When you compress a gas, is the increase in temperature proportional to the energy it took to compress the gas divided by the product of the number of particles in the gas and the Boltzmann constant?

Here is my reasoning. The temperature of a gas is equal to the energy per particle of that gas. To get the joules per particle of a gas, you multiply the temperature by the Boltzmann constant. To find the total energy of a gas, you multiply that value by the number of particles in the gas. So it seems to me that it follows that to find the increase in temperature of a gas after compression, you divide the energy it took to compress the gas by the product of the number of particles in the gas and the Boltzmann constant.

I think I'm missing something, because different substances have different thermal capacities.