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