- #1

cwill53

- 220

- 40

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

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

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.