- #1
voila
- 59
- 6
Hi all.
Suppose I have the ideal gas law $$P=\frac{RT}{v}$$If I'm asked about the partial derivative of P with respect to molar energy ##u##, I may think "derivative of P keeping other quantities (whatever those are) constant", so from the formula above I get $$\frac{\partial P}{\partial u}=0$$However, one of the consequences of the ideal gas law is that energy is only dependant on T as in ##u=cRT##. If I reintroduce this in the ideal gas law, I get $$P=\frac{u}{cv}$$ What if I know want to take the partial derivative of P with respect to u? Then $$\frac{\partial P}{\partial u}=\frac{1}{cv}\neq 0$$which doesn't match the result above.
What am I doing wrong? I've tried to show my general problem using this particular example, but you may guess this kind of confusion causes problems everywhere in thermodynamics.
Suppose I have the ideal gas law $$P=\frac{RT}{v}$$If I'm asked about the partial derivative of P with respect to molar energy ##u##, I may think "derivative of P keeping other quantities (whatever those are) constant", so from the formula above I get $$\frac{\partial P}{\partial u}=0$$However, one of the consequences of the ideal gas law is that energy is only dependant on T as in ##u=cRT##. If I reintroduce this in the ideal gas law, I get $$P=\frac{u}{cv}$$ What if I know want to take the partial derivative of P with respect to u? Then $$\frac{\partial P}{\partial u}=\frac{1}{cv}\neq 0$$which doesn't match the result above.
What am I doing wrong? I've tried to show my general problem using this particular example, but you may guess this kind of confusion causes problems everywhere in thermodynamics.