Partial Derivative Q: Is dV an Exact Differential?

[pchem1]

In determining if a function is exact, here is the question. If V=V(T,P) and PV+RT, show that dV = R/PdT - RT/P2 dP. Is dV an exact differential?
Do I go about by taking the derivative of R/PdT with respect to T, etc? I know this is not a difficult function, but I just want to make sure I'm approaching it correctly.

Thanks!

 

[HallsofIvy]

In determining if a function is exact, here is the question. If V=V(T,P) and PV+RT, show that dV = R/PdT - RT/P2 dP. Is dV an exact differential?
Do I go about by taking the derivative of R/PdT with respect to T, etc? I know this is not a difficult function, but I just want to make sure I'm approaching it correctly.

Thanks!

There are several things aobut your post that I do not understand.

I know what it means for a differential to be exact, but don't know what it means for a function to be exact. Are you talking about a differential?

Also you say "if V= V(T,P) and PV+RT". Okay, V= V(T,P) is a complete statement- it says that V is function of T and P. But "PV+ RT" is not a complete statement. What about PV+ RT? Is something missing? Was it supposed to be PV+ RT= something?

Finally, you talk about "taking the derivative of R/PdT with respect to T" but that already is a differential, Don't you mean to diffentiate an equation involving V?

If PV+ RT= some function, then P dV+ VdP+ dRT+ RdT= the derivative of that function. In particular, if PV+ RT= constant, and R is held constant, then PdV+ VdP+ RdT= 0 so that PdV= RdT- VdP and then
dV= RdT/P- (V/P)dP. Now that would be what you give IF V/P= RT/P² (I assume that your P2 was intended to be P²) or
V= RT/P.

Now, although you didn't say anything about it, I remember that for an "ideal gas", PV= nRT where R is a constant. If THAT is what you are talking about, then, yes, dV = (R/P)dT - (RT/P²)dP.