Maxwell’s Relations and Differentiating With Respect to ##N_i##

[cwill53]

I was wondering if anyone could write out Maxwell's relations for partial derivatives with respect to particle count $N_i$. Starting from the fundamental thermodynamic relation,

$$dU(S,V,N_i)=TdS-PdV+\sum_{i}\mu _idN_i$$

$$dU(S,V,N_i)=\left ( \frac{\partial U}{\partial S} \right )_{V,\left \{ N_i \right \}}dS+\left ( \frac{\partial U}{\partial V} \right )_{S,\left \{ N_i \right \}}dV+\sum_{i}\left ( \frac{\partial U}{\partial N_i} \right )_{S,V,\left \{ N_{j\neq i } \right \}}dN_i$$

I tried to write a relation by differentiating with respect to particle number, but I want to make sure all of my subscripts are correct. I wrote, for a first Maxwell relation,

$$\left ( \frac{\partial T}{\partial N_i} \right )_{S,V,\left \{ N_{j\neq i} \right \}}=\left ( \frac{\partial \mu _i}{\partial S} \right )_{V,\left \{ N_{j\neq i} \right \}}=\left ( \frac{\partial ^2U}{\partial N_i \partial S} \right )_{V,\left \{ N_{j\neq i} \right \}}$$

Are these subscripts correct? I just want to make sure this is accurate. If someone could write out the remaining Maxwell relations, that would be great. There should be seven more relations left.

 

[vanhees71]

You just take $U$ as a function of it's "natural variables", $S$, $V$, and $N_i$ and use that the partial derivatives commute. E.g.,
$$\partial_{N_i} \partial_{S} U=\partial_S \partial_{N_i} U.$$
Now you use for the left-hand side
$$\partial_{S} U=T.$$
Then you take the derivative wrt. $N_i$, and since you apply it to $U$ as a function of the given specific independent quantities this means you have to hold $S$, $V$, and $N_j$ (for $j \neq i$) constant. The same argument holds for the right-hand side. So the correct relation reads
$$\left (\frac{\partial T}{\partial N_i} \right)_{S,V,\{N_j \}_{j \neq i}} = \left (\frac{\partial \mu_i}{\partial S} \right)_{V,\{N_j \}_{\text{all}\, j}}.$$
You have to be carefull with the 2nd derivative of $U$. Writing it in the form where you specify explicitly which variables to hold constant, it becomes a bit cumbersome:
$$\partial_{N_i} \partial_S U \equiv \left [\frac{\partial}{\partial N_i} \left (\frac{\partial U}{\partial S} \right)_{V, \{N_j \}_{\text{all}\,j}} \right]_{V,S,\{N_j \}_{j \neq i}}.$$

 

[cwill53]

You just take $U$ as a function of it's "natural variables", $S$, $V$, and $N_i$ and use that the partial derivatives commute. E.g.,
$$\partial_{N_i} \partial_{S} U=\partial_S \partial_{N_i} U.$$
Now you use for the left-hand side
$$\partial_{S} U=T.$$
Then you take the derivative wrt. $N_i$, and since you apply it to $U$ as a function of the given specific independent quantities this means you have to hold $S$, $V$, and $N_j$ (for $j \neq i$) constant. The same argument holds for the right-hand side. So the correct relation reads
$$\left (\frac{\partial T}{\partial N_i} \right)_{S,V,\{N_j \}_{j \neq i}} = \left (\frac{\partial \mu_i}{\partial S} \right)_{V,\{N_j \}_{\text{all}\, j}}.$$
You have to be carefull with the 2nd derivative of $U$. Writing it in the form where you specify explicitly which variables to hold constant, it becomes a bit cumbersome:
$$\partial_{N_i} \partial_S U \equiv \left [\frac{\partial}{\partial N_i} \left (\frac{\partial U}{\partial S} \right)_{V, \{N_j \}_{\text{all}\,j}} \right]_{V,S,\{N_j \}_{j \neq i}}.$$

Thanks a lot for this verification here. I see where my mistake was, and I’ll use this to derive the other relations.