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**Heat Capacity Ratios of Gases**

A Thermodynamic Study

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**The molar heat capacity is the amount of heat absorbed by a**

substance when its temperature is raised one Kelvin degree. The measurement can occur at either constant volume or constant Pressure producing CV or CP, respectively. For monatomic gases, it is shown in statistical thermodynamics that the translational heat capacity is ½R for each degree of freedom for a mole of gas. For a mole of diatomic gas, each rotational degree of freedom also contributes ½R. The vibrational contribution at high temperature for a mole of a harmonic diatomic would be RT, or more exactly: It is of interest to compare the ratios of the two types of heat Capacity since that can be a measure of ideality. v – frequency of normal mode

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**For the constant volume situation, there is no PV work, so:**

And using the definition of the heat capacity: Similarly for the enthalpy where there is only PV-work: Thus,

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**A relationship between the two heat capacities for an ideal gas is: CP = CV + R**

For the reversible adiabatic expansion of an ideal gas where the temperature change is insufficient to affect the value of the heat capacity: Where if the gas is ideal

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**The following expression is produced:**

This indicates the ratio of the heat capacities can be obtained If the pressure and volume changes in an adiabatic expansion can be measured. The former are easy to monitor but the latter are difficult because of the need of a frictionless piston. The volume ratios can be measured by following the adiabatic step I: P1V1T1 = P2V2T2 with a constant volume process where the gas is permitted to warm to the original temperature in step II: P2V2T2 = P3V2T1

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**For one mole of an ideal gas, the second step gives:**

Which can be combined with the ideal previous ideal gas expression to give: When this is combined with the equation for the adiabatic expansion, the expression for the heat capacity ratio is:

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Experimental The goal is to obtain the heat capacity ratio for a gas and compare to ideal values. Step I is achieved by removing stopper quickly from a pressurized carboy resulting in rapid drop to atmospheric press. Step II is achieved by allowing the system to return to initial temperature.

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**The room as a whole acts as the thermostat**

Begin with least dense gases In introducing gases allow the carboy to be swept for 10 –15 minutes at 1.5 atm. Introduce gas at bottom of carboy After sweeping, reduce flow rate (by about 20 times) by adjusting control on inlet hose. While central vent is still open, open the pressure sensor to the chamber. Maintaining the slow flow rate, insert stopper.

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**Stop the gas flow when pressure is about 80 cm above atmospheric pressure.**

Allow 10 minutes for equilibration. Take pressure readings until they are constant. Remove the stopper (or open clamp) with a small sweeping motion and then reinsert (close) it quickly and tightly. Allow to come to thermal equilibrium (10-15 min); pressure will increase. When pressure is constant equilibrium is achieved.

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**Record the equilibrium pressure.**

Record atmospheric pressure. Record room temperature. Carry out procedure two more times with the gas without the flushing process. (Only needed when changing gases.) Be sure to introduce gases more dense than air by letting them flow into bottom. Use last heat capacity ratio equation to calculate values.

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Treatment of Results Calculate theoretical values of heat capacity using spectroscopic data, etc. Compare to results. Assume CP = CV + R, and compute the experimental CV values Is the deviation within experimental error? Is there any systematic deviation? If there is, explain it.

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