Lab Report 6 Heat capacity of gases PDF

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I passed the class with a B+. Lab reports worth about 80% of the class grade. I even got a D+ on the final, but still passed with a B+ because I got good grades on the lab reports....

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Expt #6: Heat Capacity of Gases Yiyeon Kim Partners: Mark Kuhlman, Gabriela Lopez Section 22 TA: Fan Yang

1. Introduction: A. Purpose - The objective of this experiment is to determine the heat capacity ratio of monatomic, diatomic, and triatomic gas molecules. The heat at constant pressure to constant volume will be examined through the method of an adiabatic expansion. B. Procedure - The carboy was initially stoppered and the a and c valves were left open. Gas flushed the carboy for fifteen minutes. Slowing the gas flow on valve a, valve c was opened. Closed valve c and then closed valve a when the appropriate pressure was attained. The gas was allowed to reach an equilibrium temperature and the pressure was recorded. The stopper was lifted and replaced on the carboy swiftly. The gas warmed back up in fifteen minutes and the pressure was recorded. At another point, the pressure and bath temperature reading were taken. These steps were repeated for the other gases. C. Discussion - Heat capacity of chemical is a property which expresses the relationship between the amount of heat or energy added to a substance and the temperature change. It shows that a chemical species with much higher specific heat will require a lot more energy in order to be raised up a single degree in temperature. One major contributing factor to heat capacity of as is degrees of freedom, which is the number of different coordinates and configurations a molecule can have. Bigger molecules that have more degrees of freedom can store more energy and have a higher heat capacity. The heat capacity can be theoretically predicted with the equipartition of energy theorem, which states that for each degree of freedom that an atom/molecule has, there is 1/2KT or 1/2RT per mole of energy. 2. Data and Calculations: A. Data

B. Calculating Heat Capacity Ratio The heat capacity for ideal gases: C P = C V + R Heat capacity ratio: γ =

CP CV

Equation for the reversible adiabatic expansion of an ideal gas: dE =− P dV =− nRT nRT 1 nRT = P2 P3 2 C P ′ V 2′ P1 l n P = C ′ln V 1 ′ V 2

With P 2 , V 2 , T 2 →P 2 , V 2 , T 1 , the new equation: V 2 =

=

Applying this to the heat capacity ratio, the equation:

=

CP ′ CV ′

Rearranged to the desired equation: γ =

=

dV V

V2 P = P1 V1 3 CP ′ P 1 C V ln ′ P3

P ln P 1 2 P ln P 1 3

C. Sample calculations ● Average heat capacity of the three trials for argon: Trial 1: 925.1 ln 740.7 925.1 ln 804.1

= 1.586

Trial 2: 937 ln 740.7 937 ln 805.4

= 1.553

Trial 3: 939.1 ln 740.7 939.1 ln 810.9

1.586+1.553+1.617 3

● Theoretical Heat Capacity Ratio: γ =

= 1.617 = 1.585

CP CV

=1+

R CV

3 degrees of translational freedom for argon, CV=3/2R: γ =

CP CV

=1+

R 3R 2

= 1 + 23 =

5 3

● Heat Capacity Ratio with and without Vibrational Degrees of Freedom for N2 : C With Vibrational: γ = C P = 1 + 7RR = 1 + 27 = 97 V

Without Vibrational: γ =

2

CP CV

=1+

R 5R 2

=1+

2 5

D. Tabulated result

3. Discussion: A. Literature Values and Result Comparison

=

7 5

The experimentally obtained heat capacity ratio values for Ar, N2, and CO2 were 1.585, 1.351, and 1.353, respectively. The theoretically calculated heat capacity ratio values for them are 1.667, 1.286, and 1.154, respectively. Values for the ratios found in literature in reference (4) are 1.66, 1.40, and 1.28, respectively. Comparing the experimental values with the literature values, it is clear that the corresponding values closely resemble one another. The percent error of the experimental values from the literature values yield 4.5% error for Ar, 3.5% error for N2, and 5.7% for CO2. Accordingly, from these percent errors, it is clear that the experimental values were quite accurate. Comparison of the experimental values with the calculated theoretical values proves some more deviation. This likely accounts for the vibrational and rotational degrees of freedom contributions to the heat capacity. It can be seen from the tabulated result above that for N2, the theoretical heat capacity ratios with and without vibrational contributions range from 1.286 to 1.4 and that the experimental value lies in between that range. One issue with the equipartition theorem is that it does not account for vibrational energy being highly quantized and dependent on the temperature. Vibrational contributions to heat capacity then varies depending on the temperature. This would explain the reason for the N2 and CO2 heat capacity ratios falling in between the theoretical ratios with and without vibrational contributions. They both have some active and inactive vibrational contributions resulting in those values. Overall, the vibration and error discussed below affects the value. B. Error Analysis Although the experimental values did closely match the literature value, there was still a slight deviation. Uncertainties that may have caused this may have been with the adiabatic expansion step in the procedure. Adiabatic expansions call for no loss or gain of heat but, realistically, it can be minimized but not completely halted. Thus, there was some heat conducted when the gas was allowed to adiabatically expand. Another error is seen with the CO2 heat capacity ratio values. The second trial value significantly differs from that of the first and third trial values. There may have been several reasons for this deviation but the major reason may be that the gas was allowed to adiabatically expand significantly longer than those in the other trials. For the comparison of the N2 and CO2 experimental values to the theoretical values, the differences are most likely due to the varying vibrational contributions but the experimental values were most likely shifted from the error mentioned earlier as well. The values were most likely affected by the adiabatic expansion. C. General Comments Overall, this experiment gave accurate results that matched those of literature and theory. The heat capacity ratios also correlate to the equipartition theorem. Some of the equipment used in the laboratory experiment may have caused mistakes and errors due to faultiness. The experimental values agree with theory. 4. Questions:

1. Estimate the effect on CP /C  V of a very high heat conductivity, as is the case for helium: The high heat conductivity would more drastically affect the temperature during the adiabatic expansion. Since temperature affects the other thermodynamic properties such as P

pressure and volume, they would change as well. Using γ =

CP ′ CV ′

=

ln P 1 2

P

ln P 1

, large P3 would 

3

make CP /C and large P2 would make CP /C V small.  V large   2. For CO2, how would the theoretical ratio be affected if the molecule were nonlinear (such as SO2) instead of linear? Could you decide between these two structures from the CP’/CV’ ratio alone?: The theoretical ratio would lose contribution from one vibrational degree of freedom. The total equipartition energy and ratio would be 11/2R instead of 13/2R. The ratios would be different so SO2 and CO2 could be distinguished from their ratio values. 5. References: 1. Y. Yin, “Supplementary Laboratory Manual.” 2. TA Fan Yang. 3. Garland and Nibler and Shoemaker, “Experiments in Physical Chemistry”, Ch. 4, Exp. 3, pp. 106-114. 4. http://www.engineeringtoolbox.com/specific-heat-ratio-d_608.html...