# 8.6: Problems

- Page ID
- 107013

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

- \(\left (\frac{\partial U}{\partial V} \right )_T=0\) for an ideal gas.
- \(\left (\frac{\partial U}{\partial V} \right )_T=\frac{a}{V^2}\) 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:

\[\left(\frac{\partial U}{\partial T}\right)_{V,n} \nonumber\]

\[\left(\frac{\partial U}{\partial V}\right)_{T,n} \nonumber\]

Be sure you can write a short sentence explaining your answers.

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:

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

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