# 3.5: Problems

- Page ID
- 106817

Problem \(\PageIndex{1}\)

Expand the following functions around the value of \(x\) indicated in each case.

In each case, write down at least four terms of the series, and write down the result as an infinite sum.

- \(\sin{(ax)}\), \(x=0\), \(a\) is a constant
- \(\cos{(ax)}\), \(x=0\), \(a\) is a constant
- \(e^{ax}\), \(x=0\), \(a\) is a real constant
- \(e^{-ax}\), \(x=0\), \(a\) is a real constant
- \(\ln{(ax)}\), \(x=1\), \(a\) is a real constant

Problem \(\PageIndex{2}\)

Use the results of the previous problem to prove Euler’s relationship:

\[e^{ix}=\cos x + i \sin x \nonumber\]

Problem \(\PageIndex{3}\)

The osmotic pressure (\(\pi\)) of a solution is given by

\[-RT \ln x_A=\pi V_m \nonumber\]

where \(V_m\) is the molar volume of the pure solvent, and \(x_a\) is the mole fraction of the solvent.

Show that in the case of a dilute solution

\[RT x_B \approx \pi V_m \nonumber\]

where \(x_B\) is the mole fraction of the solute. Remember that the mole fractions of the solute and the solvent need to add up to 1.

Note: you may use any of the results you obtained in Problem \(\PageIndex{1}\).

Problem \(\PageIndex{4}\)

The following expression is known as the Butler-Volmer equation, and it is used in electrochemistry to describe the kinetics of an electrochemical reaction controlled solely by the rate of the electrochemical charge transfer process.

\[j=j_0({e^{(1-\alpha)f\eta}-e^{-\alpha f \eta}}), ~ 0<\alpha<1 \text{ and } f>0, \eta>0 \nonumber\]

Show that \(j \approx j_0 f \eta\) when \(f \eta <<1\).

Note: you may use any of the results you obtained in Problem \(\PageIndex{1}\).

Problem \(\PageIndex{5}\)

The energy density of black-body radiation (\(\rho\)) at temperature T is given by Plank’s formula:

\[\rho(\lambda)=\frac{8\pi h c}{\lambda^5}[e^{hc/\lambda k T}-1]^{-1} \nonumber\]

where \(\lambda\) is the wavelength, \(h\) is Plank’s constant, and \(c\) is the speed of light. Show that the formula reduces to the classical Rayleigh-Jeans law \(\rho = 8\pi kT/\lambda^4\) for long wavelengths (\(\lambda \rightarrow \infty\)).

Hint: Define a variable \(\nu = \lambda^{-1}\) and solve the problem for \(\nu \rightarrow 0\).

Note: you may use any of the results you obtained in Problem \(\PageIndex{1}\).

Problem \(\PageIndex{6}\)

Use series to prove that \(\sum \limits _{k=0} ^\infty{\frac{\lambda^k e^{-\lambda}}{k!}}=1\), \(\lambda\) is a positive real constant.

Problem \(\PageIndex{7}\)

Write down the equation of a straight line that provides a good approximation of the function \(e^x\) at values close to \(x = 2\).

Problem \(\PageIndex{8}\)

Use a Taylor expansion around \(a\) to prove that \(\ln{x} = \ln{a}+\displaystyle\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n a^n}(x-a)^n\)