Adiabatic Expansion Cooling of Gases

Using simple adiabatic expansion, the ratio of the constant pressure to constant volume heat capacities of gases will be measured. This property will be correlated to the internal motion and molecular structure of the gases studied. Several gases will be compared and the contributions of the the different types internal molecular motions to the total thermal energy of the gas molecules will be gleaned.


Theory

For a perfect (ideal) gas, Cp = CV + R , where Cp and CV are the molar heat capacities at constant pressure and volume, respectively (see on-line lecture notes for a derivation of this and related formulae). For an arbitrary real gas a slightly more complicated relationship between these heat capacities may be derived from the equation of state. Essentially, however, the difference between heating a gas at constant volume and constant pressure is expansion work. Thus, the ratio Cp / CV is related to the capacity of the system to do work upon expansion. This ratio is usually given the symbol (gamma) [lower case greek gamma].

Properties of CV
The heat capacity of a molecule is clearly related to the way that the molecule can accomidate energy at a given temperature. The energy of the molecule is partitioned among the types of motion the molecule can exhibit. It is important in thermodynamics to count and categorize such molecular motion in a systematic way. The number of degrees of freedom (DOF) for a given molecule is the number of independent coordinates needed to specify all its nuclear positions (what do I ignore in this statement?). A molecule of n atoms therefore has 3n DOF. These could be assigned to the coordinates of the individual n atoms, or alternatively they can be classified as follows:

From classical statistical mechanics the 'equipartition of energy' theorem can be derived which associates an energy of RT/2 per mole with each quadratic term in the Hamiltonian or per degree of translational or rotational freedom (again, see my lecture notes for elaboration of this point). Here, R is the Molar Gas Constant (Boltzmann's constant times Avogadro's number) and T is the absolute thermodynamic temperature. Vibrational DOF have two quadratic terms: one potential energy term and one kinetic term per vibration. Therefore an energy of RT per mole is associated with each vibrational DOF. This is in contrast to rotational and translational DOF's which are 'free' motions and thus have no potential energy term.

From these considerations it is clear that for a monotonic gas (like He) no vibrational or rotational energy terms exist, but, like all gaseous molecules, energy may be contained in translational motion. Thus the molar energy of a monoatomic gas is simply 3RT/2. The constant volume heat capacity of a monotonic gas is therefore

eq.1

For diatomic or polyatomic molecules,

eq.2

Contributions from electronic states to the total internal energy have been neglected under the assumption that at room temperature electronic transitions out of the ground state are unlikely. On the other hand, the population of excited vibrational states depends strongly on temperature and thus the various vibrational modes can be at least partially active. In general, if a vibration involves a heavy atom or possesses a smaller force constant, then the normal mode will be more active and make a greater individual contribution to the heat capacity. For example, the frequencies of bending modes tend to be much lower than those of stretching modes. Since in the case of most diatomics there are only stretching modes, the vibrational contribution to CV will be very small. Indeed, N2 would have its equipartition value for CV only above about 4000 K. In contrast many polyatomic molecules, especially those containing heavy atoms,will at room temperature have significant partial vibrational contributions to the heat capacity.

At ordinary temperatures many of the excited state rotational levels are thermally accessible and hence the rotational contribution is in accord with the equipartition theorem of classical mechanics.

Thus, it is possible to calculate definite values for CV and hence, through the ideal gas equation of state, the ratio (gamma) for ideal monatomic and polyatomic gases using the above expressions. Statistical thermodynamics provides even better results, for which the partially 'frozen' vibrational contribution to the heat capacity may be evaluated exactly (you guessed it, I have some lecture notes on this, too).


Thermodynamics

A two step process can be used to experimentally determine the ratio of the heat capacities. These are:

I. An adiabatic reversible (isentropic) expansion from the initial pressure p1, to the intermediate pressure, p2

eq.3

II. Restoration of the temperature to its initial value T1, at constant volume.

eq.4

Recall that for any adiabatic process,

eq.5

The First Law states:

eq.6

So for the first step of the experiment

eq.7

At constant volume the heat capacity relates the change in temperature to the change in internal energy,

eq.8

Equating our two expressions for dU,


eq.9

Inserting the ideal gas law and integrating each side,

eq.10

Equation (9) predicts the decrease in temperature accompanying the reversible adiabatic expansion of a perfect gas. This expression can be cast in terms of the initial and final pressures and the ratio of the heat capacities by noting that for an ideal gas,

eq.11

yielding,

eq.12

or after rearranging and using Cp = CV + R

eq.13

For step II, the temperature is restored to T1, and

eq.14

Thus,

eq.15

and after a final rearrangement, we have the desired expression for the ratio of the molar heat capacities:

eq.16

This simple and classical method is a slightly modified version of that attributed to due to Clement and Desormes [ref to be found] that has even older origins [1] and uses the apparatus shown in the schematic below. A gas is initially contained in the sample vessel at a pressure (p1) slightly higher than atmospheric pressure (p2). The adiabatic reversible expansion (step I) is carried out by quickly removing and replacing the stopper in the top of the sample container. Step II occurs by simply waiting for the gas remaining in the carboy to return to its initial temperature (T1) and final pressure (p3) by heat transfer from the surroundings. The initial pressure p1, and the final pressure p3 are read using a capacitance manometer pressure gauge.

The thermodynamic expressions used to derive the heat capacity ratio apply only to the part of the gas that remains in the carboy after the stopper is replaced since molar volumes and molar heat capacities were used. Physically, the reversibility of the expansion can be justifed as follows: One can imagine an invisible surface separating the gas that remains within the carboy and the gas that escapes when the stopper is removed. The gas below this surface expands in an approximately reversible way against the surface. Work is done as the upper gas is pushed out. The change is approximately adiabatic only because it is rapid copared to heat flow but slow on the scale of the mechanical work. No appreciable amount of heat is transferred between the reservoir and the gas in the carboy during the short period of the expansion, but after the stopper is replaced, the temperature is seen to increase as thermal equilibrium is reestablished for the gas remaining in the vessel.
Experimental

Procedure

The lab is equipped with several apparati as shown above, each filled with a different sample gas. Student teams shall rotate through the stations so that each team may obtain data on all the sample gases. For each gas, the following procedure is suggested:

The choice of sample gases varies with availability but usually includes He, Ar, CO2, N2, butane, etc.

Calculations Discussion
Critically compare your results with theory (perhaps several 'levels' of theory). Do your results concur with a given theory to within experimental error? Can molecular structure be determined by thermodynamic measurement? Can these data be experimentally determined in a different way? Can statistics always determine all sources of error in an experiment? What is the significance of the heat capacity ratio in science and industry?
[1] Lord Rayleigh, "The Theory of Sound", 2d ed., vol. 11, pp. 15-23, Dover, New York (1945)

Other useful references
D. Shoemaker, C. Garland, and J. Nibler, 'Experiments in Physical Chemistry", McGraw-Hill, New York
P. Atkins, "Physical Chemistry", 5th ed., W. H. Freeman, New York (1994)


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