Why does an ideal gas satisfy ##(\partial U/\partial P)_T=0##?

Click For Summary

Discussion Overview

The discussion centers around the justification for the equation \((\partial U/\partial P)_T = 0\) for an ideal gas, exploring its theoretical basis and implications within the context of thermodynamics and kinetic theory.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant cites the definition of an ideal gas and its equations, questioning the origin of \((\partial U/\partial P)_T = 0\).
  • Another participant states that the internal energy of an ideal monoatomic gas is given by \(\frac{3}{2}RnT\) and concludes that differentiating this with respect to \(P\) at constant \(T\) yields zero.
  • A different participant expresses a belief that the equation arises from kinetic theory but seeks an alternative justification not based on that theory.
  • One participant provides a link to an external thread for further reading, suggesting additional resources may be available.
  • Another participant argues that since ideal gas particles do not interact significantly, compressing the gas at constant temperature does not change the internal energy, thus supporting the equation in question.

Areas of Agreement / Disagreement

Participants express differing views on the justification for the equation, with some relying on kinetic theory and others emphasizing the non-interaction of gas particles. The discussion remains unresolved regarding the foundational reasoning behind \((\partial U/\partial P)_T = 0\).

Contextual Notes

Some participants reference the internal energy expression and its implications, but there is no consensus on the justification outside of kinetic theory. The discussion also highlights the dependence on assumptions about particle interactions.

zenterix
Messages
774
Reaction score
84
TL;DR
Why does an ideal gas satisfy ##\left (\frac{\partial U}{\partial P}\right )_T = 0##?
The book I am reading says that by definition, the ideal gas satisfies the equations

$$PV=nRT\tag{1}$$

$$\left (\frac{\partial U}{\partial P}\right )_T = 0\tag{2}$$

where does (2) come from? In other words, what justifies this equation in the definition above?
 
Science news on Phys.org
The internal energy of an ideal (monoatomic) gas is ##3RnT/2##. Differentiating with respect to ##P## with ##T## constant is clearly zero.
 
Orodruin said:
The internal energy of an ideal (monoatomic) gas is ##3RnT/2##. Differentiating with respect to ##P## with ##T## constant is clearly zero.
The thing is, I believe that equation comes from the kinetic theory of the ideal gas right.

The chapter of the book that I am on is a few chapters before talking about that theory. The only reason I know about that equation is from looking ahead.

I am wondering about some other justification not based on that theory.
 
Ideal gas is made of particles which do not interact with each other, or more realistically, of particles for which the interaction energy is negligible. By compressing ideal gas while keeping temperature constant you increase the pressure ##P##, but since the interaction energy is negligible, ##U## does not change. This answers the OP's question.
 
  • Like
Likes   Reactions: chimay, weirdoguy, Philip Koeck and 1 other person

Similar threads

  • · Replies 1 ·
Replies
1
Views
3K
  • · Replies 23 ·
Replies
23
Views
3K
  • · Replies 6 ·
Replies
6
Views
2K
  • · Replies 2 ·
Replies
2
Views
2K
  • · Replies 19 ·
Replies
19
Views
3K
  • · Replies 23 ·
Replies
23
Views
3K
  • · Replies 6 ·
Replies
6
Views
2K
  • · Replies 35 ·
2
Replies
35
Views
5K
  • · Replies 3 ·
Replies
3
Views
2K
  • · Replies 9 ·
Replies
9
Views
3K