# Basic thermodynamics

## Main Question or Discussion Point

For an ideal gas, how to prove that:
(a) its internal energy does not change with volume at constant temperature
(b) its enthalpy does not change with pressure at constant temperature

Thanks.

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Andrew Mason
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For an ideal gas, how to prove that:
(a) its internal energy does not change with volume at constant temperature
(b) its enthalpy does not change with pressure at constant temperature

Thanks.
What have you done to try to work this out?

AM

Well for an ideal gas, they are like free particles as there are no forces or potential energies between particles. If you increase the size of the box, that doesn't change the potential energy since there is no potential energy. It doesn't change the kinetic energy because all collisions with the box are elastic, so colliding with the walls doesn't change the kinetic energy, so changing the frequency of collision with walls (which would alter the pressure) by changing how big the box is does not change the kinetic energy! So since internal energy is kinetic+potential, none of this changes with the size of the box!

As for enthalpy I have no clue what that is.

Well for an ideal gas, they are like free particles as there are no forces or potential energies between particles. If you increase the size of the box, that doesn't change the potential energy since there is no potential energy. It doesn't change the kinetic energy because all collisions with the box are elastic, so colliding with the walls doesn't change the kinetic energy, so changing the frequency of collision with walls (which would alter the pressure) by changing how big the box is does not change the kinetic energy! So since internal energy is kinetic+potential, none of this changes with the size of the box!

As for enthalpy I have no clue what that is.
Enthalpy is the total energy of a thermodynamic system - Internal Energy and the energy required to make room for it (i.e. increase the pressure of its environment to make space).
It is basicaly H=U+pV (H is Enthalpy in Joules - U is internal energy, p is pressure and V is volume). As the volume has increased, the pressure has decreased, so Enthalpy stays the same (assuming Pressure and volume change at inverse rates - i.e. no external change occurs).

That's my understanding anyway.

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Oh sorry... What I am trying to ask is, how to prove that

(a)(dU/dV)T=0
(b)(dH/dP)T=0

for an ideal gas.