Is Internal Energy of Ideal Gas Really Only Dependent on Temperature?

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Discussion Overview

The discussion centers on the relationship between internal energy and temperature for an ideal gas, exploring whether internal energy is solely dependent on temperature or if other factors, such as heat and work, play a role. The scope includes theoretical considerations and derivations related to the first law of thermodynamics and ideal gas behavior.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant asserts that the internal energy of a monatomic ideal gas is given by U=3/2nRT and questions the assertion that internal energy only depends on temperature.
  • Another participant challenges the correctness of a derived equation related to internal energy, suggesting a possible error in the formulation.
  • There is a proposal that internal energy relations can be derived from constant pressure conditions and adapted for any process, contrasting with the common emphasis on constant volume in texts.
  • A participant raises a question about deriving internal energy from the first law and ideal gas behavior during adiabatic expansion, suggesting a lack of temperature relation in their derivation.
  • One participant notes that volume is a function of temperature, implying that temperature dependence should still be present in the equations.
  • Another participant provides the ideal gas law relation V=nRT/P, indicating that pressure varies during adiabatic expansion and questioning the implications of substituting this relation into their equations.
  • A later reply suggests that despite the complexity of the equations, they should ultimately align with the simpler expression for internal energy change.

Areas of Agreement / Disagreement

Participants express differing views on the dependence of internal energy on temperature and other factors, with no consensus reached on the validity of the various equations and derivations presented.

Contextual Notes

Participants highlight potential limitations in their assumptions and derivations, particularly regarding the relationships between pressure, volume, and temperature during different processes.

s943035
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We known U=3/2nRT (monatomic ideal gas), just depends on temperature.

Most texts assert connecting U and Q with constant volume condition
and say"\Delta U = nCv\Delta T for any process because of internal energy only depends on temp".

I think that statement is very strange.
Deriving the first law to \Delta U = nCp\Delta T + nR \Delta T = n\Delta T (Cp + R). Well, this equation also just depends on temperature.

Why not both heat and work in other condition to determine internal energy? Maybe kinetic relation is derived from constant volume?
 
Last edited:
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Hi s943035, welcome to PF!

Where are you getting the equation \Delta U = nCp\Delta T + nR \Delta T = n\Delta T (Cp + R)? It does not seem correct. Maybe you mean \Delta U = nCp\Delta T - nR \Delta T = n\Delta T (Cp - R) for an ideal gas? This works out, since Cv = Cp-R.
 
Thanks for your greeting and correction!

So, we also can derive the internal energy relation from constant pressure and adapt for any process? Just texts favor start from constant volume?

Another thinking, why both energy relation (\Delta U = n\Delta T(Cv) = n\Delta T (Cp-R)) can adapt for any process?

If I assume we only remember PV^\gamma = C(C is constant, for adiabatic expansion) and forget about any internal energy description,
just from first law and ideal gas behavior in adiabatic expansion,
I got \Delta U = 0 - \int \frac{C}{V^\gamma}dv = -C\frac{V^{-\gamma +1}}{-\gamma +1} = C\frac{V^{-\frac{R}{Cv}}}{\frac{R}{Cv}}= C\frac{Cv V^{-\frac{R}{Cv}}}{R}
It seems like no any temperature relation, why?
 
Last edited:
V is a function of temperature, so there is a temperature dependence. If you keep manipulating the equations, it should work out to give the same answer as before.
 
I don't know relation between V and T ...
 
s943035 said:
I don't know relation between V and T ...

V=nRT/P for an ideal gas.
 
Mapes said:
V=nRT/P for an ideal gas.

but P is various in this case (adiabatic expansion),
using this relation substitute into, doesn't make wrong?
 
I can see the equation getting pretty complicated, but in the end it's got to be equivalent to the simpler expression \Delta U=nc_V\Delta T.
 

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