miaomiao9456.files.wordpress.com file · Web viewTo estimate the C P /C V ratio of three air molecules, Nitrogen, Carbon dioxide and Argon, the sound –velocity method was used

Cp/Cv Ratio

CHEM 366

Miaomiao Gu Student ID: 301100545

Experiment date: Feb 10th, 2011 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 λ/2(mm) ± c/(m/s) ±

1916 69 4 264 152947 46 5 271 294084 33 2 270 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 λ/2(mm) ± c/(m/s) ±1913 91 5 348 192601 67 3 349 163039 58 2 353 12

cave= 350 3Table 2. Sound wavelengths at different frequencies, and the calculated sound speed

in N2 at room temperature and room atmosphere pressure.

frequence/Hz λ/2(mm) ± c/(m/s) ±1914 84 2 322 82534 63.2 0.8 320 42923 55.1 0.9 322 5

cave= 321 1Table 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) γ(nonvib)CO2 9R/2 1.22 5R/2 1.4NO2 3R 1.33 5R/2 1.4Ar 3R/2 1.67 3R/2 1.67

Table 5. CP/CV ratios calculated by theoretical approximations.

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, 8 th

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 CO2Mol weight (kg mol-1) 44.01g/mol

Temp/C 23.00

Atm press (mm Hg) 739.80

　 Freq1 (Hz) Freq2 (Hz)Freq3 (Hz) Error freq (Hz)

mm 1916.0 2947.0 4084.0

Piston position_1 395.00 456.00 420.00

Piston position_2 300.00 417.00 386.00Error position (mm)

Piston position_3 257.00 367.00 352.00 4.00

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_11 90.00



Gas 2 N2Mol weight (kg mol-1) 28.0134g/mol

Temp/C 23.00



mm 1913.0 3039.0 2601.0

Piston position_1 373.00 397.00 400.00


Piston position_3 185.00 280.00 263.00 3.00

Piston position_4 99.00 220.00 196.00




Gas 3 ArMol weight (kg mol-1) 39.948g/mol

Temp/C 23.00



mm 1914.0 2534.0 2923.0

Piston position_1 421.00 436.00 433.00


Piston position_3 250.00 310.00 323.00 1.23

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



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miaomiao9456.files.wordpress.com file · Web viewTo estimate the C P /C V ratio of three air molecules, Nitrogen, Carbon dioxide and Argon, the sound –velocity method was used