Since entropy is about how energy is shared, it is important to be able to identify places where energy can be put, or degrees o
f 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 o' 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 mode 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.)
1. Since each molecule has three kinetic degrees of freedom (can move in three independent directions), the gas must have 3N DoFs.
2. Each molecule has the three kinetic degrees of freedom the monotonic atom has moving without rotating but it can also spin. There are three axes for it to spin around so we would expect three rotational degrees of freedom, but as were as above, the one about the diatomic molecule's axis doesn't count because of quantum. So we have two rotational DoFs and three kinetic, for a total of 5 per molecules. So the gas will have 5N DoFs.
3.When a spring vibrates it has two DoFs, its KE and its PE, so adding 1 vibration adds 2 DoFs per molecule, giving 7 per molecule and giving thegas 7N DoFs.
Since the four metallic objects are identical, and total charge must be conserved, this means that after brought simultaneously into contact so that each touches the others, once separated, total charge must be the same than before being brought in contact.
But due they are identical, after charges were able to transfer freely between them, the four objects must have the same final charge, i.e. the fourth part of the total charge, as follows:
b)
This charge will be divided between n protons, since the charge is positive.
Since each proton carries a charge equal to the elementary charge e, which value is 1.6*10⁻¹⁹ C, we can find the number of protons in excess, doing the following calculation:
