# Ratio of Specific Heats of a Gas

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Ratio of Specific Heats of a Gas EX-5531 Page of

Ratio of Specific Heats of a Gas
Equipment

 Included: 1 Heat Engine/Gas Law Apparatus TD-8572 1 Large Rod Stand ME-8735 1 45 cm Long Steel Rod ME-8736 1 Dual Pressure Sensor PS-2181 Required but not included: 1 850 Universal Interface UI-5000 1 PASCO Capstone Software UI-5400

Introduction
A cylinder is filled with air and a Pressure Sensor is attached. The piston is plucked by hand and allowed to oscillate. The oscillating pressure is recorded as a function of time and the period is determined. The ratio of specific heat capacities is calculated using the period of oscillation, according to Ruchhardt's method.
Theory
In Ruchhardt's Method, a cylinder of gas is compressed adiabatically by plucking the piston. The piston will then oscillate about the equilibrium position. Gamma, the ratio of specific heat, can be determined by measuring the period of oscillation.

If the piston is displaced downwards a distance x, there will be a restoring force which forces the piston back toward the equilibrium position.

Just like a mass on a spring, the piston will oscillate. The piston acts as the mass and the air acts as the spring. The period of oscillation of a mass on a spring (or for the piston and air) is
(1)

Figure 1: The piston is plucked by hand.
To determine the spring constant, k, for air, calculate the force when the piston is displaced a distance x. When the piston is displaced downward a distance x, the volume decreases by a very small amount compared to the total volume: dV = xA where A is the cross-sectional area of the piston.
The resulting force on the piston is given by F = (dP)A where dP is the small change in pressure. To find a relationship between dP and dV, we assume that if the oscillations are small and rapid, no heat is gained or lost by the gas. Thus the process is adiabatic and
(2)

where
= Ratio of Molar Specific Heats (3)

For a diatomic gas, CV = 5/2 R and CP = 7/2 R, so γ= 7/5.

Taking a derivative of Equation (2) gives

(4)

Solving for dP,

(5)
Since dV = xA,

(6)
Plugging into F = (dP)A gives

(7)

Comparing this to F = -kx yields

(8)
Substituting into the period equation for k gives

(9)
Solving for the volume gives
(10)
The total volume is A(h+ho), where h is the height measured on the labeled scale and ho is the unknown height below zero on the label. Substituting in for the volume and solving for the height of the piston, h, gives
(11)
Thus, if the piston height is plotted versus the square of the period, the resulting graph will be a straight line with
(12)
and y-intercept ho. Solving equation (12) for the ratio of specific heats gives

(13)
where m = mass of piston, A = cross-sectional area of piston, P = atmospheric pressure, and the slope is from the graph of h vs. T2.
Set-Up

1. Slide the Heat Engine/Gas Laws Apparatus onto the rod stand as shown in Figure 1.

1. Attach a Dual Pressure Sensor to one of the ports on the Heat Engine Apparatus. Unclamp both of the tube clamps at the bottom of the apparatus.

1. Raise the piston to the 9-cm mark and clamp it at this position with the side thumb screw at the top of the cylinder. Close the tube clamp on the open port. Loosen the side thumb screw and now the piston will stay at 9 cm.

1. Plug the Dual Pressure Sensor into Channel A on the 850 interface.

1. In PASCO Capstone, create a graph of Pressure vs. time. Also, in a table, create a User-Entered Data measurement called “Piston Height” and another User-Entered Data measurement called “Period”.

Procedure
1. Find the mass (m) of the piston (given on the apparatus label) and the cross-sectional area (A) of the piston (the piston diameter is given on the apparatus label).
2. Click on Record in the PASCO Capstone program.
3. Using the tip of your finger, pluck the top of the piston. Click Stop on the computer.
4. Using the Coordinates Tool on the graph, determine the period of the oscillation from the pressure versus time graph. Expand the area of the graph that shows the oscillation. Measure the period by measuring the time for several peaks and dividing by the number of peaks. Type this period and the corresponding piston height into the table.
5. Open the tube clamp on the open port. Lower the piston to the 8-cm mark and clamp it at this position with the side thumb screw at the top of the cylinder. Close the tube clamp on the open port. Loosen the side thumb screw and now the piston will stay at 8 cm. Repeat the procedure.

1. Then continue to lower the piston in steps of 1 cm, repeating the procedure at each piston position down to 1 cm.

1. Unless a barometer is available, assume the atmospheric pressure is 1.01 x 105 Pa.

1. Graph the piston height vs. the period. Choose a QuickCalc of the period squared on the horizontal axis. Apply a linear fit and use the slope to calculate γ for air. Compare to the ideal value.

1. If another gas is available, determine γ for that gas. NOTE: Another gas, such as Helium, can be introduced into the cylinder by moving the piston to its lowest position, attaching a rubber balloon filled with Helium to the unused port and opening the hose clamp and letting the Helium from the balloon flow into the cylinder, pushing the piston up to the top. Then the hose clamp is closed with the piston at 9 cm. Never attach a high pressure hose directly to the apparatus.

Questions
1. What is the ratio of specific heats of a diatomic gas in theory? Why?
2. What is the ratio of specific heats of a monatomic gas in theory? Why?
3. Would the slope of the graph for Helium be greater or less than the slope for air? Why?
4. Why can we assume air is diatomic? What are the main components of air?

Written by Ann Hanks

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