Statistical definition of internal energy

Question

Hello!

I read that internal energy is an extensive property, but that doesn't make sense with the statistical definition of it.

Statistically, U = SUMMATION: (P_i)(E_i). However, that only gives the average energy level the particles are occupying, not the average of the total energy of all the particles.

For example, suppose you have 3 particles all occupying the same energy level E_5. Then U = E_5 (since P_i=0 for all other E_i). Suppose you have 10 particles instead, then U = E_5 too!

I don't understand how it can be an extensive property with that statistical definition.

Answer 1

it is the average energy <U>, namely U total / N. if the total # particles N = 3, then U total = 3 <U>. if the total # particles N = 10, then U total = 10 <U>. you can see from this that U is indeed an extensive property