The enthalpy change is a state function and hence at the same temperature the enthalpy change remains the same. Enthalpy change is given by:
dH = Vdp + pdV + dU
For isothermal expansion of a gas at a constant pressure:At constant temperature dU should be equal to 0 and at constant external pressure VdP should also be equal to zero. In this way dH = pdV and is not equal to O. So how do we say that enthalpy change should be equal to zero at constant temperature?