# Confused about ideal gases, and internal energy

1. There are some assumptions we make about ideal gases, one of them on the lack of intermolecular attractive forces. Why ideal gases can have interatomic forces of attraction but not intermolecular forces of attraction? (An example of presence of interatomic forces would be in diatomic or triatomic ideal gases)

2. "Internal energy of a system is the sum of microscopic random kinetic energies and microscopic potential energies of the molecules of the system."

Since diatomic ideal gases have microscopic potential energies, which are the interatomic forces of attraction, their internal energy is not due to KE alone and so the following formula cannot be applied to diatomic ideal gas. I would like to confirm if the following formula can only be applied for monoatomic ideal gases? Any other possible applications of the following formula? (Apart from monoatomic ideal gas)

Total internal energy, $$U = (3/2) nRT = (3/2) NkT = (3/2) pV~?$$

1. Total kinetic energy = 3/2 nRT = 3/2 NkT = 3/2 pV. Since the above formula was derived from solving the following equations simultaneously: pV = nRT -----(1) pV = 1/3 Nm<c^2> -----(2) And equation (1) is derived from Boyles Law, Charles Law and Pressure Law, while equation (2) is in short, derived by making use of momentum of molecules colliding, both (1) and (2) seems to be able to be applied to diatomic gases? (Since diatomic molecules move together and we can just treat it as a single particle?)

Thus, I would like to ask if unlike for total internal energy, the equation 3/2 nRT = 3/2 NkT = 3/2 pV can be used to find total kinetic energy regardless of whether the gas is monoatomic?

Why ideal gases can have interatomic forces of attraction but not intermolecular forces of attraction?

Let's think of a gas with just two particles. The energy of the gas has the following contributions:

1. Kinetic energy of particle 1
2. Kinetic energy of particle 2
3. Potential energy between particles 1 and 2
4. Energy of the constituent elements of particle 1: for a diatomic particle this is rotational and vibrational energy. Vibrational energy is due the potential interactions between the two atoms in the molecule.
5. Energy of the constituent elements of particle 1: Same as for particle 1

Of all these energies only #3 depends on the distance between particles. When this distance is very large, this energy goes to zero. In this limit we reach the Ideal Gas State. All other energies are still present. #1 and #2 are always present, #4 and #5 may or may not be present depending on the inner structure of the molecules.

I would like to ask if unlike for total internal energy, the equation 3/2 nRT = 3/2 NkT = 3/2 pV can be used to find total kinetic energy regardless of whether the gas is monoatomic?

The correct formula is $$E = \frac{f}{2} N K T,$$ where $$f$$ is the number of degrees of freedom. For a monoatomic gas $$f=3$$, one degree of translational motion in each of the three directions in space. Then we get $$E = \frac{3}{2} N K T,$$ In the case of a monoatomic gas, energies #4 and #5 are zero.

For a diatomic molecule $$f=4$$ (Wikipedia) and in this case $$E = \frac{7}{2} N K T .$$ The heat capacity of diatomic molecule is higher because it can store energy in additional places, namely #4 and #5.