Why didn't Joule change his experiment to make $C_{\text {surroundings }} / C_{\text {system }} \approx 0.001$ to increase the sensitivity of the apparatus?

Derive the following relation,

$$\left(\frac{\partial U}{\partial V_m}\right)_T=\frac{3 a}{2 \sqrt{T} V_m\left(V_m+b\right)}$$

for the internal pressure of a gas that obeys the Redlich-Kwong equation of state,

$$P=\frac{R T}{V_m-b}-\frac{a}{\sqrt{T}} \frac{1}{V_m\left(V_m+b\right)}$$

Use $(\partial U / \partial V)_T=(\beta T-\kappa P) / \kappa$ to calculate $(\partial U / \partial V)_T$ for an ideal gas.

Starting with the van der Waals equation of state, find an expression for the total differential $d P$ in terms of $d V$ and $d T$. By calculating the mixed partial derivatives $\left(\partial(\partial P / \partial V)_T / \partial T\right)_V$ and $\left(\partial(\partial P / \partial T)_V / \partial V\right)_T$, determine if $d P$ is an exact differential.

Because $U$ is a state function, $\left(\partial / \partial V(\partial U / \partial T)_V\right)_T=$ $\left(\partial / \partial T(\partial U / \partial V)_T\right)_V$. Using this relationship, show that $\left(\partial C_V / \partial V\right)_T=0$ for an ideal gas.

Is the expression $\Delta U_V=\int_{T_1}^{T_2} C_V d T=n \int_{T_1}^{T_2} C_{V, m} d T$ only valid for an ideal gas if $V$ is constant?

What is the physical basis for the experimental result that $U$ is a function of $V$ at constant $T$ for a real gas? Under what conditions will $U$ decrease as $V$ increases?

Derive an expression for the internal pressure of a gas that obeys the Bethelot equation of state,

$$P=\frac{R T}{V_m-b}-\frac{a}{T V_m^2}$$

The molar volume of $H_2O$(l) decreases with increasing temperature near $4^\circ C$. Can you explain this behavior using a molecular level model?

Show that the expression $(\partial U / \partial V)_T=$ $T(\partial P / \partial T)_V-P$ can be written in the form

$$\left(\frac{\partial U}{\partial V}\right)_T=T^2\left(\partial\left[\frac{P}{T}\right] / \partial T\right)_V=-\left(\partial\left[\frac{P}{T}\right] / \partial\left[\frac{1}{T}\right]\right)_V$$

Is the following statement always, never, or sometimes valid? Explain your reasoning:
$q= 0$ for a cyclic process.

The Joule coefficient is defined by $(\partial T / \partial V)_U=$ $\left(1 / C_V\right)\left[P-T(\partial P / \partial T)_V\right]$. Calculate the Joule coefficient for an ideal gas and for a van der Waals gas.

Is the following statement always, never, or sometimes valid? Explain your reasoning:
a thermodynamic process is completely defined by the initial and final states of the system.

Is the following statement always, never, or sometimes valid? Explain your reasoning:
$\Delta H$ is only defined for a constant pressure process.