How to do this partial derivative?
The pressure, volume, and temperature of an ideal gas are related by the ideal gas law
PV=kT,
where k is a positive constant. Any two of the variables (P, V, or T) may be considered independent in this equation, leaving the third defined implicitly as a function of the other two.
What is ∂ P/∂ V=?
I got the answer -(kT)/V^2, but that is apparently wrong. I've tried multiple times and I still get the same answer. Can someone show me the right way to do this problem?
Thank you!!
Answers & Comments
Verified answer
PV = kT
k and T are constant soooo
d/dV (PV) = d/dV (kT)
the left is a chainrule and kT is a constant so the right is 0
dP/dV (V) +(P)(1) = 0
dP/dV (V) = -P
dP/dV = -P/V
The important question in problems of this kind is What is being held constant? You get very different answers. For example, if the problem says that the process is isothermal that means T is being held constant. In that case your answer is correct. Of course you can then solve for V^2 in terms of P and T to get:
∂P/∂V = -P/V
BUT if the process is adiabatic (no heat added or lost) then you have PV^γ = Const.
In that case ∂P/&partV = -γCV^(-γ-1). Work it all out and you have:
∂P/∂V = -γP/V
THE ANSWER IS -P/V, I'm not sure why, I just copied my roommate's homework.