Introduction.

The ratio of heat capacity at constant pressure to heat capacity at constant volume is a value that occurs frequently in thermodynamics. This constant can be measured in an isobaric or an isochoric process, to find heat capacity at constant pressure and volume, respectively. 

Objective: In this experiment the ratio between two heat capacities, g = Cp/Cv, will be measured using the adiabatic expansion method and the sound velocity method. Three gases will be studied: Ar, N2 and CO2.

The manner in which a material takes up heat can provide information about its internal molecular structure. We will study it for gases. At high temperatures (but not too high, that the gas is not electronically excited or ionized) one can employ the simplest model assuming that the heat intake is equipartioned amongst the possible modes of energy available to the gas. The gaseous molecule can gain energy by moving along the x, y or z axes, i.e., increase its translational energy. If the molecule is composed of two or more atoms, it can also rotate and vibrate. In order to describe the molecule in three dimensional space, we will need 3N coordinates, or degrees of freedom, where N is the total number of atoms in the molecule. If we specify all 3N coordinates, we will know exactly where all the atoms in the molecule are at any point in time. The motions available to the molecule couple these coordinates, however, and it is most convenient to separate them according to whether the motions fall into translational", "rotational" or "vibrational" modes. These are the three modes where energy can be "stored".

Translational Energy: One can describe this motion by specifying its components along x, y and z independently since a translational motion along any direction can be broken down into motion relative to each cartesian axis. Of course, the orientation of the coordinate system is completely arbitrary and motion (in average) is the same in any direction, provided that the gas container, itself, is at rest. Thus, one mode of translational energy is available for each axis to give a total of three translational modes per molecule.

Of course, there is actually a distribution of molecular speeds among the molecules and, as we increase the temperature, the distribution shifts to higher average velocities. However, the average kinetic energy per translational mode is the same, <E> = 1/2 kBT (equipartition theorem, see pp. 4 and 603 of Atkins). Since there are three translational modes, every gas molecule, whether it is composed of a single atom or it is polyatomic, will have a translational kinetic energy of <Etrans> = 3/2 kBT, making contribution to the internal energy Utrans = 3/2 RT per one mole of gas.

Rotational Energy: In separating out translational, rotational and vibrational modes of motion, we assume the atoms comprising the molecules are zero dimensional; they are points. Monatomic gases, therefore, cannot possess rotational energy since there is nothing around which to rotate. And it should be, since N = 1 for a monatomic gas and we have already used up all our 3N degrees of freedom on its translational motion. All other gases (with N > 2) will have either two or three rotational modes, depending on whether the molecule is linear or nonlinear. Rotational motion for a linear molecule is two dimensional and thus there are two degrees of freedom associated with it. Nonlinear molecules need all three dimensions to describe their rotational motion and, correspondingly, they need three degrees of freedom.

Like translational energy, rotational energy is purely kinetic. There will, therefore, be 1/2 RT per mole of the internal energy associated with each rotational mode. Monatomic gases will have Urot = 0, linear polyatomic molecules will have Urot = RT and nonlinear molecules will have Urot = 3/2 RT per one mole of gas.

Vibrational Energy: All degrees of freedom not associated with either a translational or a rotational mode must be vibrational in nature. Atoms are points and, as such, they do not vibrate. Thus they cannot raise their temperature by adding energy to vibrational motion. After subtracting out degrees of freedom used for tranlational and rotational modes, linear molecules have 3N - 5 degrees of freedom left over for vibrational motion and nonlinear molecules have 3N - 6. Because vibrational motion has both a kinetic energy and a potential energy component, there is 1/2kBT + 1/2kBT = kBT associated with each vibrational degree of freedom. The vibrational energy contribution to the total internal energy of the gas is: for monatomic gases Uvib = 0 , for linear polyatomic gases Uvib = (3N - 5)RT, and for nolinear molecules Uvib = (3N - 6) RT per one mole of gas.

Heat Capacities (High temperature limit): By adding up all components together and differentiating with respect to T, we can calculate the heat capacity :

(1)

for each type of gas:

You would need to recall (it is the first of the prelab questions) that Cp = Cv + nR for an ideal gas, where n is the number of moles. Thus the ratio g = Cp/Cv can be calculated (you will do it in the second prelab question).

Not that high temperatures: The classical equpartition theorem, as we already mentioned above, can work fairly well only at high temeperatures. But quantum mechanics says that each motion is actually quantized, i.e. the molecular energy increases noncontinuously, in discreat portions, dE. For translational energy, dEtrans is defined by the size of a volume and is very small, for rotational energy, dErot is greater, on the order of 1 cm-1, and for vibrational energy, dEvib, is even greater, on the order of 102 - 103 cm-1. At room temperature kBT is only ca. 200 cm-1 which is much greater than translational and rotational energy quanta but compartible or less than typical vibrational energy quanta. As a result, energy partition into vibrational and translational motion can be treated as before with no big mistake, but vibrational motion has to be evaluated using statistical quantum mechanics. So called low temperature limit for estimating the heat capacities of gases assumes that contributions to internal energy from vibrational motion is close to zero, Evib = 0 , and overall internal energy can be approximated as E = Etrans + Erot. In this approximation, our heat capacities become:

Adiabatic expansion method.

The heat capacity ratio can be measured using the adiabatic expansion method. You will use this method for the three gases: Ar, N2 and CO2. At first, a gas with the initial pressure, P1, expands, adiabatically and reversibly to atmospheric pressure P2 , and then isochorically relaxes to the original temperature, T, with the resulting pressure, P3. The pressure variation in this experiment relates to the heat capacity ratio via:

(2)

It has been argued1 that the conditions of this experiment are such that the gas in the canister, initially at pressure P1 , upon removal of the stopper, expends, irreversibly against the surroundings at constant pressure P2 . This alternative explanation leads to the folowing equation:

(3)

The two equations provide very close results when P1 - P2 << P2 .

Sound velocity method.

You will measure sound velocity for the same gases, Ar, N2 and CO2, and for air. If sound propagation is considered as a reversible adiabatic process than for a perfect gas its sound velocity, c, can be expressed via the gas temperature, T, its molecular weight, M, and g = Cp/Cv, as in the following equation:

c2 = g P/r = g RT/M (4)

By accurate measurement of all three parameters, c, T, and M, one can obtain the value of g .

Section C.

You can come up with your own project within the scope of the setup you use. For example, there are other methods for measuring g, such as Ruchardt's method and its modification, called Rinkel's method.  

References.

1. Bertrand, G.L. and McDonald, H.O. J. Chem. Educ. 1986, 63, 252

Last updated 2/1/99.