Experiment 13: Measurement of Cp/Cv
A glass tube extends from the top of a plastic jug. When a steel ball is dropped in,
the air below it is trapped because the ball’s diameter closely matches the tube’s. Air
is easily compressed or stretched, so it acts like a spring and the ball bounces. The
period is measured using a pressure sensor connected to the jug. (The computer
displays a graph of pressure versus time from which the period is read.) The air is
compressed and decompressed rapidly enough that little heat has time to flow out or
in through the poorly conducting walls. So the process is approximately adiabatic,
and therefore PVγ = constant applies. You will manipulate this equation to obtain γ,
the ratio Cp/Cv, from your measured period.
PROCEDURE:
Take care not to get dust particles on the ball or in the tube; they can wedge the ball
in place. Also, take care not to dent or scratch the ball.
Data. Measure the ball's mass with a balance and its diameter with a micrometer. Calculate the
ball's cross sectional area from its diameter. The volume, V, (of the jug and about two thirds of the
tube) was previously measured, and is listed on your data sheet. Read the pressure from the
5
barometer in the chemistry lab. (76 cm of Hg = 1.01 x 10 Pa.)
Determine the period:
1. Connect the small tube in the cork to your computer's pressure sensor.
2. In Science Workshop, drag the analog plug icon to channel A. Scroll down and click on
low pressure sensor (the bottommost of the three pressure sensors). Click ok. Drag the graph
icon to channel A. A graph should appear. Click on the interface box to get it back on top.
3. Click on REC at the upper left. When the writing on it gets faint, drop the ball in. When it
stops bouncing, click STOP.
4. Carefully tip the apparatus so the ball rolls back into your hand. Set it in the bottle cap
when not in use, so it stays clean and won't roll away.
5. On the graph, click the button closest to the origin, just below and to its left, to size the
graph's scale to the data. You can also change the scale by clicking on + or - at the lower right.
6. Read the period from the graph. The value of γ is sensitive to just a few hundredths of a
second, so for best accuracy, take the time for as many periods as possible and divide. If you
click on this:
the pointer on the screen turns into a set of cross hairs whose coordinates
appear next to the axes of the graph. To measure the difference between the coordinates of two
points, put the cross hairs on the first point, then hold down the mouse button while moving it
to the second point. The numbers will disappear when you release the button.
7. Make a few more runs, and average the results.
Put the ball back in its tube, and cork the ends so it stays clean in storage.
CALCULATIONS:
Show all details, step by step. ΔV and ΔP are small enough to approximate them with the
differentials dV and dP.
3 4
1. Take the differential of the equation PVγ = constant. (Example: Differentiation of P V = constant
3
3
2
4
γ-1
γ -1
would give P (4V dV) + (3P dP)V = 0) Hint: Noticing that V = V V simplifies the result.
2. Rearrange this to get an expression for γ.
3. Substitute Ax for dV, as shown in the diagram.
4. To obtain a substitution for dP, put the following things together:
- The definition of pressure, P = F/A. Since the additional
pressure, ΔP, is what supplies the restoring force, ΔP = Fr/A.
- The restoring force on a harmonic oscillator is given by
Hooke's law, Fr = -kx.
5. From Phy 131, the period of a harmonic oscillator is T = 2 m / k . Solve that for k, stick it in,
and you have a formula for γ in which everything is something you've measured.
6. Plug your data into this and calculate a value for γ.
7. Take T's uncertainty to be the difference between the average T and the trial that was the farthest
off. Assume A, V, m, and P contribute a total of 2% to γ's uncertainty. Since T is squared in your
expression for γ, double its percent uncertainty, then add the 2% to find the uncertainty in γ.
Recall that there are different kinds of experimental errors. The kind we have usually concentrated
on is random errors from uncertainty in measurement. Another kind is systematic errors, which
always throw you off in the same direction. This apparatus has a systematic error which typically
makes γ about 7 or 8% less than the accepted value, corresponding to a period 3½ or 4% longer than
ideal. Friction is involved in slowing down the ball, but the largest problem is that the process is not
quite adiabatic. As the ball falls, heat escaping from the air below it makes its temperature rise
slower than it should. This makes the pressure rise slower, so the ball falls longer before there is
enough pressure to turn it around. Then as it moves upward, there is a similar delay. Heat flow is
driven by a temperature difference, so this error should decrease if the air is compressed less. In
fact, if your apparatus is working well enough for you to get around fifteen periods of data, and if
you measure very carefully, you can see that the period is one or two hundredths of a second shorter
at the end where the amplitude is small.
So, compare your result to the accepted value for air (a mixture of nitrogen and oxygen). If you are
around 7% below the accepted value, you have done well. Remember to explain the reason for the
difference in your conclusion.
Report on Experiment 13: Measurement of Cp/Cv
PHY 132
DATA:
m = ____________
d = ____________
Calculate A:
P = _________________ Pa
Time
Number
cycles
of
Period
average = ____________
Calculation of γ (continue on back):
Calculate γ's uncertainty:
V = .0084 m3