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

AI Thread Summary
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
Messages
220
Reaction score
40
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.
 
Science news on Phys.org
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.
 
I need to calculate the amount of water condensed from a DX cooling coil per hour given the size of the expansion coil (the total condensing surface area), the incoming air temperature, the amount of air flow from the fan, the BTU capacity of the compressor and the incoming air humidity. There are lots of condenser calculators around but they all need the air flow and incoming and outgoing humidity and then give a total volume of condensed water but I need more than that. The size of the...
Thread 'Why work is PdV and not (P+dP)dV in an isothermal process?'
Let's say we have a cylinder of volume V1 with a frictionless movable piston and some gas trapped inside with pressure P1 and temperature T1. On top of the piston lay some small pebbles that add weight and essentially create the pressure P1. Also the system is inside a reservoir of water that keeps its temperature constant at T1. The system is in equilibrium at V1, P1, T1. Now let's say i put another very small pebble on top of the piston (0,00001kg) and after some seconds the system...
Back
Top