- #1
Hiero
- 322
- 68
- Homework Statement
- Consider a one component system with the only generalized force being pressure
- Relevant Equations
- Gibbs-Duhem relation
##SdT-Vdp+nd\mu = 0##
##v = V/n## (volume per mole)
I was just wondering what is wrong with the following logic;
From the Gibbs-Duhem relation we get,
##\frac{\partial \mu}{\partial P}\Big\rvert_T = v##
Now consider,
##\frac{\partial v}{\partial \mu}\Big\rvert_T = \frac{\partial }{\partial \mu}\Big (\frac{\partial \mu}{\partial P}\Big\rvert_T \Big )\Big\rvert_T= \frac{\partial }{\partial P}\Big (\frac{\partial \mu}{\partial \mu}\Big\rvert_T \Big )\Big\rvert_T= \frac{\partial }{\partial P}(1)\Big\rvert_T=0##
I’m pretty sure the result is not true, that it is not generally zero, so which step is flawed?
From the Gibbs-Duhem relation we get,
##\frac{\partial \mu}{\partial P}\Big\rvert_T = v##
Now consider,
##\frac{\partial v}{\partial \mu}\Big\rvert_T = \frac{\partial }{\partial \mu}\Big (\frac{\partial \mu}{\partial P}\Big\rvert_T \Big )\Big\rvert_T= \frac{\partial }{\partial P}\Big (\frac{\partial \mu}{\partial \mu}\Big\rvert_T \Big )\Big\rvert_T= \frac{\partial }{\partial P}(1)\Big\rvert_T=0##
I’m pretty sure the result is not true, that it is not generally zero, so which step is flawed?