Why is the notation for partial derivatives so prevalent in thermodynamics?

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Discussion Overview

The discussion revolves around the notation for partial derivatives in thermodynamics, specifically why the notation \(\left(\frac{\partial T}{\partial V}\right)_S\) is commonly used despite some participants arguing that it conveys the same information as \(\left(\frac{\partial T}{\partial V}\right)\).

Discussion Character

  • Debate/contested

Main Points Raised

  • One participant questions the necessity of the subscript in the notation for partial derivatives, suggesting it may be a reminder for physicists about the concept of holding other variables constant.
  • Another participant argues that the subscript is essential, as different conditions (e.g., constant entropy, pressure, internal energy, or particle number) lead to different values for the partial derivative.
  • A further contribution emphasizes that in thermodynamic systems, variables are often interdependent, providing an example to illustrate that the same variable can yield different partial derivatives depending on the held constant.
  • A later reply expresses surprise at the implications of this discussion, indicating a need for further contemplation.

Areas of Agreement / Disagreement

Participants do not reach a consensus; there is disagreement regarding the necessity and implications of the subscript in partial derivative notation.

Contextual Notes

The discussion highlights the complexity of variable interdependence in thermodynamic systems, which may not be fully addressed in the initial claims.

ehrenfest
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Why in the world is the notation [tex]\left(\frac{\partial T}{\partial V}\right)_S[/tex] so ubiquitous in thermodynamics when it means exactly the same thing as [tex]\left(\frac{\partial T}{\partial V}\right)[/tex], that is, the partial of T with respect to V. The definition of a partial is that all the other variables are held constant! Is the problem just that physicists need a constant reminder of what a partial derivative is or what?
 
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It is not the same thing. In general,

[tex]\left(\frac{\partial T}{\partial V}\right)_S\neq \left(\frac{\partial T}{\partial V}\right)_P \neq \left(\frac{\partial T}{\partial V}\right)_U \neq\left(\frac{\partial T}{\partial V}\right)_N \neq \left(\frac{\partial T}{\partial V}\right)_\mu[/tex]

and so on. The subscript is not redundant.
 
Maybe I should provide some more detail. You may be used to working with systems in which the variables are all independent, but this is not generally the case, and is certainly not the case in most thermodynamic systems. Let me give an example: consider the relationships

[tex]a=bc[/tex]

[tex]d=a-1[/tex]

It should be clear that

[tex]\left(\frac{\partial a}{\partial b}\right)_c=c[/tex]

is not the same as

[tex]\left(\frac{\partial a}{\partial b}\right)_d=0[/tex]

In other words, yes, everyone needs a constant reminder of what the partial derivative is if they care about getting the right answer.
 
That is mind-blowing! I will have to think about that.
 

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