What would be the enthalpy change for a isothermal expansion?

I was doing a question on thermodynamics and came across this one stating



> 
> **During isothermal expansion of gas of an ideal gas :**
> 
> 
> **a)** Internal energy decreases
> 
> 
> **b)** Enthalpy increases
> 
> 
> **c)** Enthalpy reduces to zero
> 
> 
> **d)** Enthalpy remains unchanged
> 
> 
> 


I cannot agree to any of these as $\Delta H = \Delta U + W$, $\Delta U = 0$ as $\Delta T = 0$ but $W \neq 0$ so $\Delta H = W$ which is $\neq 0$ and as $W \lt 0$ the enthalpy should decrease but how much how do I know and to add to the confusion the answer given is enthalpy remains unchanged in the light of the equation $\Delta H = \Delta U +\Delta n\_\mathrm gRT$ and $\Delta U$ and $\Delta n = 0$ so $\Delta H = 0$ is it right.




You have already mentioned that the change of internal energy is zero $(\Delta U=0)$ since, for an ideal gas, the internal energy $U$ only depends on amount of substance $n$ and temperature $T$, and in a closed system $n$ is constant $(\Delta n=0)$ and during an isothermal process also $T$ remains constant $(\Delta T=0)$.


Enthalpy $H$ is defined as


$$H=U+pV$$


and the ideal gas law states that


$$pV=nRT$$


Thus


$$H=U+nRT$$


Since $n$ (closed system) and $T$ (isothermal process) are constant, the product $nRT$ is constant, and therefore, according to the ideal gas law, also the product $pV$ is constant.


Furthermore, since $U$ is constant during the given process, the sum $H=U+nRT$ as well as $H=U+pV$ has to remain unchanged (answer **d**).




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Note that your assumption


$$\Delta H = \Delta U + W$$


(where $W=p\Delta V$ is the pressure-volume work) only applies to processes under constant pressure $p$, since enthalpy is defined as


$$H=U+pV$$


and thus


$$\begin{align}
\mathrm dH&=\mathrm dU+\mathrm d(pV)\\
&=\mathrm dU+V\mathrm dp+p\mathrm dV
\end{align}$$


which simplifies to


$$\mathrm dH=\mathrm dU+p\mathrm dV$$


at constant pressure $(\mathrm dp=0)$.


However, the pressure does not remain constant during the process given in the question.


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