# Free Expansion Of and Ideal Gas

We know that in free expansion of an ideal gas, no heat enters or leaves the system.

We also know that

$P_\text{initial}V_\text{initial}=P_\text{final}V_\text{final}$

is valid.

If heat exchange is zero, then we can call this process to be adiabatic.

Then why the following is not valid?

$P_\text{initial}{V_\text{initial}}^γ=P_\text{final}{V_\text{final}}^γ$

Also, if I am wrong above, are isothermal free expansion and adiabatic free expansion different?

• See Wikipedia page. For free expansion of an ideal gas initial and final temperature are same. – Tyrion Lannister Apr 19 '16 at 19:37

Your mistake is in thinking that $PV^\gamma = \text{constant}$ applies to a free expansion. That expression is for a reversible (i.e., isentropic) adiabatic process. A gas that has undergone a free expansion has more entropy after the expansion is complete than it did before the expansion started. Free expansion is not isentropic, and therefore $PV^\gamma = \text{constant}$ does not apply to free expansion.