What Are the States of V and p in an Adiabatic Process?

[Niles]

Homework Statement

I have the equation for the internal energy:

U = (f/2) * N * k * T, where f is the degrees of freedom, N is the number of molecules, k is Bolzmann's constant and T is the temperature in Kelvin.

This can be written as U = (f/2)*p*V using the ideal gas law. Differentiating this I get:

delta U = (f/2)*(delta_p*V + p*delta_V).

In this equation, I know what delta_p and delta_V are, but what about V and p? Are they the initial or final states?

Thanks in advance.

 

[Mapes]

I think you mean not delta but dU=(f/2)(Vdp+pdV). The values are those of the immediate states, and can vary depending on what you're doing to the gas. If you wanted to integrate dU to find the change in energy during a process, you would need to keep p and V inside the integrand unless they were constant, and this could make the integration complicated.

In practice, then, you wouldn't express dU this way unless p or V were constant (or a known function of other variables) during the process. For example, if you heat a gas while expanding from V1 to V2 in an isobaric process, then p is constant, dp=0, and U=(f/2)p(V2-V1).