# 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 $$k\,\mathrm{ N\, m^{-1}}$$, 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 $$k = 518.0\,\mathrm{ N\, m^{-1}}$$ 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.

$$\quad$$ (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.

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

(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 $$\displaystyle \frac{d^n}{dx^n}x^n$$

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 $$\displaystyle \frac{dp}{dV} = cV^{-2}$$ 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 $$\displaystyle s = u + \frac{1}{2}at^2$$. 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 $$\displaystyle E = A e^{i(\omega t - kx +\varphi)}$$, where $$x$$ is the distance from the source, $$k$$ the wavevector (2$$\pi/\lambda$$), $$\varphi$$ the phase, and $$A$$ is a constant and is the amplitude of the wave at $$t = 0,x = 0,\varphi=0$$.

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

(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 $$\displaystyle D_F=D\left( 1+\frac{ d\ln(\gamma) }{ d\ln(c) } \right)$$

(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 $$\displaystyle \frac{d}{dx}\int_0^{a/x}\frac{x^2}{e^{-x}+1}dx$$