# Questions 1-11

## Contents

# 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

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

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