Heat Capacity

Keith Griswold March 13, 2000

Exp. Physical Chem.

Dr. Humphrey

Heat-Capacity Ratios for Gases

Cp = Cv + R

The ratio,

Cp = 

Cv

is related to the ability of the gas to do expansion work. Heat capacity at constant volume, Cv can be described using the equipartition theory, which states that each mode of motion will contribute to a molecule or atom’s energy.

E = E(translational) + E(rotational) + E(vibrational)

Setting up a Cartesian coordinate system, translational motion can occur in any of the three directions: x, y, or z. Thus for a monatomic gas energy can be represented as 3(RT/2); it is clear that no vibrational or rotational motions contribute. Rotational motion contributes to the energy of diatomic and polyatomic molecules; they are easily accessible at room temperature therefore will significantly contribute to . Vibrations can be separated into two categories: bending and stretching, where the number of modes can be described as 3N-5 for linear, and 3N-6 for nonlinear molecules. Vibrational levels are not as accessible as rotational ones are at room temperature, so it is valid to consider them, at most, only partially active; the extent depends on certain properties of the molecule. Stretching modes tend to have very high frequencies giving way to a small contribution to heat capacity ratios. It should be noted that electronic transitions will be ignored since most molecules are in their electronic ground state at room temperature. Applying classic statistical mechanics to the equipartition theory, an expression for the energy contribution of one mole of a gas from each mode of motion is given as RT/2. Since heat capacity varies with temperature the following relationship is given:

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Cv = (E/T)v. (E is the internal energy)

Using these relationships, a theoretical value of  can be derived. In contrast, experimental values will be calculated through an adiabatic expansion method initiated by Clement and Desormes. Discrepancies between the experimental values and those predicted by the equipartition theory will be examined.

Theoretical Calculations:

Theoretical values for  can be calculated using the equipartition theory and the relationship Cp = Cv + R. The gases of interest for this experiment are argon, nitrogen, and carbon dioxide. Argon is monatomic, therefore we will only be looking at it’s translational motions. Cv will equal 3R/2 and Cp will equal 5R/2, leaving  with a value of 1.6667. Nitrogen is a linear diatomic molecule, therefore rotational modes must be accounted for; a value of  will be calculated with and with out vibrational contribution. Cv for nitrogen without vibrational acknowledgement is equal to 5R/2 and Cp is equal to 7R/2, leaving  equal to 1.4000. Taking vibrational contributions into account, Cv will equal 3R and  will then be 1.3333. Carbon dioxide is triatomic, but is still linear so Cv is 5R/2 and Cp is 7R/2, letting  equal 1.4000. With vibrational modes included,  equals 1.2222. All of these values are summarized in a table below along with experimental ones.

Experimental:

The adiabatic expansion process will be carried out using the experimental setup illustrated below.

Apparatus Scheme

A gas of interest is placed in the carboy and pressure readings are then recorded. The carboy is closed off with a rubber stopper that is removed for a brief moment and then replaced. The gas in the carboy will momentarily reach atmospheric pressure, and then reside to it’s initial temperature, at which time the pressure is recorded again. It is necessary to record the atmospheric pressure at the time of the experiment. Three trials were run on each gas to obtain the following data. The pressure transducer used here was an open tube manometer containing dibutyl phthalate, so the pressure readings were converted to mmHg before used in calculations.

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Gas Trial P1(mmdi-but.) P1 (mmHg) P3(mmdi-but.) P3 (mmHg)

Ar 1 155 773 60.0 766

2 145 773 60.0 766

3 240 780 50.0 765

N2 1 270 782 55.0 766

2 410 793 90.0 768

3 150 773 33.0 764

CO2 1 480 798 110 770

2 710 816 350 788

3 255 781 58.0 766

Barometric Pressure: 761.5 mmHg

Raw data was converted to mmHg by multiplying the recorded pressures by (1.046g/cm3)/(13.55g/cm3).

Results:

In order to calculate heat capacity ratios from the raw data it is necessary to treat this adiabatic expansion as being reversible. Upon quickly releasing the stopper, the upper and lower portions of gas form an imaginary surface between them, in which the lower portion pushes reversible against the upper portion. Work is done by the lower portion of the gas pushing the upper portion out of the carboy. The relationship,

 = ln(P1/ P2)/ln(P1/ P3)

can be derived where P2 is the barometric pressure at the time of the experiment.

Experimental Theoretical

Gas Trial  (average)  (with vibrational contributions

Ar 1 1.65

2 1.65

3 1.24 1.65 1.6667

N2 1 1.29

2 1.27

3 1.28 1.27 1.4000 1.3333

CO2 1 1.31

2 1.98

3 1.30 1.30 1.4000 1.2222

The entries that are not highlighted were left out of the average since they lack precision. This discrepancy will be attributed to uncertainties in the experimental procedure.

Conclusion:

The experimental setup applied is questionable, but it does produce a reasonable system for study. The process is considered adiabatic because it is rapid and no appreciable amount of heat is transferred. It is also not necessary to be concerned with molar concentrations since ratios are being measured. As would be expected, argon is in most agreement with the equipartition theory; small discrepancies can be attributed to experimental uncertainties. A major reason for this agreement is the fact that argon exits as a monatomic gas, where there are no vibrational or rotational modes. There is no concern that the low temperature employed would hinder it’s access to allowed degrees of freedom. Nitrogen, on the other hand is affected greatly by the fact that this experiment was conducted at room temperature. Rotation requires less energy, causing a decent level of contribution from it, but vibrational modes are, at best, partially activated at 25C. There are a number of factors that decide the of extent vibrational contribution to heat capacity ratios. nitrogen gas consists of two nitrogen atoms connected by a triple bond that bares a force constant of 2,243 N/m. This triple bond requires a lot of energy to oscillate and would require extremely high temperatures to have complete activity from this vibrational mode.

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Since nitrogen gas is a diatomic molecule, it only undergoes stretches that vibrate at low frequencies, which contribute less to  at low temperatures. It seems that nitrogen is not a good candidate for this experiment and behaves less ideally than argon under the circumstances. An aspect that is disturbing is the fact that the experimental value for it’s heat capacity falls lower that the theoretical one calculated that takes vibrational modes into account. Considering that there is sufficient evidence to suggest that vibrational modes are not fully activated, there has to be a reason for this unexpected result. The unexplained may just lie completely in the realm of experimental error, a trivial explanation none the less. It could be possible that resonance varies the behavior of the nitrogen/nitrogen bond causing it to exhibit deviated characteristics. Carbon dioxide was the last gas to be studied; it’s triatomic with two double bonds connecting oxygens to the central carbon. In a similar way to nitrogen gas, carbon dioxide does not behave according to the equipartition theorem at room temperature. Although the vibrational modes of CO2 are only partially active, they contribute more that the ones for nitrogen gas. This is a result of the bending capabilities that CO2 has, which require less energy than stretches do. The bonds in carbon dioxide have force constants of 1,857 N/m; significantly less that the force constant of nitrogen. Taking into account experimental uncertainties, the values calculated for CO2 from this procedure are what would be expected. The experimental value falls in-between both the theoretical ones derived with and without vibrational contributions, proving that for CO2, the vibrational modes are only partially active at room temperature.

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In studying carbon dioxide, we can consider how structural linearity affects it’s heat capacity ratio predicted by the equipartition theory. If it were non-linear, for example like SO2 or H2O, there would be a reduced number of vibrational modes contributing to it’s energy. Accounting for the depleted vibrational motions, (theoretical) would increase slightly to a value of 1.2500. It would be difficult to decipher between these two structures based off experimental values of , especially with such a questionable experimental setup. It is necessary to realize that the discrepancy between the two ratios, posed by the difference in structure, is small; such precision would be difficult to achieve.

Bibliography

Brucat, P.J. Adiabatic Expansion: Cooling of Gases. CHM4411L; Physical Chemistry

Laborator, Fall 1996. http://itl.chem.ufl.edu/4411L_f96/gamma/gamma,html

Hechinger, Brandon. Lab4- The Ratio of Heat Capacities. Course: Physics 2; March 31,

1997. http://www.voyager.com/~jaggy/physics/lab4/.

McQuarrie, Donald A. Physical Chemistry; A Molecular Approach. University Science

Books; Sausalito, CA: 1997. Pages 169.