Predict the change in internal energy for an isolated system at constant volume.
For isolated system there is no transfer of energy as heat, i.e., $q=0$ and there is no transfer of energy as work. i.e., $\mathrm{W}=0$. According to the first law of thermodynamics
$$\begin{aligned} \Delta U & =q+W \\ \Delta U & =0+0=0 \end{aligned}$$
Although heat is a path function but heat absorbed by the system under certain specific conditions is independent of path. What are those conditions? Explain.
The two conditions under which heat becomes independent of path are
(i) when volume remains constant
(ii) when pressure remains constant
Explanation
(i) At constant volume By first law of thermodynamics, $\Delta U=q+W$ or $q=\Delta U-W$. But $W=-p \Delta V$ Hence, $q=\Delta U+p \Delta V$. But as volume remains constant $\Delta V=0$
$\therefore q_v=\Delta V$ but $\Delta U$ is a state function.
Hence, $q_v$ is a state function.
(ii) At constant pressure As we know, $q_p=\Delta U+p \Delta V$. But $\Delta U+p \Delta V=\Delta H$.
$\therefore q_p=\Delta H$. As $\Delta H$ is a state function therefore, $q_p$ is a state function.
Expansion of a gas in vacuum is called free expansion. Calculate the work done and the change in internal energy when 1L of ideal gas expands isothermally into vacuum until its total volume is 5L?
Work done of a gas in vacuum, $W=-p_{\text {ext }}\left(V_2-V_1\right)$. As $p_{\text {ext }}=0$ so $W=-0(5-1)=0$ As internal energy of an ideal gas depends only on temperature, therefore, for isothermal expansion of an ideal gas, internal energy remains constant, i.e.,
$$\Delta U=0 .$$
It is to be remember that as $H=U+p V, \Delta H=\Delta(U+p V)=\Delta U+p \Delta V=\Delta U+n R(\Delta T)$. For isothermal process, $\Delta T=0$ and also $\Delta U=0$, as stated above, therefore, $\Delta H=0$.
Heat capacity $\left(C_p\right)$ is an extensive property but specific heat $(c)$ is intensive property. What will be the relation between $C_p$ and $c$ for 1 mole of water?
For water, molar heat capacity,
$$\begin{aligned} C_p & =18 \times \text { Specific heat, } \mathrm{c} \\ C_p & =18 \times c \text { Specific heat } \\ c & =4.18 \mathrm{Jg}^{-1} \mathrm{~K}^{-1} \quad \text{(for water)} \end{aligned}$$
Heat capacity,
$$\begin{aligned} C_p & =18 \times 4.18 \mathrm{JK}^{-1} \mathrm{~mol}^{-1} \\ & =75.3 \mathrm{JK}^{-1} \mathrm{~mol}^{-1} \end{aligned}$$
The difference between $C_p$ and $C_v$ can be derived using the empirical relation $H=U+p V$. Calculate the difference between $C_p$ and $C_v$ for 10 moles of an ideal gas.
Given that, $C_v=$ heat capacity at constant volume,
$C_p=$ heat capacity at constant pressure
Difference between $C_p$ and $C_v$ is equal to gas constant ($R$).
$$\begin{aligned} \therefore \quad C_p-C_v & =n R \quad \text{(where, $n=$ no. of moles)}\\ & =10 \times 4.184 \mathrm{~J} \\ & =41.84 \mathrm{~J} \end{aligned}$$