# Internal energy of a Real Gas

1. Oct 13, 2013

### themanonthemo

Hello everybody.

The concept of internal energy of a real vs ideal gas has perplexed me.
From what I understand, an ideal gas solely considers the kinetic energy of gas molecules (temperature) where as real gases consider kinetic energy of particles in addition to potential energy.

So logically this would be true..

ideal: U = 1.5nRT (for a monatomic gas) real: U = 1.5nRT + PE (for a monatomic gas)
from what I read from the textbook, PE is a function of Pressure or volume.

Next, the idea of heat capacities seems strange. It is said that
dQ = (n)(Cv)(dT) where Cv is the heat capacity at constant volume.

For an ideal gas at const. volume or NOT: it is logical that Cv would be 1.5R (assuming a monatomic gas)

Question: For a real gas at const volume, the formula dU = nCvdT apparently still applies, although Cv is not idealized like in the previous example.

dU = dKE + dPE (real gas) nCvdT = n(1.5R)dT + dPE dPE = ndT(Cv - 1.5R)

This suggests that PE is only a function of temperature which contradicts the idea that it is a function of pressure or volume.

Is there something wrong with my assumptions?

I'm sorry if it's not clear. I appreciate all the help!

2. Oct 14, 2013

### Staff: Mentor

If V is constant, isn't here a relationship between T and P? Also, the value of $C_V$ can be different depending on which range of pressure you are considering.

The potential energy in a real gas basically depends on the average distance between molecules, and not in a linear fashion. Therefore, any things that changes this distance will affect the internal energy of a gas, and that is the reason why you normally do not get a simple relation between the internal energy and the temperature alone.