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 Cv of a gas at a constant volume is the quantity of energy required to raise the temperature T of one mole of gas by one degree while the volume remains the same. Mathematically, Cv=1nΔEthΔT, where n is the number of moles of gas, ΔEth is the change in internal (or thermal) energy, and ΔT 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 energy equal to 12kBT, where kB is Boltzmann's constant 1.38×10^−23J/K. When summed over the entire gas, this gives 12nRT, where R=8.314Jmol⋅K is the ideal gas constant, for each molecular degree of freedom. Required:
a. Using the equipartition theorem, determine the molar specific heat, Cv , of a gas in which each molecule has s degrees of freedom. Express your answer in terms of R and s.
b. Given the molar specific heat Cv 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, C4H8 , has molar specific heat Cv=70.6Jmol⋅K . How many degrees of freedom of cis-2-butene are energetically accessible?