Cp/Cv Ratio
CHEM 366
Miaomiao Gu
Experiment date: Feb 10th, 2011
Student ID: 301100545
Submission date: Feb 24th, 2011
Abstract. The ratio of heat capacity at constant pressure to heat capacity at constant
volume for three gases: Nitrogen, Carbon dioxide and Argon were estimated and
given the values: 1.41(2), 1.29(2), and 1.673(5) by measurement of the speed of sound
through the gas. The speed of sound was obtained indirectly though wavelength
measurements with the aid of electronic instrumentation.
Purpose:
Obtain the heat capacity ratio CP/CV by the Sound Velocity method though the use
of a Kundt’s tube for three gases N2, CO2 and Ar. The results were interpreted in terms
of the contribution of the various degrees of freedom in the molecule to the heat
capacity of the compound.
Introduction:
To estimate the CP/CV ratio of three air molecules, Nitrogen, Carbon dioxide and
Argon, the sound –velocity method was used. By measuring the sound speed when
the sound wave was traveling through all the three types of gas at constant volume
and pressure, the ratio was obtained by both the experimental calculation and
equipartition theorem. The results were compared in turn between these two
extrapolation methods.
Theory:
For an ideal gas the heat-capacity ratioγ= Cp/Cv of a gas can be obtained by: (1)
γ=Mc2/RT
(1)
However, for Van-del-waal gases,
γ=c2M/RT(1-2a/PV2+2b/V)
(2)
For a theoretical value of Cv, the ratio can be approximated as:
γ=1+R/Cv*(1+2ap/(RT)2)
(3)
By using equipartition theorem, the ratio can be extrapolated by:
γ=Cp/Cv=1+R/Cv
(4)
In the special case of electromanetic waves moving through a vacuum, then v= c,
where c is the speed of sound in a vacuum, and this expression becomes:
ƒ = c/
(5)
For an ideal gas, the molar heat capacity at constant volume is obtained from
formula:
CV= (degrees of freedom/2 )*R
(6)
The Cv value for a gas was composed of three parts: the translational, rotational
and vibration. Each of the components was related to R/2 and the number of degree of
freedom of that component. The detailed method to calculate values of Cv of each gas
would be shown in the next section.
Experimental Procedure:
Most procedures were same as shown in the lab manual,
(3)
few details were
indicated in the following:
The heat capacity ratio Cp/Cv is obtained by the Sound Velocity method for three
gases N2, CO2 and Ar at 739.8mm Hg and 23°C conditions.
The frequency range was limited between 1000-4000Hz.
In order to get the pure gas in the Kundt’s tube, each gas was pumped out and
refilled several times.
Result and Calculation:
The sound wavelengths traveling in different gases were measured at different
frequency. Thereafter the speed of the sound was extrapolated as the production of the
frequency and the wavelength. Then the values were shown in Table 1-3.
frequence/Hz
1916
2947
4084
λ/2(mm)
69
46
33
c/(m/s)
±
264
15
271
29
270
16
cave=
268
4
Table 1. Sound wavelengths at different frequencies, and the calculated sound speed
in CO2 at room temperature and room atmosphere pressure.
frequence/Hz
1913
2601
3039
±
4
5
2
λ/2(mm)
91
67
58
±
c/(m/s)
±
5
348
19
3
349
16
2
353
12
cave=
350
3
Table 2. Sound wavelengths at different frequencies, and the calculated sound speed
in N2 at room temperature and room atmosphere pressure.
frequence/Hz
1914
2534
2923
λ/2(mm)
84
63.2
55.1
±
c/(m/s)
±
2
322
8
0.8
320
4
0.9
322
5
cave=
321
1
Table 3. Sound wavelengths at different frequencies, and the calculated sound speed
in Ar at room temperature and room atmosphere pressure.
From equation (5), the sound speed among the three kinds of gases was calculated.
The sound was traveled fastest in N2 with 350(3) m/s, whereas in Ar the sound
traveled at speed 321(1) m/s and in CO2 the sound had lowest speed at 268(4) m/s.
They were shown in Table 1-3.
Since the heat-capacity ratio related to the speed of sound, the experimental CP/CV
ratio could be calculated from equation (1) for ideal gas and equation (2) for
van-del-waal gases. Nevertheless, both equations were applied on all three gases, and
the obtained results from different equations were compared to see if the equation was
applicable for all gases. The results were extrapolated from equation (1) and (2) were
shown in table 4:
γ(id.gas)
γ(Vdw gas)
±
±
CO2
1.28
0.03
1.2883
0.0008
N2
1.44
0.02
1.4061
0.0004
Ar
1.673
0.005
N/A
N/A
Table4. Calculated Cp/Cv ratios by different equations of three gases.
From equation (1) where M is the molar mass as 44.01g/mol for CO2, R is the gas
constant 8.3145 J/mol/K and T is temperature at 296K, the heat-capacity ratio for CO2
was obtained as 1.284455 and shown with computed heat-capacity ratio of other two
gases in table 4.
By equation (2), the heat-capacity ratio for van der waals was determined. The van
der waals constants a and b as literatures of N2 and CO2 were given.(1) The quantity V
was approximated by the ideal gas values V= RT/P as 24.965L. P was the pressure at
739.8 mmHg. Therefore, from equation (2), the heat-capacity ratio of CO2 for van der
waals was obtained as 1.2883. Same method was used for N2 and values were shown
in Table 4. Since gas Ar had no a and b values in the reference, the heat capacity ratio
for Argon gas can only be deduced by ideal gas equation (1).
The CP/CV ratios for all three gases could also be obtained by equipartion theorem.
CV values could be calculated as the sum of three components: translational,
rotational and vibrational. Each component was related to degrees of freedom*R/2
which was dependent on the number of degrees of freedom. Each gas had 3
translational degrees of freedom. As both N2 and CO2 molecules were linear, their
rotational degrees of freedom were same as 2 and the vibrational degree of freedom
was 3N-5 where N was the number of molecule atoms. Then, the vibrational degrees
of freedom were 1 and 4 for N2 and CO2. Thus, the three component values were:
3R/2, R, R/2 for N2 and 3R/2, R, 2R for CO2. As Ar was ideal gas, it only had 3R/2 of
translation. The calculated Cv values for these gases were given in table 5.
Cv(vib)
γ(vib)
Cv(nonvib)
CO2
9R/2
1.22
5R/2
NO2
3R
1.33
5R/2
Ar
3R/2
1.67
3R/2
Table 5. CP/CV ratios calculated by theoretical approximations.
γ(nonvib)
1.4
1.4
1.67
In table 5, the CP/CV ratios were calculated from expected CV values by equation
(4). When the vibrational component accounted as shown in the first two columns, the
CP/CV ratio of N2 indicated significant difference from obtained value in experiment.
However, the expected values for Ar and CO2 by equiparition theorem were highly
consistent with the extrapolated values by experiment. The last two columns showed
the heat capacity ratios when vibrational component was not accounted. The heat
capacity ratio of CO2 showed indicated significant difference from the obtained value
in experiment. But the expected values of N2 and Ar were highly consistent with the
extrapolated experiment values.
Error Analysis:
The average 1/2 wavelength of CO2 at 1916Hz was calculated as 69 mm and the
standard deviation on this wavelength has been calculated from σ=√(Σ(xi-xave)2/(n-1))
where xave= 69 mm and n= 5, thus, σST = 4 . The result should be reproted as
69(4) mm and was shown in table 1. The other errors of wavelengh and sound speed
were calculated by the same method and shown in table 1-3.
The heat capacity ratio of ideal gas was related to the sound speed. By equation (1)
the heat capacity ratio of CO2 was obtained as 1.284455. Since except the sound
speed, other variables in equation (1) were constant, the error of heat capacity ratio
was extrapolated from the error of sound speed. Therefore, % eCp/Cv =√ (% ec)2 +(%
ec)2 , then σCp/Cv = 0.03. As a result, the heat capacity ratio of CO2 obtained as an ideal
gas was reported as 1.28(3) in table 4. The error of heat capacity ratio of other two
gases with both ideal gas and van der waals methods were extrapolated by the same
way and shown in table 4.
Discussion:
The expected CP/CV ratio of N2 obtained from equipartion theorem with the
vibrational was significantly different from the experimental value for van del waal
gas, but the value obtained without vibrational was consistent with the experimental
value. This is because when nitrogen atoms stretched along the N-N bond; energy
could be absorbed and affected the heat capacity.
In this case, N2 molecules could
store more energy in the bond compared to Ar which had no intermolecular bonds and
thus had no rotational or vibrational energy. This could be proved that when no
vibrational was considered, the CP/CV ratio of N2 was highly consistent with the
experimental value.
The expected CP/CV ratio of CO2 with the vibrational was consistent with the
experimental value, but without the vibrational was significantly different from the
experimental value. In O=C=O bonds, the bond energies were cancelled with each
other, thus it did not affected the heat capacity. Therefore, CP/CV ratio of CO2 was
highly consistent with the experimental value when vibrational were considered.
If CO2 was non-linear, both of the rotational and vibrational would change to 3R/2.
Thus, CV= 9R/2 and Cp/Cv ratio (equation (4)) was 1.22. This value was same with
the heat capacity ratio of linear CO2. Therefore, equipartion theorem would not be
very useful in determining whether a molecule is linear or non-linear.
Since the kinetic energy of particles equaled to 3/2kT= 1/2 mv2, the average speed
of a gas could be calculated as v= √(3kT/m) = √(3RT/M) where T= 296K and R=
8.3145 J/mol/K, thus, the average speed of N2, CO2 and Ar were: 513.4, 504.6,
429.9(m/s), compared to sound speed in each gas, the gas molecules moved faster. In
addition, as c=f, the sound speed depended on the wavelength and frequency, and
neither of them could be affected by pressure. Thus the sound speed was independent
of pressure.
For van-del-waal gases, equation (2) gave more precise results than equation (1)
did, as the uncertainty indicated. For the results obtained from equation (2), an
uncertainty of 1*10-4 scale was given where as the other method gave accuracy to
only 2 decimal places. Thus the results given in table 4 justified equation (2) over
equation (1) as for van-del-waal gases. Moreover, the van del waal equation corrected
the ideal gas equation from inter-molecule collision.
Conclusion:
The CP/CV ratio was estimated for three gases: N2, CO2 and Ar. The expected
values for these gases deduced by equipartion theorem were not consistent with
experimental values except for the ideal gas Ar. This was because the intermolecular
bonds of van-del-waal gases absorbed energy and thus affected the heat capacity of
gases.
Reference:
1. Carl W. Garland; David P. Shoemaker; Experiments in Physical Chemistry, 8th
edition; McGraw-Hill Higher Education, New York, 2009. P106-118
2. Physical Chemistry lab manual.
3. Handbook of Chemistry and Physics. 91st Edition, 2010-2011.
4. Daniel C.Harris; Quantitative Chemical Analysis, 7thed; W.H.Freeman and
Company, New York, 2007.P39-71.
Appendices:
Cp/Cv experimental results
Feb, 10th , 2011
MIAOMIAO GU
301100545
Gas 1
-1
Mol weight (kg mol )
CO2
44.01g/mol
Temp/C
23.00
Atm press (mm Hg)
739.80
Freq3
Freq1 (Hz)
mm
Piston position_1
Freq2 (Hz)
(Hz)
1916.0
2947.0
4084.0
395.00
456.00
420.00
Error freq (Hz)
Error position
Piston position_2
300.00
417.00
386.00
(mm)
Piston position_3
257.00
367.00
352.00
Piston position_4
190.00
330.00
322.00
Piston position_5
120.00
273.00
287.00
Piston position_6
50.00
234.00
253.00
Piston position_7
180.00
225.00
Piston position_8
135.00
190.00
Piston position_9
97.00
156.00
Piston position_10
46.00
123.00
Piston position_11
90.00
Piston position_12
57.00
Piston position_13
25.00
Gas 2
-1
4.00
N2
Mol weight (kg mol )
28.0134g/mol
Temp/C
23.00
Atm press (mm Hg)
739.80
Freq3
Freq1 (Hz)
mm
Piston position_1
Freq2 (Hz)
(Hz)
Error freq (Hz)
1913.0
3039.0
2601.0
373.00
397.00
400.00
Error position
Piston position_2
280.00
340.00
330.00
Piston position_3
185.00
280.00
263.00
Piston position_4
99.00
220.00
196.00
Piston position_5
164.00
130.00
Piston position_6
107.00
62.00
Piston position_7
49.00
0.00
Gas 3
-1
(mm)
3.00
Ar
Mol weight (kg mol )
39.948g/mol
Temp/C
23.00
Atm press (mm Hg)
739.80
Freq3
Freq1 (Hz)
mm
Piston position_1
Freq2 (Hz)
(Hz)
Error freq (Hz)
1914.0
2534.0
2923.0
421.00
436.00
433.00
Error position
Piston position_2
335.00
372.00
378.00
Piston position_3
250.00
310.00
323.00
Piston position_4
167.00
247.00
269.00
Piston position_5
85.00
184.00
213.00
Piston position_6
0.00
120.00
159.00
Piston position_7
57.00
103.00
Piston position_8
0.00
47.00
(mm)
1.23