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EXPERIMENT 4
THE RATIO OF HEAT CAPACITIES
Air contained in a large jar is subjected to an adiabatic expansion then an isochoric process. If the initial, final, and atmospheric pressures are known, then the ratio of heat capacities, , for air can be found.
THEORY
The heat capacity of a material is defined as the amount of heat that is absorbed or liberated by a material for a given change In temperature. Mathematically, this can be expressed as
Heat capacity =
Q
This expression is dependent upon the amount of material present. However, it is possible to eliminate any dependence upon the amount of material by dividing (1) by the mass m. or the number of moles m of the material. When divided by the mass, (1) becomes the specific heat of the material. or
Specific heat = c = ,
T
Q
m
and when divided by the number of moles. (1) becomes the molar heat capacity:
molar heat capacity = C = Mc = ,
T
Q
m
where M is the molecular mass of the material.
For gases, the heat that is exchanged between the system and the environment is greatly dependent upon whether the exchange of heat takes place isobarically or isochorically. Because of this a different value of the molar heat capacity must be specified for each process. In the former case, the molar heata capacity is written as Cp. and the latter as Cv. The ratio of the two values is known as the ratio of heat capacities. y. That is,
v
p C
C
In this experiment, for air is determined from an observation of the pressure changes that take place when air undergoes an adiabatic expansion and an isochoric pressure increase.
Consider a gas enclosed in container at an initial pressure p 1. which is slightly greater than atmospheric pressure p 0. The initial temperature of the gas is room temperature T (^) R. Suppose that by momentarily operung a valve, the gas is allowed to attain atmospheric pressure. The change In pressure takes place so rapidly that no heat is considered to be transferred to or from the environment, and the expansion is assumed to be adiabatic. Because compressed gas In the container must perform'work forcing some of the gas out of the container, the temperature of the gas that remains in the container drops below room temperature. If the gas is now allowed to warm up to room temperature, the pressure increases to a final value of p 2.
In order to analyze this quantitatively. consider a container of volume V 0 , divided into two parts by an imaginary membrane. (Refer to Figure 1.) Volume V 1 , represents the volume that contains the N molecules that will remain in the container after the valve is opened. The Pressure in the container iS P2, and the temperature is room temperature TR.
Figure 1. The container with the two volumes separated by an imaginary membrane.
Suppose that the valve is momentarily opened allowing the release of the gas above the membrane, as shown in Figure 2. The volume occupied by the N molecules is now V 0. The pressure in the container drops to atmospheric pressure p 0 , and the temperature drops to T. below room temperature.
Figure 2. The container immediately after the adiabatic expansion.
Since the process is adiabatic,
Figure 4. The experimental arrangement showing the pump, jar, and manometer. The clamp is placed on the tubing between the pump and the jar.
APPARATUS
o five gallon jar with stopper o pump o U-tube manometer o 1 hose clamp
PROCEDURE
a) Connect the pump and the manometer to the stopper on the flve gallon jar. Slip the clamp on the tubing between the pump and the Jar.
b) Record the atmospheric pressure P,, and Its uncertainty.
c) Pump air into the Jar until the manometer shows a difference in heights of approximately 8 to 9 centimeters. Clamp the tube between the pump and the jar.
CAUTION: Hold the rubber stopper so that it will not POP Off the jar while the ressure inside the Jar is being increased. Also be very careful not to tip the manometer over. Mercury spills
Wait until the mercury columns stop moving and record the height h, and h of each column and the uncertainty in the readings Ah.
d) Remove and replace the rubber stopper quickly. This is the adiabatic process. This is also the most difficult step in the procedure. If done too slowly, then heat transfer occurs; and if done to rapidly, then the gas in the container does not drop to atmospheric pressure.
e) Wait for a couple Of minutes to a.Ilow the temperature of the gas to reach room temperature. then record the height of each mercury column.
f) Loosen the clamp between the PUMP and the Jar.
g) Repeat the above procedures four more times for a total of five trials.
h) Repeat the experiment five more times, but pump air out of the container.
ANALYSIS
Instead of finding y and its uncertainty directly, the maximum value, y, and the minimum
value, min, are found. When air is pumped into the jar, the expressions are
ln /
ln / 1 1 2 2
1 1 0 0 max (^) p p p p
p p p p
and
ln /
ln / 1 1 2 2
1 1 0 0 min (^) p p p p
p p p p
However, when air is pumped out of the jar, (10) represents Y,,, and (11) represents y. Notice that the signs associated with the pressure uncertainties are chosen so as to maximize or minimize the quotients. The uncertainties in pressure Zip, and ap, are related as
p 1 p 2 ^ ^ h h 0 ^ p 0 2 p. (12)
The best value for the ratio of heat capacities is
max^ min (13)
and the uncertainty is
max^ min (14)
For each trial calculate y from (13) and the corresponding uncertainty Ay from (14). Report these values together with the best known value in a table. Also plot the values for each trial and the book value on a single one-dimensional graph with the values displaced vertically from each other.
