Thermal Physics

A heat pump is a device that pumps heat from a body at a temperature $T_{\text {cold }}$, to a body at a temperature $T_{\text {hot }}$. An example of a heat pump is a domestic refrigerator. The heat is pumped from the inside of the refrigerator to the outside. The pump will cool the inside of the refrigerator until a predetermined temperature is reached.

In some countries a heat pump is used as a domestic heater in winter and as an air conditioner in summer, i.e. cooling the air in the room and pumping the heat to the outside.

Figure 2.1 is a block diagram of the heat pump/refrigerator. A is the hot body at temperature $T_{\text {hot }}$. B is the cold body at temperature $T_{\text {cold }}$. $\Delta Q_{\text {hot }}$ and $\Delta Q_{\text {cold }}$ are the quantities of heat per second absorbed by A, and extracted from B. The pump delivers power $P$ to achieve this.
a) Write down an equation relating $\Delta Q_{\text {hot }}, \Delta Q_{\text {cold }}$ and $P$. In addition it can be shown that an ideal heat pump satisfies the equation:

$$\frac{\Delta Q_{\text {hot }}}{\Delta Q_{\text {cold }}}=\frac{T_{\text {hot }}}{T_{\text {cold }}} .$$

b) Find the heating power that can be delivered into a room by an ideal heat pump if the pump power, $P$, is 1.0 kW , when the outside temperature is 280 K and the temperature of the room is 293 K . Comment on your answer.
c ) In an experiment using 2.0 kW heat pump the cold body B is a tank with 10 kg of water initially at 290 K . Ignore the thermal capacity of the empty tank. The rate of heat leakage into the tank from its surroundings only is $20 \theta \mathrm{~W}$, where $\theta$ is the temperature difference between the tank and its surroundings that are at a constant 290 K . Assume that $T_{\text {hot }}$ is constant at 305 K . Write down an equation for the rate of fall of temperature of the tank. Calculate the quantity of heat reaching the hot body per second:
(i) initially, when the tank is 290 K
(ii) when the tank has reached 280 K .

Calculate rate of fall of temperature of the tank:
(iii) initially
(iv) at 280 K .

Calculate the approximate speed of Nitrogen molecules at a temperature of $20^{\circ} \mathrm{C}$.
Calculate the height at which the density of the Nitrogen molecules becomes one per cent of the pressure at ground level. It has been suggested that if a plane achieved a suborbital flight to Australia then a great deal of fuel could be saved. Estimate the energy needed to reach a suitable height so that the resistance of the air would be very small.

The "Dreamliner" has a mass of $2.27 \times 10^{5} \mathrm{~kg}$.

(a) The flow of thermal energy through a conductor due to a temperature difference between two faces is described by an equation of the form,

$$\frac{\mathrm{dQ}}{\mathrm{~d} t}=-\lambda A \frac{\Delta T}{\Delta x}$$

in which $\frac{\mathrm{dQ}}{\mathrm{d} t}$ is the rate of heat flow, $A$ is the cross sectional area of the conductor, and $\frac{\Delta T}{\Delta x}$ is the temperature gradient along the line of heat flow. The thermal conduction properties of the medium are given by the coefficient of thermal conductivity, $\lambda$.
i) A similar flow through a conductor may be ascribed to conduction of electricity. Write down an equation of the same form for electrical conduction through a material, stating what the corresponding term is in each equation.
Comment on the presence of the minus sign.
ii) In an example in which two cylindrical conductors of heat energy with the same cross sectional areas and similar lengths are placed end to end in series, so that the thermal energy flows through each in turn, what is the equivalent thermal conductivity of the series combination of conductors, $\lambda$, in terms of $\lambda_{1}$ and $\lambda_{2}$ ? (No heat is lost through the sides of the conductors).
(b) A circuit is set up with the current flowing radially through a uniform disc of conducting material of resistivity, $\rho$, and thickness, $t$. Current enters a centre metal core of radius $r_{0}$ which has a very high conductivity and flows out of the rim of the disc, which has radius $r$, and has a ribbon of highly conducting metal around it. In terms of the linear dimensions and the resistivity, obtain an expression for the resistance through the disc.

(c) Physicists often have to explain effects seen in nature for which there may be a number of contributing explanations. Observation is a key skill of the physicist, being able to see what is in front of him and spot the important relevant features.

In Figures 1.2 and 1.3 below, water was seen flowing down a road from the constant source of a small stream at the side of the road. The smooth flow of the stream breaks up further up the road. The water flows down the uniform slope of the road towards the observer. State what you can observe in the photographs and comment on them from the point of view of a physicist. You are not required to do any calculations or explain all the physics of this complicated situation.

Figure 4.1 represents a cylinder with thermally insulated walls containing a movable, frictionless, thermally insulated piston. On each side of the piston are $n$ moles of an ideal gas. The initial conditions of pressure, $P_{0}$, volume $V_{0}$, and temperature $T_{0}$, are the same on both sides of the piston.
$C_{P}$ is the molar heat capacity (i.e. specific heat capacity per mole) of the gas at constant pressure and $C_{V}$ is the molar heat capacity (i.e. per mole) at constant volume.
The value of $\gamma$ is 1.5 where $\gamma$ is the ratio of the molar heat capacities, $\frac{C_{P}}{C_{V}}$.
By means of a heating coil in the gas on the left side of the piston, heat is supplied slowly to the gas on this side. It expands and compresses the gas on the right side until its pressure has increased to $\frac{27}{8} P_{0}$.

Hint: for a gas undergoing an adiabatic change, $P \times V^{\gamma}=$ constant, and also the ideal gas law still applies.

In terms of $n, R, C_{V}$, and $T_{0}$,
(a) how much work is done on the gas on the right side?
(b) what is the final temperature of the gas on the right?
(c) what is the final temperature of the gas on the left?
(d) how much heat flows into the gas on the left?

The first law of thermodynamics can be written as

$$\delta U=\delta Q+\delta W$$

where $\delta U$ is the change in internal energy of a given system, $\delta Q$ is the heat supplied to the system and $\delta W=-p \delta V$ is the work done on the system.
(a) Show that for a monatomic ideal gas, the heat capacity at constant volume is given by $C_{V}=\frac{3}{2} n R$, and also that the heat capacity at constant pressure is given by $C_{p}=C_{V}+n R=\frac{5}{2} n R$. $n$ is the number of moles of gas.
(b) When a monatomic ideal gas undergoes an adiabatic change, no heat is supplied to the gas leading to $p V^{\gamma}=$ const., where $\gamma=\frac{C_{p}}{C_{V}}$. Determine the corresponding relationships between $V$ and $T$, and between $p$ and $T$.
(c) Two thermally isolated (i.e. insulated) cylinders, L and R , of equal volume, both equipped with pistons, are connected by a valve. Initially L has its piston fully withdrawn and contains an ideal monatomic gas at temperature $T_{0}$, while R has its piston fully inserted and the valve is closed as shown in the diagram below:

Calculate the final temperature of the gas in each cylinder after the following operations are performed, each with the same initial configuration as already described. Assume that the thermal capacity of the cylinders is negligible.
(i) The valve is fully opened and the gas is slowly drawn into R by pulling out piston R ; piston L remains stationary.
(ii) Piston R is fully withdrawn and the valve opened slightly. The gas is then driven as far as it will go by pushing in piston L at such a rate that the pressure in L remains constant. The cylinders are in thermal contact.
(iii) Piston R is fully withdrawn and the valve opened slightly. The gas is then driven as far as it will go by pushing in piston L at such a rate that the pressure in L remains constant. The cylinders are thermally isolated from each other.
(iv) Piston R is fully withdrawn and the valve is then opened. The cylinders are in thermal contact.
(v) Piston R is fully withdrawn and the valve is then opened. The cylinders are thermally isolated from each other.

Ben often drinks coffee, but finds that he rarely finishes his cup completely; the coffee is normally too cold by the end for him to drink it all up. This situation puzzles him and, as he is a physicist, he decides to model it. As the coffee cools to a more comfortable temperature, Ben can drink it faster, but when it goes below a critical temperature, $T_{F}$, it is too cold and he stops drinking it. He therefore models the rate of drinking as

\begin{equation*} \frac{\mathrm{d} V}{\mathrm{~d} t}=-\frac{k}{T-T_{F}} \quad \text { for } \quad T \geq T_{F} \tag{1} \end{equation*}

where $V(t)$ is the volume and $T(t)$ the temperature at time $t . k$ is a positive constant parameterising how fast he is drinking.
(a). Explain how Ben's model accounts for the variation in his rate of drinking and suggest, with reasoning, a value for $k$.
(b). A picture of Ben's cup is shown below on the left. Identify and discuss the physical factors of the coffee and mug you would expect the rate of heat loss to depend on, and the relative contributions they might make. Express the rate of cooling as a function of these factors, defining any symbols that you use. Hint: assume the simplest reasonable dependence on the variables in question. You may also assume his mug is cylindrical.

(c). Show that Ben's model produces behaviour which accounts for his observations and estimate the amount of coffee he wastes with each mug he drinks, explaining your reasoning carefully. Hint: $\frac{\mathrm{d}^{2} V}{\mathrm{~d} t^{2}}=\frac{\mathrm{d} V}{\mathrm{~d} t} \frac{\mathrm{~d}}{\mathrm{~d} V}\left(\frac{\mathrm{~d} V}{\mathrm{~d} t}\right)$
(d). Ben travels to a conference where he notices that, in an effort to get participants to drink their coffee up during the break, rather than taking it with them into the lecture theatre, it is being served in cups like the one on the right in the image above. The cups are made of polystyrene. Ben does not wish to accept a cup of coffee unless he is certain that he can drink it all up before it goes cold, otherwise he will be stuck with liquid in a cup and nowhere to put it. Explain whether or not you would advise him to accept a coffee. As a physicist, Ben will only be convinced by coherent quantitative reasoning.

Write you answer starting on a new page for this question.

The atmosphere is composed of $78 \%$ nitrogen, $21 \%$ oxygen, $0.9 \%$ argon and about $0.1 \%$ trace gases. Water vapour varies between about $0 \%-4 \%$ depending on location and time of day.
(a) Consider a fluid at rest in a gravitational field $g$. Show that, if frictional forces within the fluid can be neglected, the pressure variation across a sufficiently small cube of incompressible fluid varies linearly with the height of the cube. This is referred to as hydrostatic pressure.
(b) In a simple model, the atmosphere is treated as an ideal gas.
(i) Show that the variation in pressure with height is exponential. You may assume that the pressure variation across a very thin layer is hydrostatic, and that temperature is constant.
(ii) Why is it safe to assume that $g$ is constant too?
(iii) What do you think is the biggest issue with the formula found in (i)?
(c) In another model, the gas (still ideal) is assumed to expand adiabatically as it rises. An adiabatic change is one for which the pressure, $P$, of the gas and its volume, $V$, are related by $P V^{\gamma}=$ constant, where $\gamma=\frac{C_{P}}{C_{V}}$ with $C_{P}$ and $C_{V}$ the heat capacities of the gas per mole at constant pressure and volume respectively.
(i) Find the rate of change of temperature with height, $\frac{\mathrm{d} T}{\mathrm{~d} h}$ - known as the temperature lapse rate - under these circumstances. You may assume without proof that $C_{P}= C_{V}+R$ with $R$ the molar gas constant.
(ii) Hence or otherwise find a more realistic formula for the pressure variation with height in the atmosphere.

A diatomic molecule can store energy in the form of vibration and rotation of the molecule. The energies can be shown to be quantised i.e. there are discrete energy levels similar to the energy levels encountered in the visible spectrum of an atom.
(d) An HCl molecule can be modelled as two masses, $m_{1}$ and $m_{2}$ with mean separation $\ell$ at the ends of a stiff spring, with spring constant $k$, representing the bond. The frequency of vibration can be expressed in terms of $k$ and $\mu$, where $\mu\left(m_{1}, m_{2}\right)$ is function of $m_{1}$ and $m_{2} . \mu$ has dimensions of mass and is known as the reduced mass. By considering the centre of mass of the system or otherwise, derive an expression for the frequency of vibration $f$ of the masses on the spring in terms of $m_{1}$ and $m_{2}$ and $k$. Hence write down the expression for $\mu$.

The IR spectrum of HCl is shown in Fig. 3. The wavenumber on the horizontal axis is $1 / \lambda$ measured in $\mathrm{cm}^{-1}$. A low pressure gas of HCl is illuminated with IR radiation and the absorption measured over a range of frequencies. The HCl molecule will rotate corresponding to a number of close, discrete energy levels designated by the letter $J=0,1,2,3 \ldots$ shown in Fig. 4. The two vibrational energy levels labelled with $n=0$ and $n=1$ correspond to a very much larger energy level difference.

(e) For the wavenumber value $2900 \mathrm{~cm}^{-1}$ calculate the energy $E$ of the transition and calculate the corresponding temperature using $E=k_{\mathrm{B}} T$.

Only transitions between states corresponding to $\Delta J= \pm 1$ are allowed. So the $\Delta J=0$ transition is missing. In Figs. 3 and 4 the set of transitions in which the upper level has the higher $J$ is called the $\mathbf{R}$ branch, while the $\mathbf{P}$ branch has this reversed. The transitions within a branch are labeled by the $J$ of the lower level; $\mathbf{R ( 0 ) , P ( 1 )}$ would be the labeling for the lowest $J$ transitions.
(f) Draw a labelled energy level diagram, similar to Fig. 4, showing the first five rotational levels in the ground and first vibrational states ( $n=0 \rightarrow 1$ ). Show, using arrows as in Fig. 4, all of the transitions allowed between these states (and only these transitions), and label each transitions with the $\mathbf{P}, \mathbf{R}$ notation.
(g) An estimate of the $J=0$ transition energy can be obtained from the average of the $\mathbf{R ( 0 )}$ and $\mathbf{P}(\mathbf{1})$ transition energies. If this corresponds to $8.66 \times 10^{13} \mathrm{~Hz}$, using your result from (d), calculate the value of the spring constant $k$ for the HCl bond.

Data: Atomic mass of hydrogen is 1.0078 u Atomic mass of chlorine- 35 is 34.9688 u

This question explores the formation of clouds in the lower atmosphere.
(a) Consider a sealed flask, connected to a pressure gauge, containing dry air (no water vapour) at atmospheric pressure. A small volume of liquid water is injected into the flask.
(i) With reference to evaporation and condensation, explain why the pressure inside the flask is observed to increase over time, then eventually level-off.
(ii) The flask is now cooled externally. With the aid of a sketch graph, explain qualitatively the subsequent readings on the pressure gauge.

Figure 5 shows the so-called phase diagram of water.
(b) Sketch out the phase diagram on your answer sheet and mark the regions showing the three possible phases - solid, liquid and vapour - with the letters the $S, L$ and $V$ respectively. Now add arrows to indicate the names of the variety of different phase transitions possible, e.g., melting from $S$ to $L$.

The coexistence curve $A-C$ shows the possible combinations of pressure $p$ and temperature $T$ for which water exists in equilibrium as a mixture of liquid and vapour phases.

The Clausius-Clapeyron equation describes the gradient of this co-existence curve:

\begin{equation*} \frac{d p}{d T}=\frac{L M p}{R T^{2}} \tag{4} \end{equation*}

where $L=2.50 \times 10^{6} \mathrm{Jkg}^{-1}$ is the specific latent heat of vaporisation of water, $M$ is the molar mass of water and $R$ is the molar gas constant. At any given temperature $T$, the equilibrium pressure exerted by water in its vapour phase is called the saturation pressure, $p_{\mathrm{sat}}(T)$.
(c) Using the triple point $A$, indicated on Figure 5, to provide the relevant initial conditions, show that:

\begin{equation*} \ln \left(\frac{p_{\mathrm{sat}}}{p_{0}}\right)=\frac{L M}{R}\left(\frac{1}{T_{0}}-\frac{1}{T}\right) \tag{5} \end{equation*}

where $p_{0}$ and $T_{0}$ are the pressure and absolute temperature at the triple point, respectively.
(d) Calculate the saturation pressure of water at $20^{\circ} \mathrm{C}$.

Our atmosphere can be effectively modelled as an air/water-vapour mixture of ideal gases. The contribution to the total atmospheric pressure due to the water vapour is called the partial pressure of water, denoted $p_{\mathrm{w}}$.

The relative humidity $\phi$ of a parcel of air is defined as the ratio of the partial pressure of the vapour to the saturation pressure, with all quantities evaluated at the temperature of the parcel:

\begin{equation*} \phi(T)=\frac{p_{\mathrm{w}}(T)}{p_{\mathrm{sat}}(T)} \tag{6} \end{equation*}

(e) With reference to Eq. 6, explain what happens to the relative humidity of an air parcel that rises from the Earth's surface and hence explain how convective clouds form in the lower atmosphere.

For the rest of the question, take the air at sea level to be at $\mathbf{2 0} \boldsymbol{\circ} \mathbf{C}$ and $\mathbf{3 5 \%}$ relative humidity.
(f) Calculate the partial pressure of water vapour in air at sea level. Hence determine the number of water molecules in $1 \mathrm{~m}^{3}$ of air under these conditions.
(g) Determine the mass fraction of water in the air, making assumptions about the chemical composition of the atmosphere.
(h) Use the following atmospheric data to estimate the altitude at which cloud formation will occur.

(i) Using your understanding developed in the previous parts of the question, and without calculation, suggest the probable atmospheric conditions in Britain when air approaches from each of the three directions, A, B and C, indicated in Figure 6.

(a). Explain why a heavy balloon inflated with helium- 4 gives a lower reading on an accurate weighing scale than the same balloon when deflated, despite having a larger mass.

A party balloon of mass 10 g is inflated with helium-4.
(b). Show that, under typical atmospheric conditions, the volume of the balloon $V$ should satisfy $V \gtrsim 1.0 \times 10^{-2} \mathrm{~m}^{3}$ if it is to float.

The balloon is initially inflated to a volume $V_{0}=2.0 \times 10^{-2} \mathrm{~m}^{3}$, then released. It floats up gently before coming to rest on the ceiling. Unfortunately, there is a leak, and the balloon slowly deflates through a very small hole of area $A$.
(c). Show that the root mean-square momentum $P_{\mathrm{rms}}$ of a particle in a pure ideal gas of temperature $T$ is

$$P_{\mathrm{rms}}=\sqrt{3 m k T}$$

where $m$ is the particle mass. Hence reason why the typical momentum in any given direction is

$$P=\sqrt{m k T}$$

(d). The pressure of a gas can be loosely thought of as the total momentum flowing through unit area per unit time. Making the approximation that all particles travelling in a given direction have momentum $P$, explain why $N$, the number of atoms of helium-4 in the balloon, decreases over time as:

$$\frac{d N}{d t}=-\frac{p A}{\sqrt{m k T}}$$

(e). Hence obtain an expression for the rate of change of the volume, $\frac{d V}{d t}$, in terms of $A, k, T, m$.
(f). If after 1 hour the balloon has shrunk by $5 \%$ in volume, estimate the time until the balloon drops back to the floor, and estimate the area of the leaky hole. How does this compare to the initial surface area of the balloon?
(g). Suggest why the model presented above would break down if the hole was made significantly larger.
(h). The balloon was itself filled from a rigid, pressurised cylinder of helium-4 of fixed volume $v$. If this cylinder also develops a small leak (of area $a$ ), derive an algebraic expression for the gas pressure inside the cylinder as a function of time. Take the initial pressure to be $p_{0}$. No numerical calculations are needed.

This question explores the motion of individual particles within a gas. This is an example of a random walk.
(a) Calculate the typical (root mean-square) speed of nitrogen molecules ( $N_{2}$ ) at room temperature. Assume that $N_{2}$ acts as an ideal gas. The molar mass of $\mathrm{N}_{2}$ is $28 \mathrm{~g} \mathrm{~mol}^{-1}$.
(b) Show that the density $\rho$ of any substance composed of $N$ particles occupying a total volume $V$ is given by

\begin{equation*} \rho=\frac{M_{R} N}{N_{A} V}, \tag{2} \end{equation*}

where $M_{R}$ is the molar mass and $N_{A}=6.022 \times 10^{23} \mathrm{~mol}^{-1}$ is Avogadro's constant. Hence, find an expression (in terms of density) for the average volume occupied by each particle.
(c) Under standard conditions, liquid nitrogen $\left(N_{2}\right)$ has a density of $807 \mathrm{~kg} \mathrm{~m}^{-3}$. Estimate (crudely) the diameter of a molecule of $\mathrm{N}_{2}$, stating any assumptions you make.

For an ideal gas of $N$ particles occupying a total volume $V$, the typical distance moved between collisions by molecules of diameter $d$ is known as the mean free path $\lambda$. This is given by:

\begin{equation*} \lambda=\frac{V}{\sqrt{2} \pi d^{2} N} . \tag{3} \end{equation*}

We can also define the collision time $\Delta t$ as the typical time from one collision to the next.
(d) Determine $\Delta t$ for nitrogen molecules $\left(N_{2}\right)$ at room temperature and pressure $p=1.01 \times 10^{5}$ Pa.

With each collision the direction of travel of a given molecule is effectively randomised. As such, any individual molecule executes a random walk through the gas, an example of which is shown in Fig. 2.

For simplicity, consider gas molecules moving in two dimensions (2D) and make the approximation that the speed $v$ of any given molecule is constant in time. Moreover, assume that

the time between collisions $\Delta t$ is constant from one collision to the next. Due to the exchange of momentum and the random nature of collisions, these assumptions are clearly false in detail. Nevertheless, they will yield instructive results.
(e) Suppose a given molecule undergoes $M$ collisions in a total time $t$. After collision $i$, its velocity vector $\mathbf{v}_{i}$ points at an angle $\theta_{i}$ measured anticlockwise from the positive $x$ axis, as shown in Fig. 3. Write down an expression for $\mathbf{v}_{i}$ in terms of $\theta_{i}$, the molecular speed $v$ and the conventional unit vectors $\hat{\mathbf{x}}, \hat{\mathbf{y}}$, which are aligned with the positive horizontal and vertical axes, respectively.
(f) Briefly explain why, after time $t$ and $M$ collisions, the vector displacement of the molecule $\mathbf{r}(t)$ is given by

\begin{equation*} \mathbf{r}(t)=\sum_{i=1}^{M} \mathbf{v}_{i} \Delta t \tag{4} \end{equation*}

(g) With reference to your answer to (e), argue why on average the vector displacement is zero.
(h) By considering the average value of $\mathbf{r} \cdot \mathbf{r}$, determine an expression for the expected magnitude of the molecular displacement after $M$ collisions.
(i) Assuming the expression you have just derived also applies in three dimensions (3D), estimate the time taken for a nitrogen molecule to diffuse a distance of 5.00 m from its starting point. Assume room temperature and pressure $p=1.01 \times 10^{5} \mathrm{~Pa}$, and leave your answer in a sensible unit.