Heat Capacity Ratios for Gases as Determined by Sound Velocity Method Lab Report

Heat Capacity Ratios for Gases as Determined by the Sound

Velocity MethodObjective: In this lab you will experimentally determine ratio of CP/CV (designated as ?) for three gases. Youwill then compare your ? values to the accepted values for these gases, as well as perfect gas values predictedfrom equipartition theory.Background:The First Law of Thermodynamics may be used to deduce that the difference in the constant volume andconstant pressure heat capacities for a perfect gas is given by:CP ? CV ? R (1)The ratio of CP/CV is given the designation ?, and is experimentally measurable. In this lab you will measurethis parameter for three gases by measuring the speed of sound in each gas using a Kundt’s tube. It can beshown that for a perfect gas, ? is related to the speed of sound in the gas, c, by the following equation:RTMc 2? ? (2)wherec ? ?f (3)and ? and f are the wavelength and the frequency respectively of the sound wave. In this lab, you will measurethe value of c directly in N2, He, and CO2 by determining the peak amplitudes in a series of standing waves inthe Kundt’s tube as a function of the frequency.Additional background information can be found in the associated documents for this lab.Procedure:A detailed procedural description can be found in: Experimental Physical Chemistry, 8th ed. by Shoemaker,Nibler, and Garland. Essential procedural components are summarized here.The apparatus for this determination consists of a section of PVC tube with a small speaker at one end and amicrophone at the other. The speaker is connected to a sine wave generator and the output of the microphone isconnected to the Y-input on an oscilloscope. The output from a square wave is connected to a second Y-inputon the oscilloscope. This square wave has the same frequency as the sine wave used to drive the speaker andwill be used as a reference.microphone speakerThe sound produced by the speaker propagates down the tube in the form of a compression wave which isdetected by the microphone. The waveform at the microphone is phase-shifted relative to the input waveformby an amount ?. The degree of this shift will depend on the frequency and speed of the wave which in turn aredependent on the identity of the gas. Here, the phase angle is given by:???2 d? (4)When it reaches the end of the Kundt’s tube, the sound wave is reflected back towards the speaker. Thisreflected wave will interfere with the incoming wave. If the frequency of the sinewave is such that the length ofthe tube is an integral number of wavelengths, constructive interference will occur and the output waveformfrom the speaker will have maximum amplitude. This happens if the phase angle between the incoming andreflected waveform is 0°. If the reflected waveform is 180° out of phase with the incoming waveform, thendestructive interference occurs, and the amplitude of the signal produced by the speaker is minimized.When the input waveform is plotted against the output waveform on the oscilloscope, geometric shapes resultwhich, assuming equal amplitudes on the waves, are determined by ?. These are called Lissajous figures.Several Lissajous figures are given below, along with the corresponding phase angles which produce them.????0 ??????? ??????? ???????? ?????The important Lissajous figures for this experiment are the first one and the last one. These indicate conditionssuch that the two waveforms are either exactly in phase or 180° out of phase. Alternatively, you can simply lookfor frequencies which result in maximum amplitude in the output waveform. This is the recommendedprocedure. For the cases where ? is either 0° or 180°, the wavelength must satisfy the following relationship:n ? ? d? ????? 2, n=1,2,3,… (5)where d is the distance between the speaker and the microphone, ? is the wavelength of the sound wave, and nis an integer. If we make the substitution for frequency as described by equation (3), this becomes:nc df2? (6)Thus, a plot of the frequency of the sound wave as a function of n should be a straight line whose slope containsthe speed of sound in the gas.Filling the Kundt’s TubePrior to taking any data, have the laboratory instructor review the overall operation of the apparatus and theuse of compressed gases with you.Hook up the line from the gas to be studied to the inlet port of the Kundt’s tube. Make sure the valve on theinlet port is shut. Open the outlet port on the Kundt’s tube. With the regulator shutoff valve closed, adjust theoutlet pressure on the regulator to approximately 20 psi, then open the regulator shutoff valve. Using the valveon the inlet port, gradually allow gas to flow into the Kundt’s tube and vent into the room through the outletvalve.The Kundt’s tube should never be pressurized. This can potentially damage the components, or injure theuser. Make sure the inlet flowrate is kept low, and that the outlet port is closed!!Observe the output waveform on the oscilloscope while gas is being flushed through the Kundt’s tube. Whenthe waveform ceases to shift in phase and amplitude, the tube has been completely charged with the gas to bestudied. When this condition is met, close (1) the inlet port valve, then (2) the outlet port valve. The apparatus isnow ready to acquire data.Acquiring the DataYou must first determine d, the length of the Kundt’s tube. This is most easily done with a meter stick.Set the frequency on the sinewave generator to around 1 kHz for N2 and CO2 (2 kHz for He). Set theoscilloscope to display both channels simultaneously and slowly adjust the frequency of the input sinewave.Observe the changes that occur with respect to both the amplitude of the waveform produced by the microphoneand the phase shift between the two waveforms. You should be able to see that there are certain frequencies atwhich the amplitude of the output waveform is maximized. Make sure that the scope is set to trigger on theinput waveform (channel 1). The trigger synchronizes the sweep rate on the scope with the frequency of thewaveform. If it is not set directly the display will appear to “travel” on the display.Starting around 1 kHz, increase the frequency and record those frequencies where 0° and 180° phase shifts areindicated. (Frequencies can be determined from the scope display. Get the instructor to show you how to dothis.) You should be able to identify ten or twelve frequencies. These frequencies will be associated with anarbitrary integer, n, as given in equation (6).Open the outlet port and connect the next gas to be studied to the inlet port of the Kundt’s tube. Slowly flush thetube with the next gas in the same manner described above. When the old gas has been completely replaced,close the valves on the Kundt’s tube as was done before. Do not overpressurize the tube! Determine thefrequencies at which the 0° and 180° phase shifts occur in the same manner. Obtain complete data sets for He,CO2 and N2.Calculations and Data Analysis:Make plots of frequency vs. n for Ar, CO2 and N2. Determine the speed of sound in each gas from the slopes ofthese lines, then use equation (2) to calculate ? for each of the three gases. Compare your measured speeds ofsound and ? values for the three gases to the known values. Quantitatively assess the uncertainty in allexperimentally-determined values. Within the uncertainty, do your values agree with the accepted ones? Are the? values distinguishable from one another within experimental error? Compare the ? values to those predictedbased on the assumptions of ideality and equipartition theory. Do they agree? If not, is there a valid physicalreason for it? Heat Capacity Ratios for Gases as Determined by the Sound
Velocity Method
Objective: In this lab you will experimentally determine ratio of CP/CV (designated as γ) for three gases. You
will then compare your γ values to the accepted values for these gases, as well as perfect gas values predicted
from equipartition theory.
Background:
The First Law of Thermodynamics may be used to deduce that the difference in the constant volume and
constant pressure heat capacities for a perfect gas is given by:
CP  CV  R
(1)
The ratio of CP/CV is given the designation γ, and is experimentally measurable. In this lab you will measure
this parameter for three gases by measuring the speed of sound in each gas using a Kundt’s tube. It can be
shown that for a perfect gas, γ is related to the speed of sound in the gas, c, by the following equation:

Mc 2
RT
(2)
where
c  f
(3)
and λ and f are the wavelength and the frequency respectively of the sound wave. In this lab, you will measure
the value of c directly in N2, He, and CO2 by determining the peak amplitudes in a series of standing waves in
the Kundt’s tube as a function of the frequency.
Additional background information can be found in the associated documents for this lab.
Procedure:
A detailed procedural description can be found in: Experimental Physical Chemistry, 8th ed. by Shoemaker,
Nibler, and Garland. Essential procedural components are summarized here.
The apparatus for this determination consists of a section of PVC tube with a small speaker at one end and a
microphone at the other. The speaker is connected to a sine wave generator and the output of the microphone is
connected to the Y-input on an oscilloscope. The output from a square wave is connected to a second Y-input
on the oscilloscope. This square wave has the same frequency as the sine wave used to drive the speaker and
will be used as a reference.
microphone
speaker
The sound produced by the speaker propagates down the tube in the form of a compression wave which is
detected by the microphone. The waveform at the microphone is phase-shifted relative to the input waveform
by an amount φ. The degree of this shift will depend on the frequency and speed of the wave which in turn are
dependent on the identity of the gas. Here, the phase angle is given by:

2d
(4)

When it reaches the end of the Kundt’s tube, the sound wave is reflected back towards the speaker. This
reflected wave will interfere with the incoming wave. If the frequency of the sinewave is such that the length of
the tube is an integral number of wavelengths, constructive interference will occur and the output waveform
from the speaker will have maximum amplitude. This happens if the phase angle between the incoming and
reflected waveform is 0°. If the reflected waveform is 180° out of phase with the incoming waveform, then
destructive interference occurs, and the amplitude of the signal produced by the speaker is minimized.
When the input waveform is plotted against the output waveform on the oscilloscope, geometric shapes result
which, assuming equal amplitudes on the waves, are determined by φ. These are called Lissajous figures.
Several Lissajous figures are given below, along with the corresponding phase angles which produce them.
0




The important Lissajous figures for this experiment are the first one and the last one. These indicate conditions
such that the two waveforms are either exactly in phase or 180° out of phase. Alternatively, you can simply look
for frequencies which result in maximum amplitude in the output waveform. This is the recommended
procedure. For the cases where φ is either 0° or 180°, the wavelength must satisfy the following relationship:

n   d ,
2
n=1,2,3,…
(5)
where d is the distance between the speaker and the microphone, λ is the wavelength of the sound wave, and n
is an integer. If we make the substitution for frequency as described by equation (3), this becomes:
f 
c
n
2d
(6)
Thus, a plot of the frequency of the sound wave as a function of n should be a straight line whose slope contains
the speed of sound in the gas.
Filling the Kundt’s Tube
Prior to taking any data, have the laboratory instructor review the overall operation of the apparatus and the
use of compressed gases with you.
Hook up the line from the gas to be studied to the inlet port of the Kundt’s tube. Make sure the valve on the
inlet port is shut. Open the outlet port on the Kundt’s tube. With the regulator shutoff valve closed, adjust the
outlet pressure on the regulator to approximately 20 psi, then open the regulator shutoff valve. Using the valve
on the inlet port, gradually allow gas to flow into the Kundt’s tube and vent into the room through the outlet
valve.
The Kundt’s tube should never be pressurized. This can potentially damage the components, or injure the
user. Make sure the inlet flowrate is kept low, and that the outlet port is closed!!
Observe the output waveform on the oscilloscope while gas is being flushed through the Kundt’s tube. When
the waveform ceases to shift in phase and amplitude, the tube has been completely charged with the gas to be
studied. When this condition is met, close (1) the inlet port valve, then (2) the outlet port valve. The apparatus is
now ready to acquire data.
Acquiring the Data
You must first determine d, the length of the Kundt’s tube. This is most easily done with a meter stick.
Set the frequency on the sinewave generator to around 1 kHz for N2 and CO2 (2 kHz for He). Set the
oscilloscope to display both channels simultaneously and slowly adjust the frequency of the input sinewave.
Observe the changes that occur with respect to both the amplitude of the waveform produced by the microphone
and the phase shift between the two waveforms. You should be able to see that there are certain frequencies at
which the amplitude of the output waveform is maximized. Make sure that the scope is set to trigger on the
input waveform (channel 1). The trigger synchronizes the sweep rate on the scope with the frequency of the
waveform. If it is not set directly the display will appear to “travel” on the display.
Starting around 1 kHz, increase the frequency and record those frequencies where 0° and 180° phase shifts are
indicated. (Frequencies can be determined from the scope display. Get the instructor to show you how to do
this.) You should be able to identify ten or twelve frequencies. These frequencies will be associated with an
arbitrary integer, n, as given in equation (6).
Open the outlet port and connect the next gas to be studied to the inlet port of the Kundt’s tube. Slowly flush the
tube with the next gas in the same manner described above. When the old gas has been completely replaced,
close the valves on the Kundt’s tube as was done before. Do not overpressurize the tube! Determine the
frequencies at which the 0° and 180° phase shifts occur in the same manner. Obtain complete data sets for He,
CO2 and N2.
Calculations and Data Analysis:
Make plots of frequency vs. n for Ar, CO2 and N2. Determine the speed of sound in each gas from the slopes of
these lines, then use equation (2) to calculate γ for each of the three gases. Compare your measured speeds of
sound and γ values for the three gases to the known values. Quantitatively assess the uncertainty in all
experimentally-determined values. Within the uncertainty, do your values agree with the accepted ones? Are the
γ values distinguishable from one another within experimental error? Compare the γ values to those predicted
based on the assumptions of ideality and equipartition theory. Do they agree? If not, is there a valid physical
reason for it?
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LABORATORY REPORT FORMAT
There are seven total labs scheduled in the course. These labs will be completed in one or two lab periods and will
be done in short form report format which must include the following components:
1. A clear statement of the objective
2. A detailed description of how you got the data. (Points will be taken off if this is copied from the
provided lab materials.)
3. A tabulation of the raw data
4. A section containing results (includes graphs, calculations, and the final answer)
5. Analysis of error (if applicable)
Grading in these lab reports will focus largely on (1) the presence of all required components, (2) the quality of the
data (3) the accuracy of the answer and (4) the clarity of the writing. The writing must be concise and accurate in
the description of what was determined. An error analysis will be required for any lab in which the result is a
quantifiable value resulting from measurable parameters. The lab reports will be worth 30 points each. Lab
notebooks are not required, but are encouraged.
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