Heat Capacity Ratio of Gases

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Experiment 1
The Heat Capacity Ratio of Gases

Purpose
The purpose of this experiment is to calculate the heat capacity ratio of gases, Helium, Nitrogen and Carbon Dioxide, and compare with their theoretical values.

Introduction
Thermodynamics is the study of heat as it relates to energy and work. There are various properties which all relate to each other when determining the characteristic of a certain substance. One of such properties is heat capacity, which is the amount of heat energy required to raise the temperature of a substance by one degree Kelvin. Mathematically, it is ∁ = q∆T , where q is the amount of heat absorbed by a substance and ΔT is the change in temperature measured. When substances absorb heat, their molecules translate, rotate and vibrate due to the rise in temperature . As a result of the motion of movement of molecules in these modes, there is a contribution of energy towards determining the heat capacity of that substance. The heat capacity is, however, defined through constant volume (Cv) or constant pressure (Cp) with a relationship, Cp = Cv + nR and CpCv , the heat capacity ratio for ideal gases which is further determined by obtaining the pressure difference with atomospheric pressure in adiabatic conditions. That is lnp1-lnp2lnp1-lnp3. The energy contribution through the modes of movement of molecules is the total of their, translational, rotational and vibrational energies. For ideal gases, this can be calculated theoretically as a result of their classes, Monatomic, Diatomic and Linear polyatomic. Monatomic gases such as Helium, move in translation with the energy 32RT. Diatomic gases such as Nitrogen, move in all 3 modes with the energy 72RT. And the linear polyatomic gases such as CO2 move with the energy 132RT. The constant volume heat capacity for these ideal gases can be determined as a result of its relationship with these energies as the energy U = nRT and Cv is the derivative with respect to volume. i.e Cv = ∂U∂Tv . This leads to the following Cv for the 3 classes of gases; 12.5 Jmol*K for monatomic, 29.1 Jmol*K for diatomic, and 54.0 Jmol*K for linear polyatomic.

Data
Room Temperature = 16.2 oC ± 0.1 oC
p2 = Room Pressure
Room Pressure = 760.84 mmHg ± 0.22 mmHg
Helium
Trial| P1 (mmHg)(±0.3)| P3 (mmHg)(±0.3)|
1| 300.4| 75.6|
2| 275.7| 69.0|
3| 281.9| 74.8|

Carbon Dioxide
Trial| P1 (mmHg)(±0.3)| P3 (mmHg)(±0.3)|
1| 290.3| 34.1|
2| 277.8| 25.3|
3| 283.1| 40.1|

The values for Helium and Carbon dioxide were gotten from the other group who performed the experiment. Nitrogen
Trial| P1 (mmHg)(±0.3)| P3 (mmHg)(±0.3)|
1| 278.7| 63.7|
2| 286.6| 89.7|
3| 270.5| 58.9|
4| 294.2| 85.0|
5| 285.5| 89.7|
6| 291.4| 70.0|
7| 268.1| 54.1|
8| 289.0| 64.8|
9| 281.5| 65.8|
10| 265.3| 59.7|

Values in bold are the 3 best trial obtained.

Answers to Questions
1) C, mathematical defined as C = qΔT , is the heat capacity, the amount of energy required to raise the temperature of a substance by one degree Kelvin. Cv, is the heat capacity per unit volume while, Cp , is the heat capacity per unit pressure. Both are related mathematically by the equation Cp = Cv + nR. The expected heat capacity for the three classes of gases are as follows Monatomic = 3R2=12.5 Jmol.K

Diatomic = 7R2=29.1 Jmol.K
Linear triatomic = 13R2=54.0 Jmol.K
The equations leading to the heat capacity ratio, γ, is summarized by CpCv= ln(p1p2)ln⁡(p1p3)
The vibrational contribution to Cv can be determined once the vibrational frequencies of the molecule is known. That is Rx2e-x where is x>5 . x = (NA hRT)v Where NA = Avogadro's number, h = Planck's constant and v = vibration frequency.

2) Data obtained in the experiment is presented in the data section above. 3) Sample error calculation
2300.42*0.32+760.842*0.222 =817.9962127
2817.99621271061.242+0.22760.842 = 0.770793...
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