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=(1/n)(du/dt)

where n is the number of moles of gas, du is the change in internal energy, and dt 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.

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.

cv =

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?