# 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).

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

(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.

• Yes. The enthalpy of an ideal gas is a function only of temperature, no matter what kind of process is imposed. – Chet Miller Aug 17 '16 at 13:17
• @chester Miller if it were so then all isothermal process would be isoenthalpic which is not the case – Nitro phenol Aug 18 '16 at 3:33
• @PrateekChauhan For a single phase ideal gas, of course it is. – Chet Miller Aug 18 '16 at 13:14
• @ChesterMiller If I have an ideal gas, then will the change in enthalpy be zero, for an isothermal path? Does it matter whether the process is carried out reversibly or not? – SmarthBansal Sep 28 '18 at 6:39
• If the initial and final equilibrium temperatures are the same (irrespective of what happens along the path), the change in enthalpy of an ideal gas is zero. The enthalpy and internal energy of an ideal gas are functions only of temperature. – Chet Miller Sep 29 '18 at 1:23