Questions 1-11

Q1 Gas pressure change

A gas occupies a volume $V{\mathrm{m}}^3$ at pressure $p$ bar and at some temperature $pV=1000$ Joules. Express $V$ in terms of $p$ and $\delta V$ in terms of $p$ and $\delta p$. What is the change in volume when the pressure is increased from $1\to 1.1$ bar?

Strategy: Write $V=1000/p$, increment $V$ and $p$ then calculate $\delta V$ as $(V+\delta V)-V$.

Q2 Diatomic vibrational frequency

A diatomic molecule has a vibrational frequency of $v,{\mathrm{s}}^{-1}$, a force constant $k\mathrm{N}{\mathrm{m}}^{-1}$, reduced mass $\mu$ kg, and the vibrational frequency is given by ${\displaystyle v=\frac{1}{2\pi }\sqrt{\frac{k}{\mu }}}$.

(a) If, by isotopic substitution $\mu$ is increased by $1$%, show that the relative or fractional change is

$${\displaystyle \frac{\delta v}{v}=1-\sqrt{\frac{1}{1+\delta \mu /\mu }}}$$

which can be approximated as ${\displaystyle \frac{\delta v}{v}=\frac{\delta \mu }{2\mu }}$. Justify the approximation you make. You will need the series expansion ${\displaystyle (1+x{)}^{-1/2}=1-x/2+3x^2/8-\cdots }$.

(b) If $k=518.0\mathrm{N}{\mathrm{m}}^{-1}$ and $v=5889.0{\mathrm{cm}}^{-1}$ what is the absolute change in frequency in s${}^{-1}$ or Hz?

Q3 Thermometer

In the common thermometer, the thermal expansion of a liquid is measured and calibrated to the temperature rise. Mercury or ethanol is often used. If this is held in a $1$ ml reservoir and the capillary of the thermometer has a $0.12$ mm diameter, work out the sensitivity of this thermometer if $\beta$ is the coefficient of volume expansion. The liquid’s volume expands as $V=V_0(1+\beta \delta T)$ for a temperature rise of $\delta T$. Sensitivity is the change in length of the liquid for a 1 K rise in temperature. The constants are $\beta \text{( Hg )}=1.81\cdot {10}^{-4}{\mathrm{K}}^{-1}$, and $\beta \text{( EtOH )}=1.08\cdot {10}^{-3}{\mathrm{K}}^{-1}$.

Strategy: The sensitivity you need to work out is $\delta L/\delta T$. Use the volume of the capillary to work out its length.

Q4 Differentiate

(a) Differentiate with respect to $x$, a being constant.

$$ (i) $3x^4+\sin (x)$, (ii) $10x^{-4}+5x^2+a$, (iii) $\ln (x)+(ax{)}^{-1}$, (iv) $e^{ax}+x^2$

(b) Differentiate with respect to $a$, $x$ being constant.

$$ (i) $3x^4+\sin (x)$, (ii) $e^{ax}+x^2$, (iii) $y=a{x}^2$.

(c) Differentiate $\sin (ax)$, $n$ times with respect to $x$ when $n$ is even and when $n$ is odd. Differentiate up to $n$ = 5 before deciding on the pattern of equations, then use the identity $\cos (x)=\sin (x+n\pi /2)$.

Q5 Differentiate $n$ times

Differentiate $x^n$, $n$ times, $n$ being a positive integer, and find ${\displaystyle \frac{d^n}{d{x}^n}{x}^n}$

Strategy: In situations like this, where there are repeated operations and $n$ is undefined, it is best to try to get an answer by induction. Start with $n=1,2,\cdots$ and so forth, then build up a pattern. Find the answer for some general or intermediate term, such as the $m^{\text{th}}$, then finally make $m=n$.

Q6 Differentiate $n$ times

Differentiate $y=e^{-ax}$, $n$ times.

Q7 Ideal gas

A gas of volume $V{\mathrm{m}}^3$ has a pressure $p$ bar at a constant temperature $T$ K.

(a) Using the ideal gas law, show that ${\displaystyle \frac{dp}{dV}=c{V}^{-2}}$ where $c$ is a constant.

(b) Find ${\displaystyle \frac{dV}{dp}}$ .

(c) What is the relationship between ${\displaystyle \frac{dV}{dp}}$ and ${\displaystyle \frac{dp}{dV}}$ ?

Q8 Throwing a ball

(a) If a ball is thrown with an initial velocity $u$, the distance it travels in time $t$ is ${\displaystyle s=u+\frac{1}{2}a{t}^2}$. Find the velocity $v$ at any time $t$.

(b) What is the meaning of parameter $a$?

Q9 Electric field of light wave

(a) The electric field of a laser or other plane light-wave of frequency $\omega$ is ${\displaystyle E=A{e}^{i(\omega t-kx+\phi )}}$, where $x$ is the distance from the source, $k$ the wavevector (2$\pi /\lambda$), $\phi$ the phase, and $A$ is a constant and is the amplitude of the wave at $t=0,x=0,\phi =0$.

Show that the $n^{\text{th}}$ derivative with respect to time of $E$ is ${\displaystyle \frac{d^{n}E}{d{t}^n}=(i\omega {)}^{n}E}$

(b) Calculate the similar derivative for distance $x$.

Strategy: Start with the first derivatives, look for a pattern and substitute for $E$ into your answer. This equation for the electric field represents a general wave because $e^{i\theta }=\cos (\theta )+i\sin (\theta )$. This equation is described in more detail in Chapter 1.

Q10 Diffusion

The rate of diffusion / area / time in a solution of ions is described as

$${\displaystyle D_F\frac{dc}{dx}=cD\frac{d}{dx}\ln (c\gamma )}$$

where $D_F$ is the Fick’s law diffusion coefficient and $D$ is the ( kinematic ) diffusion coefficient, $c$ is the concentration of ions in solution, and $\gamma$ the activity coefficient. This is given by the modified Debye - Huckel expression ${\displaystyle \ln (\gamma )=-\frac{A\sqrt{c}}{1+B\sqrt{c}}}$ where $A$ and $B$ are constants that depend on the temperature and the solvent and $B$ on the size of the ions.

(a) Show that ${\displaystyle D_F=D(1+\frac{d\ln (\gamma )}{d\ln (c)})}$

(b) Evaluate the derivative.

Strategy: Do not let the unusual form of the differential put you off. Use $d\ln (c)=dc/c$

Q11 Differentiate the integral

Use eqn. 15 to evaluate ${\displaystyle \frac{d}{dx}{\int }_0^{a/x}\frac{x^2}{e^{-x}+1}dx}$