8.6: Problems

Problem $$\PageIndex{1}$$

Given a generic equation of state $$P = P(V, T, n)$$, explain how you can obtain the derivative

$\frac{\partial V}{\partial T}_{P,n} \nonumber$

using the properties of partial derivatives we learned in this chapter.

Problem $$\PageIndex{2}$$

The thermodynamic equation:

$\left (\frac{\partial U}{\partial V} \right )_T=T\left (\frac{\partial P}{\partial T} \right )_V-P \nonumber$

shows how the internal energy of a system varies with the volume at constant temperature.

Prove that

1. $$\left (\frac{\partial U}{\partial V} \right )_T=0$$ for an ideal gas.
2. $$\left (\frac{\partial U}{\partial V} \right )_T=\frac{a}{V^2}$$ for one mole of van der Waals gas (Equation \ref{c2v:eq:vdw})

Problem $$\PageIndex{3}$$

Consider one mole of a van der Waals gas (Equation \ref{c2v:eq:vdw}) and show that

$\left (\frac{\partial^2 P}{\partial V\partial T}\right )=\left (\frac{\partial^2 P}{\partial T\partial V} \right) \nonumber$

Problem $$\PageIndex{4}$$

Consider a van der Waals gas (Equation \ref{c2v:eq:vdw}) and show that

$\left (\frac{\partial V}{\partial T}\right )_{P,n}=\frac{n R}{\left( P-\frac{n^2a}{V^2}+\frac{2n^3ab}{V^3} \right)} \nonumber$

Hint: Calculate derivatives that are easier to obtain and use the properties of partial derivatives to get the one the problem asks for. Do not use the answer in your derivation; obtain the derivative assuming you don’t know the answer and simplify your expression until it looks like the equation above.

Problem $$\PageIndex{5}$$

From the definitions of expansion coefficient ($$\alpha$$) and isothermal compressibility ($$\kappa$$):

$\alpha=\frac{1}{V}\left(\frac{\partial V}{\partial T}\right)_{P,n} \nonumber$

and

$\kappa=-\frac{1}{V}\left(\frac{\partial V}{\partial P}\right)_{T,n} \nonumber$

prove that

$\left(\frac{\partial P}{\partial T}\right)_{V,n}=\frac{\alpha}{\kappa} \nonumber$

independently of the equation of state used.

Note: A common mistake in this problem is to assume a particular equation of state. Use the cycle rule to find the required relationship independently of any particular equation of state.

Problem $$\PageIndex{6}$$

Derive an equation similar to Equation \ref{c2v:eq:calculus2v_chain1}, but that relates

$\left ( \frac{\partial f}{\partial y} \right )_x \nonumber$

with

$\left ( \frac{\partial f}{\partial r} \right )_\theta \nonumber$

and

$\left ( \frac{\partial f}{\partial \theta} \right )_r \nonumber$

Problem $$\PageIndex{7}$$

(Extra-credit level)

The expression:

$\nabla^2=\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2} \nonumber$

is known as the Laplacian operator in two dimensions.

When applied to a function $$f(x,y)$$, we get:

$\nabla^2f(x,y)=\frac{\partial^2f}{\partial x^2}+\frac{\partial^2f}{\partial y^2} \nonumber$

Express $$\nabla^2$$ in polar coordinates (2D) assuming the special case where $$r=a$$ is a constant.

Problem $$\PageIndex{8}$$

Calculate $$\int_{0}^{1}\int_{1}^{2}\int_{0}^{2}{\left( x^2+yz \right)\, dx\, dy\, dz}.$$ Try three different orders of integration an verify you always get the same result.

Problem $$\PageIndex{9}$$

Calculate $$\int_{0}^{2\pi}\int_{0}^{\pi}\int_{0}^{\infty}{e^{-r}r^5\sin{\theta}\, dr\, d\theta\, d\phi}.$$ Use only the formula sheet.

Problem $$\PageIndex{10}$$

How would Figure $$8.5.2$$, reproduced below, look like for an ideal gas? Sketch the potential energy as a function of the distance between the atoms.

Problem $$\PageIndex{11}$$

From everything we learned in this chapter, and without doing any math, we should be able to calculate the sign (>0, <0, or 0) of the following derivatives:

For an ideal gas:
$\left(\frac{\partial U}{\partial T}\right)_{V,n} \nonumber$
$\left(\frac{\partial U}{\partial V}\right)_{T,n} \nonumber$

For a van der Vaals gas:
$\left(\frac{\partial U}{\partial T}\right)_{V,n} \nonumber$
$\left(\frac{\partial U}{\partial V}\right)_{T,n} \nonumber$

Problem $$\PageIndex{12}$$

The critical point is the state at which the liquid and gas phases of a substance first become indistinguishable. A gas above the critical temperature will never condense into a liquid, no matter how much pressure is applied. Mathematically, at the critical point:

$\left(\frac{\partial P}{\partial V} \right)_{T,n}=0 \nonumber$

and

$\left(\frac{\partial^2 P}{\partial V^2} \right)_{T,n}=0 \nonumber$

Obtain the critical constants of a van der Waals gas (Equation \ref{c2v:eq:vdw}) in terms of the parameters $$a$$ and $$b$$.

Hint: obtain the first and second derivatives of $$P$$ with respect to $$V$$, make them equal to zero, and obtain $$T_c$$ and $$V_c$$ from these equations. Finally, replace these expressions in Equation \ref{c2v:eq:vdw} to obtain $$P_c$$.

Note

As derived in Section 8.3,

$\label{c2v:eq:calculus2v_chain1} \left(\dfrac{\partial f}{\partial x}\right)_y=\cos{\theta}\left(\dfrac{\partial f}{\partial r}\right)_\theta-\dfrac{\sin{\theta}}{r}\left(\dfrac{\partial f}{\partial \theta}\right)_r$

As defined in Section 8.5, the Van der Waals is defined as:

$\label{c2v:eq:vdw} P=\frac{nRT}{V-nb}-a \left(\frac{n}{V}\right)^2$

1. If you are not familiar with this you need to read about it before moving on