Isentropic Process, General Results for dU and dH

[Kushwoho44]

TL;DR

I'm having trouble understanding intuitively the relation between the LHS and RHS of
dU = n*c_v *dT = -pdV.

Hello forumites,

I've been working with the following expression for the change in internal energy in an isentropic scenario.
$$dU = n*c_v *dT = -pdV$$

However, I'm a bit stumped here, the left hand side of the expression (or middle rather), states the change in internal energy is the product of the specific heat for constant volume and temperature, but this is equal to the work done on the system, which is the product of pressure and the volume differential.

This is confusing to me. We first invoke a constant-volume argument and then on the right hand side, state that it's equal to an expression dependent on a change in volume.

Any help would as always be appreciated.

 

[Chestermiller]

For an ideal gas, internal energy is a function only of temperature, and is independent of pressure and volume. We use the heat capacity at constant volume, because this parameter is defined precisely in terms of the partial derivative of internal energy with respect to temperature at constant volume:
$$c_v\equiv \frac{1}{n}\left(\frac{\partial U}{\partial T}\right)_v$$
For an ideal gas, this reduces to:$$c_v= \frac{1}{n}\frac{d U}{dT}$$