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Heat Capacity Ratios for Gases

Teaching Assistant: TA Name

Chem 374-005

Group #6

10/24/96

Performed: 10/10/96

Lab Partner: Their Name

Abstract

The ratio of heat capacity at constant pressure to heat capacity at constant volume for several gases was determined by measurement of the speed of sound through the gas. The speed of sound was obtained indirectly through wavelength measurements with the aid of electronic instrumentation.

Introduction & Theory

The purpose of this experiment was to determine the heat capacity ratios (constant pressure to constant volume) of gases. Through the use of a Kundt's tube, a piston, a frequency generator, and an oscilloscope, wavelength measurements were made at two frequencies (1 kHz and 2 kHz) for carbon dioxide, helium, and nitrogen. This wavelength was used to find the speed of sound in each gas, which was in turn used to determine the heat capacity ratios. All electronic measurements were taken from the oscilloscope, and the change in the piston's displacement (corresponding to wavelength) was measured with a meter stick.

Experimental

The procedure from Experiments in Physical Chemistry 1 pages 114-116 was followed, with an amplifier being used between the microphone and the oscilloscope. The only deviation from this procedure was the viewing of speaker output and microphone input superimposed on one another on the oscilloscope screen rather than creating Lissajous figures.

Ambient temperature was 22.1C, and ambient pressure was 752.0 mm Hg. As the gas flow rate was low, the pressure of the gas in the Kundt's tube may be assumed to be the same as ambient pressure.

The outlet temperature was 24.0C for all six runs.

Results

The period of the generated sound was measured on the oscilloscope and converted to frequency with the following equation:

This straightforward conversion was done at the time of data collection, and frequency data were recorded, not period. The speed of sound was determined from the measurement of one-half wavelength by the following relationship:

It should also be noted that the heat capacity ratio is defined as follows:

As the experiment was not carried out at low temperatures or high pressures, and the gases used were simple, pure, monatomic and diatomic gases, the heat capacity ratio () may be determined as for an ideal gas with the equation:

A more accurate result may be obtained by treating the gas as a van der Waals gas with the equation:

Where M is the molecular weight of the gas and V tilde is the molar volume. The molar volume may be obtained from the van der Waals equation of state; however numerical methods must be used as the equation does not solve for the molar volume neatly.

Raw Data

Gas Used Frequency (Hz) Amplitude Max (cm) Amplitude Min (cm) Frequency (Hz) Amplitude Max (cm) Amplitude Min (cm)
Helium 1190 59.0 81.5 2326 27.6 41.9
Carbon Dioxide 1020 38.3 45.0 2174 33.5 36.9
Nitrogen 1316 36.5 43.1 2273 25.1 29.2

Table 1

Using eq. (2), and substituting the displacements between the amplitude maximum and minimum for one half wavelength, the speed of sound for each case can be obtained. For helium at 1 kHz:

By averaging the speed of sound found at both frequencies, a more accurate result may be obtained. The average speed in helium was 601 m/s.

By substitution into eq. (4), for helium may be calculated as follows:

By this procedure, was determined for all three gases, with the following results:
Ideal Gas (heat capacity ratio) [unitless]
Helium 0.585
Carbon Dioxide 0.360
Nitrogen 0.368

Table 2

From the equipartition theorem, molar heat capacity at constant volume for an ideal monatomic gas should be equal to 12.47 J K-1 mole-1. This is obtained from the following formula:

As the molar heat capacity at constant pressure is defined to be R (8.31451 J K-1 mole-1) greater than that at constant volume, it can be found to be 20.78 J K-1 mole-1. These formulas are used to create the following table of theoretical values:



No vibrational considerations With vibrational considerations










Gas
Degrees of freedom Molar heat capacity at constant volume [J K-1 mole-1] [unitless] Degrees of freedom Molar heat capacity at constant volume

[J K-1 mole-1]

[unitless]
Helium 3 12.47 1.666 3 12.47 1.666
Carbon Dioxide 6 24.94 1.333 10 41.57 1.200
Nitrogen 5 20.78 1.400 6 24.94 1.333

Table 3

Through the use of tables 2 the van der Waals coefficients for helium were found to be a=0.03457 (atm L 2 mol -2) and b=2.370 (10 -2 L mol -1). Using a programmable calculator, the molar volume was determined from the van der Waals equation of state:

Using the outlet temperature and ambient pressure, the molar volume was calculated to be 27.01 L/mol.

With the molecular weight of helium being 4.00 g/mol, and R=8314.51 (g m 2)/(s 2 K mol), the ratio of heat capacities was found [using eq. (5)] to be:

Using the preceding calculations, the following results were obtained:
van der Waals Gas (heat capacity ratio) [unitless]
Helium 0.584
Carbon Dioxide 0.362
Nitrogen 0.368

Table 4

The theoretical value of may be obtained from the following formula, provided a theoretical value for the molar heat capacity at constant volume is provided:

From this equation, the following table of values may be constructed:


Gas
Without vibrational considerations With vibrational considerations
Cv,m [J K-1 mole-1] [unitless] Cv,m [J K-1 mole-1] [unitless]
Helium 12.47 1.667 12.47 1.667
Carbon Dioxide 24.94 1.349 41.57 1.214
Nitrogen 20.78 1.407 24.94 1.339

Table 5

The obtained values for are consistently quite low, with an average percent error of 69.2%.

In the discussion, a formula for a average speed of molecules in a gas is required. For this purpose, the following formula is presented:

Evaluation of the formula for the gases in question leads to the following table:

Gas Average speed (m/s)
Helium 1572
Carbon Dioxide 143.0
Nitrogen 224.6

Table 6

Discussion

Due to the extremely large errors in the results of this experiment, it is impossible to draw any meaningful conclusions. The consistently low values obtained indicate a systematic error in data acquisition. A comparison of the obtained speed of sound in helium of 601 m/s to a known value 3 of ~965 m/s also indicates that the collection of data, at least for helium, was indeed flawed..

A primary cause of this error may have been recording of displacements at inappropriate times; nodes and antinodes may be indicated by the movement of the superimposed oscilloscope images moving in and out of phases, rather than at minimums and maximums. The proposal to modify the experiment to use superimposed images rather than Lissajous figures was suggested by the teaching assistant at the time of the experiment, and perhaps should have been avoided.

Another source of error was an instability of the oscilloscope reading. On a five volt scale, the amplitude of the signal would fluctuate by as much as 0.5 volts, making determinations of maximums and minimums difficult. The cause of this fluctuation could not be determined, even with the help of the teaching assistant. When the calibration of the oscilloscope was checked, an internal 1 kHz signal produced a reading of 952.4 Hz, a difference of 4.8%.

The treatment of the gases as van der Waals gases instead of ideal gases leads to only minor changes, the largest being that for carbon dioxide, with a difference of only 0.6%. This is to be expected from the conditions used, as the gases were at reasonably high temperatures and low pressures.

The technique of determining experimental values would not be very useful in determining whether a molecule is linear or non-linear, as non-linear molecules have only additional R contribution to their heat capacities. As a molecule must be polyatomic to be non-linear, the addition of R to its already relatively high heat capacity would be insignificant. As an example, if carbon dioxide was linear, it would lose 1 degree of freedom. As a result it would have a heat capacity ratio of 1.222 [unitless] instead of 1.200 [unitless], a difference of only 1.8%.

The average speed of molecules in a gas is equal to the average velocity of the molecules in any particular direction. The reason for this is that speed is simply the magnitude of velocity for each molecule. By assigning a direction to the speed, it becomes a velocity, and is therefore average speed in a given direction is equal to average velocity in the same direction.

As sound energy is transmitted through the motion of molecules, it follows that the rate that sound travels through a gas will be similar to the speed of the molecules themselves. Furthermore, as the speed of molecules in a substance is not a function of pressure, but temperature, it is apparent that the speed of sound in a gas is independent of the pressure.

References

  1. D. P. Shoemaker, C. W. Garland, J. W. Nibler, Experiments in Physical Chemistry, 6th ed., chap. 8, experiment 14, The McGraw-Hill Companies (1996).
  2. P. Atkins, Physical Chemistry, 5th ed., p. 217, W. H. Freeman and Company (1994).
  3. D. Halliday, R. Resnick, J. Walker, Fundamentals of Physics, 4th ed., p. 505, John Wiley & Sons, Inc. (1993)