First law of thermodynamics and the TdS equation

[Chronos000]

Homework Statement

dU = dQ + dW

so dU = Cv dT... but only if dW is zero right? as dQ = CvdT

so then the central equation is

TdS = dU + PdV

which is then TdS = CvdT + PdV according to my notes. surely PdV should be zero no?

 

[L-x]

What is pdV at constant volume?

 

[Chronos000]

it is zero right?

 

[L-x]

yes, but why?

 

[Chronos000]

dV means change of volume, no change, dV = 0

 

[Chronos000]

my notes have further manipulated this term though, which does't seem right if its zero

 

[L-x]

So pdV is 0 if volume is held constant, does that mean it is always 0?

remember we may write any function of state as an exact differential, for example

$$dU = (\frac{\partial{U}}{\partial{S}})_{V}dS + (\frac{\partial{U}}{\partial{V}})_{s}dV$$

obviously at constant S one of these terms is 0, and at constant V the other is 0, but for a general change in U we need both