for an isothermal expansion of an ideal gas
W=0
​change in E=0
​q=0
change in V=0
 

W=0
For an adiabatic process, in which no heat flows into or out of the gas because its container is well insulated, . If there is also no work done, i.e. a free expansion, there is no change in internal energy. For an ideal gas, this means that the process is also isothermal.