Thermodynamics deals with the macroscopic properties ofmaterials. Scientists can make quantitative predictions about thesemacroscopic properties by thinking on a microscopic scale. Kinetictheory and statistical mechanics provide a way to relate molecularmodels to thermodynamics. Predicting the heat capacities of gasesat a constant volume from the number of degrees of freedom of a gasmolecule is one example of the predictive power of molecularmodels.
The molar specific heat C_v of a gas at a constant volume is thequantity of energy required to raise the temperature T of one mole of gas by one degree while thevolume remains the same. Mathematically,
C_{\rm v}= \frac{1}{n}\,\frac{dU}{dT},
where n is the number of moles of gas, dU is the change in internal energy, anddT is the change in temperature.
Kinetic theory tells us that the temperature of a gas isdirectly proportional to the total kinetic energy of the moleculesin the gas. The equipartition theorem says that each degree offreedom of a molecule has an average kinetic energy equal to\frac{1}{2}k_{\rm B}T, wherek_B is Boltzmann's constant1.38 \times 10^{-23} \rm {J/K}. Whensummed over the entire gas, this gives \frac{1}{2}nRT, where R=8.314\;{\rm \frac{J}{mol\cdot K}}is the ideal gas constant, for each molecular degree offreedom.
Part A
Using the equipartition theorem, determine the molar specific heat,C_v, of a gas in which eachmolecule has s degrees of freedom.
Express your answer in terms ofR and s.
Part B
Given the molar specific heat C_v of a gas at constant volume, you candetermine the number of degrees of freedom s that are energetically accessible.
For example, at room temperature cis-2-butene,\rm C_4 H_8, has molar specific heatC_v=70.6\;{\rm \frac{J}{mol \cdot K}}. How many degrees of freedom of cis-2-butene areenergetically accessible?
Express your answer numerically tothe nearest integer.