# Euler's equation of thermodynamics in free expansion (Joule expansion)

• I
Hi everyone,
I am confused when I apply Euler's equation on the free expansion of an ideal gas.

Consider a free expansion (expansion of gas in vaccum) where the volume is doubled (V->2V)
The classical free expansion of an ideal gas results in increase in entropy by an amount of nR ln(2), a decrease in pressure (P->P/2), and the temperature T is constant.

The Euler equation of thermodynamics writes U=-PV+TS.
Before free expansion S=(U+PV)/T.
After free expansion S=(U+(P/2)(2V))/T.
It looks like that from Euler equation the entropy should remain unchange.
However it must not be the case from what we know about free expansion.

Can anyone give me some clue where am i wrong?

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Chestermiller
Mentor
Where did you get U = -PV+TS, because, according to my understanding, the Gibbs free energy is defined as G=U-TS+PV and it is not zero nor is its change zero in free expansion. In free expansion, its change is ##-nRT\ln{2}##

Where did you get U = -PV+TS, because, according to my understanding, the Gibbs free energy is defined as G=U-TS+PV and it is not zero nor is its change zero in free expansion. In free expansion, its change is ##-nRT\ln{2}##
The equation U=-PV+TS is called Euler's equation, and is derived from the homogenous property of extensive variables. Start with a chamber of gas with a state of (P,V,T), consider the first law of thermodynamics, dU=-PdV+TdS,
when we homogeneously increase volume and entropy (extensive variables) by 10% and keeping pressure and
temperature (intensive variables) constant, the internal energy U, an extensive variable, should change accordingly by 10%. Then U(final)=1.1U(initial), 0.1U=-P(0.1V)+Td(0.1S), U=-PV+TS.

Chestermiller
Mentor
The equation U=-PV+TS is called Euler's equation, and is derived from the homogenous property of extensive variables. Start with a chamber of gas with a state of (P,V,T), consider the first law of thermodynamics, dU=-PdV+TdS,
when we homogeneously increase volume and entropy (extensive variables) by 10% and keeping pressure and
temperature (intensive variables) constant, the internal energy U, an extensive variable, should change accordingly by 10%. Then U(final)=1.1U(initial), 0.1U=-P(0.1V)+Td(0.1S), U=-PV+TS.
The state of a closed system is determined by specifying 2 parameters. Once the pressure and temperature are specified, none of the other parameters can change. The equation of state for the gas, P=P(V,T) (for fixed mass) tells you once pressure and temperature are fixed, the volume can't change.

Ron19932017 and etotheipi
The state of a closed system is determined by specifying 2 parameters. Once the pressure and temperature are specified, none of the other parameters can change. The equation of state for the gas, P=P(V,T) (for fixed mass) tells you once pressure and temperature are fixed, the volume can't change.
Thank you for your reply! I checked again Euler's equation writes U=-PV+TS+##\mu## N, previously I missed the chemical potential term, it turns out that in the case of free expansion of V->2V, although ##\Delta## (-PV)=0, while ##\Delta (TS)=nrT ln(2)##, the chemical potential decreases! ##\Delta (\mu N)= -nRT ln(2)##, exactly cancels out the## \Delta (TS) ##term such that the internal energy indeed remains unchanged after the free expansion.