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"# Questions 1-11"
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"## Q1 Gas pressure change\n",
"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?\n",
"\n",
"**Strategy:** Write $V = 1000/p$, increment $V$ and $p$ then calculate $\\delta V$ as $(V + \\delta V) - V$.\n",
"\n",
"## Q2 Diatomic vibrational frequency\n",
"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}} $.\n",
"\n",
"(a) If, by isotopic substitution $\\mu$ is increased by $1$%, show that the relative or fractional change is \n",
"\n",
"$$\\displaystyle \\frac{\\delta v}{v}=1-\\sqrt{\\frac{1}{1+\\delta \\mu/\\mu}}$$\n",
"\n",
"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 $\\displaystyle (1+x)^{-1/2}=1-x/2+3x^2/8-\\cdots$.\n",
"\n",
"(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?\n",
"\n",
"## Q3 Thermometer\n",
"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 $\\beta \\text{( EtOH )} = 1.08 \\cdot 10^{-3}\\,\\mathrm{ K^{-1}}$.\n",
"\n",
"**Strategy:** The sensitivity you need to work out is $\\delta L/\\delta T$. Use the volume of the capillary to work out its length.\n",
"\n",
"## Q4 Differentiate\n",
"(a) Differentiate with respect to $x$, a being constant.\n",
"\n",
"$\\quad$ (i) $3x^4 + \\sin(x)$, (ii) $10x^{-4} + 5x^2 + a$, (iii) $\\ln(x) + (ax)^{-1}$, (iv) $e^{ax} + x^2$\n",
"\n",
"(b) Differentiate with respect to $a$, $x$ being constant. \n",
"\n",
"$\\quad$ (i) $3x^4 + \\sin(x)$, (ii) $e^{ax} + x^2$, (iii) $y = ax^2$.\n",
"\n",
"(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 $\\cos(x) = \\sin(x + n\\pi /2)$.\n",
"\n",
"## Q5 Differentiate $n$ times\n",
"Differentiate $x^n$, $n$ times, $n$ being a positive integer, and find $\\displaystyle \\frac{d^n}{dx^n}x^n$\n",
"\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 $m = n$.\n",
"\n",
"## Q6 Differentiate $n$ times\n",
"Differentiate $y = e^{-ax}$, $n$ times.\n",
"\n",
"## Q7 Ideal gas\n",
"A gas of volume $V\\,\\mathrm{ m^3}$ has a pressure $p$ bar at a constant temperature $T$ K.\n",
"\n",
"(a) Using the ideal gas law, show that $\\displaystyle \\frac{dp}{dV} = cV^{-2}$ where $c$ is a constant.\n",
"\n",
"(b) Find $\\displaystyle \\frac{dV}{dp}$ .\n",
"\n",
"(c) What is the relationship between $\\displaystyle \\frac{dV}{dp}$ and $\\displaystyle \\frac{dp}{dV}$ ?\n",
"\n",
"## Q8 Throwing a ball\n",
"(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 $t$.\n",
"\n",
"(b) What is the meaning of parameter $a$?\n",
"\n",
"## Q9 Electric field of light wave\n",
"(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 $t = 0,x = 0,\\varphi=0$. \n",
"\n",
"Show that the $n^\\text{th}$ derivative with respect to time of $E$ is $\\displaystyle \\frac{d^nE}{dt^n}=(i \\omega)^n E$\n",
"\n",
"(b) Calculate the similar derivative for distance $x$.\n",
"\n",
"**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.\n",
"\n",
"## Q10 Diffusion\n",
"The rate of diffusion / area / time in a solution of ions is described as \n",
"\n",
"$$\\displaystyle D_F\\frac{dc}{dx}=cD\\frac{d}{dx}\\ln(c\\gamma)$$\n",
"\n",
"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.\n",
"\n",
"(a) Show that $\\displaystyle D_F=D\\left( 1+\\frac{ d\\ln(\\gamma) }{ d\\ln(c) } \\right)$\n",
"\n",
"(b) Evaluate the derivative.\n",
"\n",
"**Strategy:** Do not let the unusual form of the differential put you off. Use $d \\ln(c) = dc/c$\n",
"\n",
"## Q11 Differentiate the integral\n",
"Use eqn. 15 to evaluate $\\displaystyle \\frac{d}{dx}\\int_0^{a/x}\\frac{x^2}{e^{-x}+1}dx$"
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