# Maxwell's Relations and Differentiating With Respect to Nᵢ

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,

\begin{align} \mathrm{d}U(S, V, N_i) &= T\mathrm{d}S - P\mathrm{d}V + \sum_{i}\mu _i\mathrm{d}N_i\\ \mathrm{d}U(S, V, N_i) &= \left(\frac{\partial U}{\partial S} \right)_{V, \{N_i\}}\mathrm{d}S + \left(\frac{\partial U}{\partial V} \right)_{S, \{N_i\}}\mathrm{d}V + \sum_{i}\left(\frac{\partial U}{\partial N_i} \right)_{S, V, \{ N_{j\neq i}\}}\mathrm{d}N_i \end{align}

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,

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

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.

• It seems to me what you did is correct. Oct 29, 2021 at 18:26
• @ChetMiller Should the middle partial derivative have all $N_i$ held constant or just $N_{j\neq i}$? Oct 29, 2021 at 19:27
• You're right. I missed that. It should be all of them. Oct 29, 2021 at 20:03
• @CalebWilliamsUC, if you got your answer, feel free to self answer it. Self answer are always appreciated and it also help future readers. Oct 30, 2021 at 1:00
• @NilayGhosh Alright! I will self-answer a bit later. Oct 30, 2021 at 3:37

Just now getting back around to this.

\begin{align} \mathrm{d}U(S, V, \left \{ N_i \right \}) &= T\mathrm{d}S - P\mathrm{d}V + \sum_{i}\mu _i\mathrm{d}N_i\\ \mathrm{d}U(S, V, \left \{ N_i \right \}) &= \left(\frac{\partial U}{\partial S} \right)_{V, \{N_i\}}\mathrm{d}S + \left(\frac{\partial U}{\partial V} \right)_{S, \{N_i\}}\mathrm{d}V + \sum_{i}\left(\frac{\partial U}{\partial N_i} \right)_{S, V, \{ N_{j\neq i}\}}\mathrm{d}N_i \end{align} $$\left(\frac{\partial T}{\partial V} \right)_{S, \{N_{i}\}} = -\left(\frac{\partial P}{\partial S} \right)_{V, \{N_{i}\}} = \left(\frac{\partial ^2U}{\partial S \partial V} \right)_{ \{N_{ i}\}}$$

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

\begin{align} \mathrm{d}H(S, P, \left \{ N_i \right \}) &= T\mathrm{d}S + V\mathrm{d}P + \sum_{i}\mu _i\mathrm{d}N_i\\ \mathrm{d}H(S, P, \left \{ N_i \right \}) &= \left(\frac{\partial H}{\partial S} \right)_{P, \{N_i\}}\mathrm{d}S + \left(\frac{\partial H}{\partial P} \right)_{S, \{N_i\}}\mathrm{d}P + \sum_{i}\left(\frac{\partial H}{\partial N_i} \right)_{S, P, \{ N_{j\neq i}\}}\mathrm{d}N_i \end{align} $$\left(\frac{\partial T}{\partial P} \right)_{S, \{N_{i}\}} = -\left(\frac{\partial V}{\partial S} \right)_{P, \{N_{i}\}} = \left(\frac{\partial ^2H}{\partial P \partial S} \right)_{ \{N_{ i}\}}$$

$$\left(\frac{\partial T}{\partial N_i} \right)_{S, P, \{N_{j\neq i}\}} = \left(\frac{\partial \mu _i}{\partial S} \right)_{P, \{N_{i}\}} = \left(\frac{\partial ^2H}{\partial N_i \partial S} \right)_{P, \{N_{j\neq i}\}}$$ $$\left(\frac{\partial V}{\partial N_i} \right)_{S, P, \{N_{j\neq i}\}} = \left(\frac{\partial \mu _i}{\partial P} \right)_{S, \{N_{i}\}} = \left(\frac{\partial ^2H}{\partial N_i \partial P} \right)_{S, \{N_{j\neq i}\}}$$ \begin{align} \mathrm{d}F(T, V, \left \{ N_i \right \}) &= -S\mathrm{d}T - P\mathrm{d}V + \sum_{i}\mu _i\mathrm{d}N_i\\ \mathrm{d}F(T, V, \left \{ N_i \right \}) &= \left(\frac{\partial F}{\partial T} \right)_{V, \{N_i\}}\mathrm{d}T + \left(\frac{\partial F}{\partial V} \right)_{T, \{N_i\}}\mathrm{d}V + \sum_{i}\left(\frac{\partial F}{\partial N_i} \right)_{T, V, \{ N_{j\neq i}\}}\mathrm{d}N_i \end{align} $$\left(\frac{\partial S}{\partial V} \right)_{T, \{N_{i}\}} = -\left(\frac{\partial P}{\partial T} \right)_{V, \{N_{i}\}} = \left(\frac{\partial ^2F}{\partial V \partial T} \right)_{ \{N_{ i}\}}$$

$$-\left(\frac{\partial S}{\partial N_i} \right)_{T, V, \{N_{j\neq i}\}} = \left(\frac{\partial \mu _i}{\partial T} \right)_{V, \{N_{i}\}} = \left(\frac{\partial ^2F}{\partial N_i \partial T} \right)_{V, \{N_{j\neq i}\}}$$ $$-\left(\frac{\partial P}{\partial N_i} \right)_{T, V, \{N_{j\neq i}\}} = \left(\frac{\partial \mu _i}{\partial V} \right)_{T, \{N_{i}\}} = \left(\frac{\partial ^2F}{\partial N_i \partial V} \right)_{T, \{N_{j\neq i}\}}$$ \begin{align} \mathrm{d}G(T, P, \left \{ N_i \right \}) &= -S\mathrm{d}T + V\mathrm{d}P + \sum_{i}\mu _i\mathrm{d}N_i\\ \mathrm{d}G(T, P, \left \{ N_i \right \}) &= \left(\frac{\partial G}{\partial T} \right)_{P, \{N_i\}}\mathrm{d}T + \left(\frac{\partial G}{\partial P} \right)_{T, \{N_i\}}\mathrm{d}P + \sum_{i}\left(\frac{\partial G}{\partial N_i} \right)_{T, P, \{ N_{j\neq i}\}}\mathrm{d}N_i \end{align} $$-\left(\frac{\partial S}{\partial P} \right)_{T, \{N_{i}\}} = \left(\frac{\partial P}{\partial T} \right)_{P, \{N_{i}\}} = \left(\frac{\partial ^2G}{\partial T \partial P} \right)_{ \{N_{ i}\}}$$

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