Thermodynamics deals with the macroscopic properties of materials. Scientists ca

Question

Thermodynamics deals with the macroscopic properties of materials. Scientists can make quantitative predictions about these macroscopic properties by thinking on a microscopic scale. Kinetic theory and statistical mechanics provide a way to relate molecular models to thermodynamics. Predicting the heat capacities of gases at a constant volume from the number of degrees of freedom of a gas molecule is one example of the predictive power of molecular models.
The molar specific heat of a gas at a constant volume is the quantity of energy required to raise the temperature of one mole of gas by one degree while the volume remains the same. Mathematically,
,
where n is the number of moles of gas, E_int is the change in internal energy, and T is the change in temperature.
Kinetic theory tells us that the temperature of a gas is directly proportional to the total kinetic energy of the molecules in the gas. The equipartition theorem says that each degree of freedom of a molecule has an average kinetic energy equal to (1/2)k_bT , where k_b =1.38 X10^(-23) J/Kis Boltzmann's constant . When summed over the entire gas, this gives 0.5nRT , where R= 8.314 J/(mol K) is the ideal gas constant, for each molecular degree of freedom.
Using the equipartition theorem, determine the molar specific heat, of a gas in which each molecule has "s" degrees of freedom.
Express your answer in terms of R and s.
=

..........

Part B
Given the molar specific heat of a gas at constant volume, you can determine the number of degrees of freedom "s" that are energetically accessible.
For example, at room temperature cis-2-butene, , has molar specific heat of 70.6(J/molK). How many degrees of freedom of cis-2-butene are energetically accessible?
Express your answer numerically to the nearest integer.
=

Explanation / Answer

C_v=(s/2)R

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