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# Counting degrees of freedom

last edited by 7 years, 5 months ago

7.2.3.P3

Prerequisite:

Since entropy is about how energy is shared, it is important to be able to identify places where energy can be put, or degrees of freedom. For example, a monoatomic atom moving in three dimensions in free space has three places to put energy: kinetic energy for motion in three independent directions, so it has three degrees of freedom. (Because of quantum mechanics, a perfectly spherical atom cannot have any energy of rotation.) A rigid diatomic molecule can have three kinetic energy degrees of freedom and two rotational degrees of freedom since it can rotate about two different axes. (Rotation about the axis through the center line of the molecule doesn't have any rotational energy, again because of quantum mechanics and the fact that it doesn't change when it makes that rotation.) Both kinetic and potential energies can serve as degrees of freedom. To see more about these issues, see the reading Example: Degrees of freedom. For this problem assume that you can ignore gravity.

A. Consider a gas consisting of a box of N argon atoms. How many degrees of freedom do you expect the gas to have?

B. Consider a gas consisting of a box of N carbon-monoxide molecules. Assuming that you can treat the bond as rigid, how many degrees of freedom do you expect the gas to have?

C. If you consider a gas consisting of a box of N carbon-monoxide molecules. Assuming that you can treat the bond is a spring that can only vibrate along the axis joining the two atoms (and not sideways), how many degrees of freedom do you expect the gas to have?

D. Consider a gas consisting of water vapor with N molecules having three atoms rigidly bonded in an angled shape (not a straight line). How many degrees of freedom do you expect this gas to have?

E. On average, when thermal energy is added to a substance, the energy is shared equally among the degrees of freedom. Since the temperature of a substance is proportional to the average energy in a degree of freedom, which of the substances do you expect would have the largest specific heat? (That is, which substance would change temperature the least in response to the addition of a given amount of thermal energy.) Explain.

Joe Redish 2/3/17