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

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Maxwell's relations for partial derivatives with respect to particle count \(N_i\) are derived from the fundamental thermodynamic relation, \(dU(S,V,N_i)=TdS-PdV+\sum_{i}\mu_idN_i\). The discussion emphasizes the importance of correctly applying the chain rule and ensuring subscripts are accurate when differentiating. A key relation presented is \(\left (\frac{\partial T}{\partial N_i} \right)_{S,V,\{N_j \}_{j \neq i}} = \left (\frac{\partial \mu_i}{\partial S} \right)_{V,\{N_j \}_{\text{all}\, j}}\). The conversation also highlights the need for caution when dealing with second derivatives of \(U\) and the complexity of specifying which variables to hold constant. This clarification aids in accurately deriving the remaining Maxwell relations.
cwill53
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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.
 
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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:
$$\partial_{N_i} \partial_S U \equiv \left [\frac{\partial}{\partial N_i} \left (\frac{\partial U}{\partial S} \right)_{V, \{N_j \}_{\text{all}\,j}} \right]_{V,S,\{N_j \}_{j \neq i}}.$$
 
vanhees71 said:
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.
 
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