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

In summary, the relations are as follows:- partial derivatives with respect to particle number are given by:$$\left ( \frac{\partial T}{\partial N_i} \right )_{S,V,\{N_{j\neq i}}}\equiv \left ( \frac{\partial \mu_i}{\partial S} \right )_{V,\{N_{j\neq i}}}\left ( \frac{\partial ^2U}{\partial N_i \partial S} \right )_{V,\left \{ N_{j\neq i} \right \}}$$- the first Maxwell relation is given by:$$\left
  • #1
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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  • #2
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:
$$\partial_{N_i} \partial_S U \equiv \left [\frac{\partial}{\partial N_i} \left (\frac{\partial U}{\partial S} \right)_{V, \{N_j \}_{\text{all}\,j}} \right]_{V,S,\{N_j \}_{j \neq i}}.$$
 
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  • #3
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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Related to Maxwell’s Relations and Differentiating With Respect to ##N_i##

What are Maxwell's relations?

Maxwell's relations are a set of equations in thermodynamics that relate the partial derivatives of thermodynamic properties such as temperature, pressure, and volume. They are derived from the fundamental laws of thermodynamics and are used to simplify calculations and equations in thermodynamics.

What is the significance of differentiating with respect to ##N_i## in Maxwell's relations?

Maxwell's relations involve partial derivatives with respect to various thermodynamic properties, including ##N_i##, which represents the number of particles in a system. Differentiating with respect to ##N_i## allows for the calculation of other thermodynamic properties such as entropy, enthalpy, and chemical potential.

How are Maxwell's relations derived?

Maxwell's relations are derived from the first and second laws of thermodynamics, along with the relations between thermodynamic properties such as temperature, pressure, and volume. They can also be derived using the concept of thermodynamic potentials, which are functions that describe the state of a thermodynamic system.

What is the physical interpretation of Maxwell's relations?

The physical interpretation of Maxwell's relations is that they represent the symmetry of thermodynamic properties. This means that the values of thermodynamic properties are not affected by the order in which they are measured or calculated.

How are Maxwell's relations used in practical applications?

Maxwell's relations are used in various practical applications, such as in the design and analysis of thermodynamic systems, in chemical engineering processes, and in the study of phase transitions. They are also used in the development of equations of state for gases and liquids, and in the calculation of thermodynamic properties for different substances.

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