maxwell.

What is measured experimentally is the distribution of speeds. We can convert the formula to the number of molecules per unit speed. and so The distribution expression is then or

(see next slide) With these expressions one can calculate averages for functions of the energy or the speed. We will write down some averages. Students should go through the calculations in detail. Mean speed

Root-mean-square speed Most probable speed is that speed for which See Table 11.2 for some values. The Maxwell speed distribution has been checked by careful experimental measurements and the agreement is excellent.

Equipartition of energy The principle of equipartition of energy states that: At a temperature T, the average energy of any quadratic degree of freedom is (1/2)kT If a system has N molecules, each with f degrees of freedom, and there are no other (non-quadratic) temperature-dependent forms of energy, then the total thermal energy is This is not the total energy of the system. There are “static” terms, such as the energy stored in chemical bonds, mass energy, etc. A gas in a gravitational field would have an energy contribution mgΔz, which is not quadratic. {Note: The concept of degree of freedom is different (broader) in thermodynamics than in classical mechanics.}

The principle does not take into consideration quantum effects, such as the quantization of rotational and vibrational energies. It is applicable to indistinguishable particles in equilibrium. The essential condition for validity is that the energy levels must be continuous (classical!). It does not work for liquids and solids: in these cases the interactions among the particles must be taken into consideration when calculating the partition function. The quadratic forms of energy include translational kinetic energy (along three axes), rotational kinetic energy and vibrational terms. We will then write the energy of a molecule as and there will be f terms.

The partition function is then Because of the properties of exponentials, this is a product of terms, each term corresponding to one degree of freedom. Since we are dealing with a continuous energy distribution, the sums are replaced by integrals. We make a change of variables Since there are f terms in the product The important point to notice is that the do not contain Taking the logarithm of Z,

We have, for the internal energy, The average internal energy is Hence The average value of each independent quadratic energy term is then

This is a classical result. In a strictly correct quantum mechanical description there is a discrete set of energy states and in general sums cannot be replaced by integrals. At temperatures for which the spacing between levels is <<kT, the equipartition theorem is valid. This condition is not valid at low temperatures. In using the equipartition theorem, it is necessary to decide upon the appropriate number of degrees of freedom. This is simple for a monatomic molecule, which has only translational motion and so f=3. A diatomic molecule ( like a dumbbell) has an additional two degrees of rotational motion and so f=5. A diatomic molecule can also vibrate which introduces two additional degrees of freedom (one for kinetic energy and one for potential energy). However these vibrational modes do not contribute to a molecule’s thermal energy except at high temperatures. We say that these vibrational modes are “frozen out” at room temperature. Clearly a complete discussion of the thermal properties ( such as the specific heat) of gases requires the use of quantum mechanics.

If we confine ourselves to “normal” temperatures and use we obtain: monatomic ideal gas: diatomic ideal gas:

These values for gases are in accord with experimental results at room temperature, but these classical results are incorrect at low and high temperatures. The next slide shows experimental values for the specific heat of a gas as a function of temperature. The need for a quantum mechanical discussion is obvious. Chapter 15 of the textbook is devoted to this important topic.

Experimental results for the specific heat at constant volume for H2 as a function of temperature. θrot (85.5K) and θvib (6140K) are characteristic temperatures at which rotational and vibrational modes are activated.