PHYSICS 1004

ENVIRONMENTAL & LIFE SCIENCES

STRUCTURE AND PROPERTIES OF MATERIALS

Thermal physics

Lecture Notes

By

Joe Khachan

1. Introduction

Matter may exist in the solid, liquid or gaseous states. Although on the microscopic level all matter is made up of atoms or molecules, everyday experience tells us that the three states have very different properties. This set of lectures is aimed at examining these properties and the underlying physics behind them.

Solids are composed of atoms held together by attractive or cohesive forces. If the cohesive forces are strong, the atoms are tightly bound to one another and the matter is in the solid state. If the cohesive forces are weak and the atoms have considerable movement with respect to each other, the matter is in the liquid or gaseous state. So the further apart the atoms are from each other, the smaller the cohesive forces.

Temperature greatly affects the average position of the particles (atoms or molecules) with respect to each other and so determines whether they are going to be solid, liquid or gas.

For example:

If water is cooled in the fridge it turns to ice because the kinetic energy of the water molecules becomes less than the cohesive bond energy and so water turns to a solid (ice). If on the other hand we "heat up" the water, which means we make the kinetic energy much greater than the cohesive energy, then water turns to gas or water vapour.

Many solid properties depend on:

(i) Temperature

(ii) Interatomic bonds

For example:

Diamond conducts heat four times faster than copper at room temperature. If you can make a defrosting board out of a giant slab of diamond, then your frozen chicken will defrost four times faster (in theory) !

Another example:

Liquid nitrogen boils at a temperature of -196 ºC. If a fresh rose is dunked into the liquid nitrogen and then removed, it can be smashed so that it will shatter like glass. That is, it's mechanical properties are temperature dependent.

In these lectures we will look at the thermal and mechanical properties of matter and how they relate to the behaviour of atoms on the microscopic level. At this point we should briefly describe the difference between mechanical and thermal properties.

Mechanical properties are the response of matter to applied forces. These properties are controlled largely by the interatomic forces or the interatomic potential energy. Thermal properties are the response of matter to applied heat or sources of different temperature. These properties are controlled largely by interatomic motions or kinetic energy. Most of the mechanical and thermal properties of matter are adequately described by classical mechanics, such as potential and kinetic energy. So it is assumed that you have some knowledge of these concepts.

2. Thermal properties of materials and temperature measurement

Before discussing the thermal properties of matter, we'll need to define some general thermal concepts, such as: the difference between heat and temperature, thermal equilibrium, the zeroth law of thermodynamics, and the absolute temperature scale.

Temperature and Heat

In everyday language we use the terms heat and temperature loosely as if they had the same meaning. In physics they have different meanings. Consider the following example:

Example:

Take a beaker half filled with water and place some ice in it. Put a thermometer in the water and wait till the temperature of the water becomes stable so that the temperature of the water and the ice are the same. Now place the beaker over a Bunsen burner and start heating it. You'll notice that the temperature of the water stays the same as long as there is ice left. We all agree that the flame is heating the water but the thermometer says that the temperature does not change. Once all the ice melts, the temperature of the water starts to rise. From this we can see that we'll need to closely examine our ideas about the meanings of heat and temperature.

We'll examine these concepts in more detail later but for the moment, in a nutshell:

(1) Temperature is related to the average kinetic energy of the particles (atoms or molecules).

(2) Heat is the amount of energy transferred to a system of particles

In the above examples, we were transferred heat to the system, which in turn melted the ice but the temperature did not change!

Thermal Equilibrium

Thermal equilibrium is simply another way of saying that two or more objects are at the same temperature.

Example: You and I have never met. Not even shaken hands. Yet if we are in good health you can bet that our body temperatures are at 37 ºC. We are both in thermal equilibrium. Ignoring the fact that our extremities (e.g hands, feet and nose!) may be colder than the rest of our body.

This is sometimes called the zeroth law of thermodynamics. The reason for this is that physicists first found the first and second laws, then realised that there is a more fundamental law so they decided to give it the number zero. More formally the law can be quoted as follows:

zeroth law of thermodynamics: If object A and object B are in thermal equilibrium with object C, then they are in thermal equilibrium with each other.

Absolute Temperature scale

There is a physical lower temperature limit of matter. Nothing can be cooled below -273.15 ºC. So for convenience, scientists have devised the absolute temperature scale which starts with -273.15 ºC and called it 0 Kelvin (not degrees Kelvin!). So the relationship between Celsius and Kelvin is:

TK=TC+273.15                                                             (1)

where TK is the temperature in Kelvin, and TC is the temperature in Celsius.

Example: Ice freezes at 0 ºC or  TK= 0 +273.15= 273.15 Kelvin.

Normal room temperature is at 20 ºC or TK = 20 + 273.15 = 293.15 Kelvin

How to make a thermometer

To measure temperature we have to measure another macroscopic quantity that is directly influenced by temperature. There are many ways to do this. Thermometers use physical properties ranging from electrical resistance to radioactivity. But the oldest of all is the mercury thermometer. All materials change their physical dimensions when heated or cooled. The change in length is a direct measure of temperature.

To make a thermometer, we fill a thin glass tube with mercury (some use alcohol). Place the tube in ice that's been sitting in the room for a while and starting to melt. We then note the position of the mercury column on the glass tube. We can call this level whatever we like, but we choose to call it 0 ºC. We then place the tube in boiling water and watch the mercury column expand to another length. We then call this level 100 ºC for convenience. We can then divide the length between  the two positions into 100 equal segments and call each one 1 ºC. Note that there is nothing special about this way of putting marks on a thermometer. It is simply for human convenience. We could have equally chosen a material other than water to calibrate our thermometer. We could have chosen wax! That it, we could have chosen 0 ºC to be the temperature when wax just starts to melt, and 100 ºC when it boils. There is also no need to call one 0 and one 100 such as the Fahrenheit scale which we will not be using here.

Thermal Expansion

When a material is heated or cooled, it changes its dimensions. Generally, it expands when heated and contracts when cooled although there can be exceptions to this rule.

For Example:  The Eiffel Tower can extend by as much as 12 centimetres if the temperature increases by 40 ºC. So if you go to Paris in the summer and stand on top of the Eiffel Tower, you will get a little more height for your money than if you had gone in winter.

Another example: If water is gradually cooled, it shrinks in size as expected. But at 3.98 ºC it begins to expand again until it turns to ice at 0 ºC. This expansion is peculiar to water and is associated with the unusual shape of the water molecule. This behaviour explains why lakes freeze from the top downwards in winter. The colder water is at the top of the lake because it expands and becomes less dense. So when this water freezes it insulates the water below it from the outside cold air like a blanket. It is because of this property many fish can survive the winter rather than becoming part of a giant Popsicle.

Yet another Example: When you first turn on a hot water tap, the water rushes out but is still cold. When it starts to become hot, the flow of water starts to become less and in some cases it stops. This can be explained as follows: the hot water heats the metal valve inside the tap which expands to block off any more flow of water.

The change in length of a solid is related to :

(1) the original length and

(2) the change in temperature

(2)

where DL is the change in length, L0 is the original length before the change, ∆T is the change in temperature, and  is the linear thermal coefficient of expansion, which is different for different materials. The following table gives expansion coefficients for some common materials.

Table 1

 Substance (10-6/ °C) Ice (at 0 °C) 51 Lead 29 Aluminium 23 Brass 19 Copper 17 Steel 11 Glass (ordinary) 9 Glass (Pyrex) 3.2 Invar 0.7 Fused quartz 0.5

For example, bone inserts such as pins, screws etc, used to treat broken limbs should ideally have the same expansion coefficient as bone. Do they?

Demonstration: Two strips of metal are riveted together. They both have different thermal expansion coefficients. This means that if they are heated to the same temperature they will expand to different lengths. But since they are riveted together the result will be that they bend. This is a useful property and is used in many applications that involve electricity and heat such as electric stoves where the current is switched off at a certain temperature. Another example is the automatic hot water kettle where it switches itself off when the water boils.

Expanding concrete: A concrete slab has a length of 12 m at -5 ºC on a winter's day. What is the change in length from winter to summer, when the temperature is 35 ºC ? The linear expansion coefficient of concrete is 1 X 10-5 ºC-1.

Solution:  Change in length is given by equation 2

= (1 X 10-5 º C-1)(12 m)(40 ºC)

= 5 X 10-3 m

= 5 mm

Adjacent concrete slabs in highways and sidewalks are often separated by pliable spacers to allow for this kind of expansion.

In reality, materials expand in 3 dimensions so equation 2 is modified to the following:

(4)

where  is the change in volume, V0 is the original volume, DT is the change in temperature,  is the volume expansion thermal coefficient and is related to  by:

(5)

Demonstration: The ball and ring. A metal ball barely fits through a metal ring when both are at room temperature. If the temperature of the ball alone is increased, it will not fit through the ring. If the temperatures of both the ball and the ring are increased, the ball again fits through the ring. This shows that when the ring expands, the size of the hole increases.

Thermal Expansion at the atomic level

At the atomic level, thermal expansion means there is an increase in the average spacing between atoms. As a particular atom oscillates about its equilibrium position it experiences an asymmetric potential energy as shown in figure 2. If it moves towards another atom it experiences a very steep rise in the potential energy. Whereas if it moves away from the other atom, it experiences a relatively slow increase in the potential energy and so travels much further. The asymmetry in the potential energy curve leads to a shift in the average position of the atom.

3. Concept of heat and heat transfer in the environment

Direction of heat transfer

Heat is the energy transferred to an object and is measured in joules. If two objects at different temperatures are placed in contact, heat will flow from the higher to the lower temperature object. This is called Conduction.

This is sometimes not obvious: Like when you shake hands with a person with cold hands. The conclusion that many people make is that cold has travelled from that person to you. It is only heat that travels. The coldness that you feel is simply the heat leaving your hand.

Simple Experiment:  Put a block of wood and a bowl of water in the fridge. Allow the water to freeze. Then take both of them out and feel them. Which feels "colder"? Most will say the ice. So which has the lowest temperature. If you say the ice, then you are wrong! They both have the same temperature. It feels colder because the ice conducts heat faster than wood. What you feel as "colder" simply means there is more heat leaving your hand every second than when touching the wood.

So our concept of hot or cold does not just depend on temperature but also on how fast heat travels in different materials.

So how fast does heat travel?

Heat travels at different rates in different materials. The quantity of heat transferred per unit time (in other words the rate of heat transfer) is given by:

(7)

where k is the thermal conductivity, A is the cross-sectional area, L is the length of the object, TH is the higher temperature at one end of the solid, Tc is the lower temperature at the other end.

Demonstration: Three metal strips of the same length are heated by the same flame at the same time. Matches placed at the end of these strips do not light up at the same time. The reason is that the three metal strips are made from 3 different materials: stainless steel (k=14 W/mK), copper (k=401 W/mK) and Brass 220 (W/mK). Since copper is the most conducting, the match on it will light up first and so on.

When is something neither Hot nor Cold?

Answer: When there is no heat transfer between you and the object. That is when H = 0 i.e when the object is at the same temperature as your hand.

Example: The "Wonder Defrosting Board" is made of metal that conducts heat towards the food faster than a block of wood.

Example: An aluminium pot contains water that is kept steadily boiling (100 ºC). The bottom surface of the pot, which is 12 mm thick and  in area, is maintained at a temperature of  by an electric heating unit. Find the rate at which heat is transferred through the bottom surface. Compare this with a copper based pot. The thermal conductivities for aluminium and copper are kAl = 235 Wm-1K-1 and kCu = 401 Wm-1K-1 respectively.

Solution:

The following is a schematic diagram of the pot.

The rate of heat conduction across the base is given by equation 7.

For the aluminium base:

TH = 102 ºC, TC = 100 ºC, L=12 mm = 0.012 m, k = kAl = 235 Wm-1K-1

Base area = A =  = 0.015 m2.

Substituting these into the above equation:

Js-1 (or Watts)

For the copper base k = kCu = 401 Wm-1K-1. So the rate of heat conduction across the base is

Js-1 (or Watts)

So the copper based pot transfers 1.7 times more energy every second compared with the aluminium pot. Generally copper bottom pots are more expensive. Are their prices 1.7 times those of similar aluminium pots? Or this a simplistic way of looking at it?

Conduction across composite materials

What if the thickness of a solid was made of several layers of different materials (i.e a composite material). How do we work out the rate of heat transfer H? We can make our job easier by defining another term: thermal resistance R.

where L is the length of the solid, and k is the thermal conductivity. So equation (7) now becomes:

If we have multiple slabs in series

The rate of heat transfer, H, can now be written as:

where R1, R2, etc are the thermal resistances of materials 1, 2. etc. This makes calculations for composite slabs so much simpler.

Convection

Convection is the transport of heat by the movement of liquids or gases. People make use of this when they go hot air ballooning. Hot air rises because it expands when heated and therefore becomes less dense. The hot air is then captured by the balloon. The volume of the balloon is chosen so that the buoyancy force on it is larger than the weight of the balloon and the weights attached to it (that includes people). So the balloon rises.

Rising hot air eventually cools, which means now it is more dense and can start falling again. But it can't go straight down since there is rising hot air below it. So it shifts sideways then starts to fall. Air circulating in this way is called a convection current. This is only one special case. The same phenomenon occurs in liquids.

Energy is transferred by electromagnetic radiation. All of the earth's energy is transferred from the Sun by radiation. Our bodies radiate electromagnetic waves in a part of the spectrum that we can't see called the infra-red. However, there are some cameras that can actually see this radiation.

The colour and texture of different surfaces determines how well they absorb the radiation.

(1) Black objects absorb more radiation than white objects.

(2) Matt and rough surfaces absorb more than shiney and smooth surfaces.

If you are ever in the snow, take a black and a white piece of cardboard, both the same size. Lay them down on the snow side by side. Over time you will notice that the black cardboard sinks deeper into the snow because it absorbs more heat from the sun and therefore melts more snow underneath it. You will notice this effect if you wear a black jumper and sit in the sun. You become warm more quickly than if you wore other coloured jumpers.

Curiosity: It is interesting to note that aluminium foil has two different surfaces. One side is shiney and the other is matt. If you want to heat something evenly and quickly then you wrap it up with the matt side on the outside. If you want to keep something cold then the shiney side must be on the outside to lessen the effect of heating by radiation. Do you think that's what the manufacturers intended? Is it really that important, or is it a small effect?

Greenhouse effect.

In a greenhouse, the incident radiation from the sun readily passes through the glass. Inside, the radiation is absorbed by objects as well as the air, but the air cannot escape since it is all enclosed. Consequently, its temperature starts to rises. More radiation coming in means higher air temperature. This can also happen in a car if all the windows are closed. In this case temperatures can reach incredibly high values. People and pets who have been left in cars with the windows closed have died from the high temperatures. So mainly the glass prevents convective losses by stopping the upward flow of warm air.

In a sense, the earth is a huge greenhouse, with the atmosphere replacing the glass. The water vapour and carbon dioxide in the air are good absorbers of infra-red radiation. Sunlight passes through the atmosphere and heats up the Earth and the atmosphere. The Earth then emits the radiation out again, but in a different part of the electromagnetic spectrum i.e the infra-red. As it turns out, the infra-red does not readily pass through certain gases such as carbon dioxide, water vapour, and methane. So these gases trap the energy from the emitted radiation which in turn warms up the atmosphere to the comfortable temperatures that we experience. This is the Greenhouse effect and it actually is a beneficial phenomenon that helps to support life on Earth. Otherwise the Earth will be as cold and baron as the moon. The problem arises when you put a lot of these greenhouse gases into the atmosphere (too much of a good thing in this case is a bad thing for life forms such as us). The burning of fossil fuels increases the carbon dioxide level. This means that more of the infra-red, which is radiated by the earth, will be absorbed by the atmosphere. This could increase the average temperature of the earth and lead to major climatic changes. Some of which may not be in our best interests!

4 & 5. Concept of heat capacity, specific heat, and heat of transformation

Heat Capacity

If you light a match and put the flame to a small object such as a needle or a pin then you know that after a few seconds you will not be able to hold the needle because it has become too hot. On the other hand, if you put the flame of the match to a massive object such as a truck (well away from the petrol tank!!), then instinctively you know it will not make one bit of difference to its temperature. Although in both cases the temperature of the flame is the same. So somehow the mass of an object is related to the temperature it will reach in a certain time.

In the olden days people defined the concept of heat capacity because they thought that somehow an object can have a capacity for being "filled" with heat just as a bucket has a capacity for holding water. This is not a correct point of view since it would be theoretically possible to keep transferring  heat to the object without limit. Although in practice the object might eventually vaporise.

However, we will define it here anyway:

The ratio of the energy transferred and the change in temperature is called the heat capacity

where DT=Tfinal - Tinitial. Where Q is the amount of heat transferred.

Specific Heat

It's a little cumbersome to use heat capacity since this "constant" keeps changing as the mass of the object changes. So people went further and defined something slightly better. The Specific heat. This is the number of joules required to raise 1kg 1 Kelvin. Specifically:

where c is the specific heat and m is the mass. So now the amount of heat transferred Q is given by

where the c is expressed in units of J .kg-1 K-1

Another way of expressing this relationship is in terms of moles. That is

where n is the number of moles and c, in this case, is the molar specific heat expressed in units of J mole-1 K-1.

The specific heat is a constant for a particular material. It does NOT depend on mass. We can use this relationship to check out our microwave ovens.

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Demonstration: Manufacturers claim that microwaves give out a certain amount of power, usually around 700 or 800 watts. By placing a litre of water in the microwave and heating it for a certain time, by measuring the temperature of the water before and after heating we can work out whether the manufacturer's claim is true.

Power = (Energy Transferred)/time

for 1 litre of water the mass is 1 kilogram, specific heat is c = 4190 J/kg.K.

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Example: 3 equal masses of Al (c=900 J/kg.K), Cu (c=386 J/kg.K), Pb (c=128 J/kg.K). All three masses are heated to the same temperature. If each mass was plunged in a separate beaker containing the same amount of water as the other beakers, then wouldn't we expect the temperature of the water in the three beaker to be raised to the same final temperature. Well many people would, however a glance at the equation

will show that the amount of heat transferred, Q, is dependent on the size of the specific heat c. The smaller the c, the smaller the amount of heat transferred and since all the beakers contain the same amount of water, m, the smaller the temperature change.

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Demonstration: The specific heat of Aluminium is 900 J/kg.K, Copper 386 J/kg.K, Iron (Fe) 447 J/kg.K. This means that less heat is required to heat a copper cooking pan than a steel or aluminium one of equal mass. If you recall that copper is a much better thermal conductor than either aluminium or iron. So it is much more energy efficient to buy an all copper pan. So take note of this the next time you are shopping for cookware.

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Heats of Fusion and Vaporisation

From the equation containing the specific heat, it seems that we can keep on transferring heat to an object and raising the temperature indefinitely. In fact, the equation seems to suggest that if we put in an infinite amount of heat we well get an infinite increase in temperature. However, instinctively we know that this is not totally true since at 100 ºC water starts to boil and stays at this temperature until it changes to a gas. So the equation doesn't fully account for changes of state, which are also known as phase changes.

In general, the temperature stays constant during any phase change. That is, from solid to liquid and liquid to gas, although energy (i.e heat) is still transferred.

There are some definitions we must add at this stage. These are:

Melting point - temperature at which a solid turns to a liquid or vice versa.

Boiling point - temperature at which a liquid turns to a gas (or vapour) or vice versa.

These points can be changed by adding impurities to the water / ice.

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Example: Put a piece of string on an ice cube, which is straight out of the fridge. Sprinkle some salt on the string/ice cube. Wait a few seconds, you will then be able to lift the ice cube by the string. The string seems to have become glued to the cube.

The explanation is this: The salt lowers the freezing temperature of ice. That is, the ice has a good chance of melting if it isn't too cold. So the melted ice soaks into the string. The salt now becomes less concentrated as it diffuses out of the region of the string so the freezing temperature is raised again. Since the rest of the ice cube is still at a temperature below freezing, the water will re-freeze including that which has soaked into the string. So the string will essentially be frozen to the cube.

An almost similar thing happens when you try and lick an ice tray. The tray is say at -15 ºC so it freezes the moister on your tongue.

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Phase diagram description of water

Consider the phase diagram of water given below. This type of diagram is very common in materials science and helps us determine the phase of the material for different temperatures and pressures. There are three regions: solid, liquid, and vapour, separated by boundaries. How do you read this diagram? As an example, say you were trying to find out whether water is a solid, liquid or gas at a temperature of 50 ºC and pressure of 50 kPa. Draw a line up from 50 ºC and a line across from 50 kPa; the intersection is in the liquid region of the diagram.

Using the same diagram, draw a vertical line up from 0 ºC. You'll notice that you will eventually cross from the solid to the liquid region. This means that as you increase the pressure on ice, it will eventually turn to liquid. This is useful if you're an ice skater because it means that the increase in pressure, due to your body weight, melts the ice underneath the blade which helps you glide more smoothly.

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Believe or not: Mount Everest is the highest structure and there is a rumour that it can't be any higher. The reasoning is as follows: It is 10 km high, if it was any higher the pressure at the base of it will be enough to turn the rock into a liquid and so prevent it becoming higher. Do you believe that?

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How pressure cookers work: These are essentially cooking pots with an air tight lid. As water is heated and turns to vapour, the pressure builds up because the vapour cannot escape. Water usually boils and turns to steam (vapour) at 100 ºC at a pressure of 1 atmosphere (i.e 101 kPa). By increasing the pressure we can see from the phase diagram that the water can be heated to higher temperatures before it enters the vapour region. This means that anything being cooked inside the pot will experience higher temperatures and therefore the cooking time will be less. But will it taste as good?

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Tea on Mount Everest

The air pressure on top of Mount Everest is about half an atmosphere (i.e 50 kPa). The phase diagram, tells us that water will boil (and turn to vapour) at about 70 ºC. Unfortunately this water temperature makes a very poor cup of tea!

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Energy is acquired or released when a material changes phase. For example, energy is required to melt ice and vaporise water. However, energy is given out if water vapour condenses or water freezes. The heat acquired or released is called the latent heat. During a phase change there is no change in temperature so we cannot use the equation containing the specific heat to determine the amount of heat transferred. The formal definition of latent heat is the energy given out or absorbed without a change in temperature and is given by

Q is the heat, m is the mass, L is a constant for a certain material and is called the Latent heat of fusion (for melting) or vaporisation (for boiling).

Latent heat of fusion is the energy required to melt 1 kg of a solid. The units are J.kg-1.

Latent heat of vaporisation is the energy required to evaporate 1 kg of a liquid. The units are J.kg-1.

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Example:

A styrofoam picnic chest has dimensions 0.5 x 0.3 x 0.35 m with walls 0.02 m thick.

(a) If the temperature difference across the walls is 35 ºC, at what rate will the heat be conducted out of or into the chest.

(b) How many kilograms of ice do you need to put in the chest to keep the beer at 0 ºC for 5 hours?. Assume the beer was initially cooled to 0 ºC.

The thermal conductivity of Styrofoam  ksty = 0.010 Wm-1 K-1

Latent heat of fusion of ice  Lv = 333 kJ.kg-1

Solution:

(a) We are told there is a temperature difference across the walls of the chest. It doesn't matter which side of the chest is at the higher temperature since the rate of heat transfer only depends on the temperature difference.

Rate of heat transfer  H =

where A is the surface area of the box,  is the temperature difference across the walls, x is the wall thickness.

The box has six sides, we must now calculate the total area

A = 2 x (0.5 x 0.3) + 2 x (0.3 x 0.35) + 2 x (0.35 x 0.5) = 0.86 m2

is given as = 35 ºC.

The thickness is given as x = 0.02 m.

Substitute these into the rate of heat transfer equation above:

= 15.1 Js-1

So the rate of heat transfer across the walls of the chest is 15.1 Js-1,  or equivalently we can say 15.1 Watts.

(b) In other words, the question is saying "what is the least amount of ice that would melt after 5 hours".

From part (a) we know that 15.05 Joules of energy can enter the chest in 1 second. So we must work out how much energy enters in 5 hours. Then calculate how much ice this can melt.

Note that 5 hours = 5 x 3600 seconds = 18000 seconds.

So total energy in 5 hours = 18000 x 15.05 Joules = 270900 J = 271 kJ

The energy, Q,  required to melt ice is given by the latent heat of fusion equation:

Q = m Lv

where m is the mass of the ice that will melt. We can now calculate this mass by rearranging the above equation and substituting.

m =

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___________________________________________________________________________Another example:

How many 20 g ice cubes, whose initial temperature is -10 ºC, must be added to 1.0 L of hot tea, whose initial temperature is 90 ºC, in order that the final temperature of the mixture be 10 ºC? Assume all the ice melts in the final mixture and the specific heat of tea is the same as that of water.

Latent heat of fusion of ice = Lv = 333 kJ. kg-1 = 333000 J. kg-1

Specific heat of water = cwater = 4190 J. kg-1 K-1

Specific heat of ice = cice = 2100 J. kg-1 K-1

Assume that the tea has the same properties as water. Note that 1 litre of water has a mass of 1 kg.

Solution:

Let mice be the mass of ice required, mtea be the mass of tea = mass of water with the same volume = 1kg . Use conservation of energy. The energy required to melt the ice, then heat it to 10 ºC, must come from the tea.

The following are the three stages the ice must go through to reach the final temperature of 10 ºC:

(i) ice heats up from -10 ºC to 0 ºC

(ii) ice melts at 0 ºC

(iii) melted ice heats up from 0 ºC to 10 ºC

(iv) energy for stages (i), (ii), and (iii) must come from tea.

Now conserve energy: i.e

Energy required for (i) + (ii) + (iii) = Energy lost by tea

The expressions for the different stages are given by

(i) mice cice  DT(i) = mice (2100 J. kg-1 K-1)  (0 ºC - (-10 ºC)) = mice 2100 x 10 = 21000 mice

(ii) mice Lv  = mice 333000

(iii) mice cwater DT(iii) = mice (4190 J. kg-1 K-1) (10 ºC - 0 ºC) = mice41900

(iv) mtea cwater DTtea= (1 kg) (4190 J. kg-1 K-1) (90 ºC - 10 ºC) = 335200

Using the conservation of energy equation above we have

21000 mice + mice 333000 + mice41900 = 335200

Now solving for mice we get mice = 0.847 kg = 847 g of ice are needed. Divide this by the mass of one ice cube (20 g) to find out how many cubes are needed.

847  20 » 43 ice cubes.

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6. First Law of Thermodynamics and its applications to thermal processes

In the late 1700s, Sir Benjamin Thompson, (otherwise known as Count Rumford) who was an army officer, noticed that a great deal of heat was generated when boring out the hole in a cylinder of iron in order to make a cannon. In fact, enough heat was generated to boil water. From his private research on heat generated by friction he came to the conclusion that heat was a form of motional energy, as opposed to the then current view that it was a material substance. He was right, we now know that it is kinetic energy transferred to the atoms and molecules of a solid, liquid or gas. As any boy or girl scout will tell you that it is at least in principle possible to start a fire by twisting the end of a dry stick on dry leaves placed on a dry log.

There is something different about generating heat by friction. Up till now we have been dealing with energy transfer from a high to low temperature region. It became evident that energy could not be transferred without a temperature difference. But now we find that we can heat up a material by friction without the need to talk about temperature differences. All this means that we must broaden our view on where the kinetic energy of the atoms comes from. In the case of starting a fire by the frictional heating of a stick, it is the work done by the boy/girl scout on the stick/leaves/log system. Work here is as physicists define it i.e force multiplied by distance.

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Curiosity

When you pump up the tyre of a bicycle very quickly, you will feel  that the valve of the tyre is very hot. That energy came from the work that you did in pushing air through the valve. In this case, the system is the valve/tyre, and the work was done by you. If however you press on the valve to let the air out of the tyre, you will notice that the valve becomes very cold. In this case the valve/tyre system was itself doing work on the air inside it.

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Absurdity

Theoretically, it should be possible to boil a jug of water by vigorously beating it (i.e doing work) with an egg beater. But I don't think anyone is that eager to try it considering the length of time it will take.

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At this stage we should define something called The System. The system is a general term which refers to an object or collection of objects either solid, liquid, or gas, or a combination of these. In the case of the bicycle tyre being heated by the pump, the system was the valve and tyre. In the case of making a fire by friction with a stick, the system was the stick, leaves, and log.

Conversely, the system can do work on its environment such as letting air out of the bicycle tyre by pressing on the valve. You'll notice that the valve becomes cold as energy is transferred from it to the expanding gas.

The First Law of Thermodynamics essentially states that the energy transferred to or from a system must be conserved. This is generally encapsulated by the following relationship:

DEint = Q - W

where DEint is the internal energy, Q is the heat added to a system, and W is the work done by the system. In other words, the net energy transferred to the system, Q - W, equals the change in the internal energy, DEint. Internal energy is simply the sum of the kinetic and potential energies of all the molecules of the system. For the applications that we will be looking at, an increase in the internal energy of a system means that its temperature will rise. A decrease in the internal energy corresponds to a decrease in temperature. Just keep in mind that in a more general situation, these last two statements may not necessarily be true.

Note that work, W, can be done on or by the system. Heat, W, can be added to or given out by the system. To take these into account we adopt a sign convention to be used with the above equation. That is:

Q > 0 if heat is added to the system

Q < 0 if heat is given out by the system

W > 0 if work is done by the system

W < 0 if work is done on the system

Let's make this a little more meaningful by applying it in a qualitative way to the following:

Example:

When you pop open a champagne bottle, a little fog develops in the opening. This is because the gas inside expands so quickly that heat from the outside can't get to it fast enough to supply the energy needed for expansion, so it uses up it's own internal energy. This means that the kinetic energy of the gas particles has been reduced, which implies that the temperature has also been reduced. The reduction in temperature can be low enough to condense the moisture in the air in the opening of the bottle, which forms the little fog.

In terms of the first law of thermodynamics, we say that the gas in the bottle has done work on its environment by expanding. This means that W has a positive value in the above equation. Since no heat enters in this short time we can set Q = 0. So the first law of thermodynamics equation now becomes

DEint= 0 - W = - W

This means that the change in internal energy is negative i.e a reduction in internal energy, therefore a reduction in temperature.

The process whereby no heat enters or leaves a system is known as an adiabatic process and is very important in many applications.

For example it is the principle on which refrigerators works. Your refrigerator has a system of pipes in the back that contain a gas (which used to be freon before the ozone depletion problems) that is compressed in narrow pipes in the back of the refrigerator. Just like the heat produced when you compress air through a bicycle tyre valve, the narrow pipes heat up when the gas is compressed in them by the compressor of the refrigerator (just feel the pipes in the back and you'll find that they are warm). The compressed gas is then aloud to expand quickly through a larger diameter pipe that is embedded in the back of the freezer. This adiabatic expansion reduces the internal energy of the gas and therefore its temperature. How cold does the pipe become? That pipe is what cools the top chamber of the refrigerator i.e the freezer. The result is the sub-zero temperatures in the freezer.

7. Concepts of temperature and pressure in terms of molecular properties

An ideal gas - a macroscopic approach

In the last section we talked about the work done on a system, but what if the system is a gas? What is the work done in compressing a gas? The parameters associated with a gas are pressure and volume. So how do we get force times distance out of these? Answer: work done on a gas is simply pressure times volume and is related to the temperature of the gas by the ideal gas equation

PV = nRT

where P is the pressure, V the volume, R is a constant = 8.314  Jmol-1K-1, T is the temperature in Kelvins, and n is the number of moles defined as number of gas particles (atoms or molecules) divided by Avogadro's number (NA = 6.023 X 1023).

The above equation is counterintuitive since it says that all gases contain the same number of atoms or molecules for the same pressure, volume, and temperature regardless of the type of gas. This is definitely not the case for solids and liquids. Recall such thermal properties as linear expansion for solids and liquids. The expansion coefficient depends strongly on the type of material. However, we find that for gases this is not the case since the volume occupied by the gas does not rely on the type of gas. So for the first time we see that there are material properties which are material

independent.

Although the product of pressure and volume gives the work done on or by a gas, in practice this may not be a constant quantity throughout a particular experiment. For example a  plot of pressure versus volume may look something like this

So how can we measure the work done on a gas from this graph? Answer: the work is the area underneath the graph.

Just be careful to give the correct sign to the work. As we saw in the section on the first law of thermodynamics, the work done on a system is negative, and the work done by the system is positive. In this case, the gas is expanding, as indicated by the arrow on the curve, so the gas itself is doing work, which means the work done by it is given a positive sign.

Although the ideal gas equation is quite convenient for solving many problems, it is still a macroscopic approach and does not give us any deeper understanding of what pressure and temperature actually are. A little bit like driving a car but not knowing how it works. Let's take a quick look under the bonnet (or hood).

An ideal gas - a microscopic approach

In 1827 the English botanist Robert Brown discovered that pollen suspended in water shows a continuous random motion when viewed under a microscope (known as Brownian motion). At first these motions were considered a form of life, but it was soon found that small inorganic particles behave similarly. Until that time, there was a great deal of debate as to whether atoms actually existed. The random motion of the pollen gave indirect evidence that matter is made from discrete particles (i.e atoms) and that they are colliding in continuous random motion. An exact quantitative description of Brownian motion was finally given by Albert Einstein.

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Interesting experiment:

Suspend a mirror from a string in a chamber that has all the air taken out (i.e a vacuum). Now shine a light beam on the mirror so that it reflects onto a screen. Look at the light beam spot on the screen and you'll find that nothing out of the ordinary happens. Now let a very small amount of air into the chamber. You'll find that the spot on the screen is now moving in a random motion. What we are looking at is simply the random collisions of the air molecules with the mirror. We don't see this phenomenon when we do the experiment at normal air pressure because there are so many collisions from all sides of the mirror that it remains stationary. At low pressures there are not many molecules around so that there is an imbalance in the number of particles striking both sides of the mirror, so that it develops a net force in a particular direction. This experiment was carried out in 1827 by Kappler.

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So on the atomic level, if all that is happening is the random motion of atoms, then how can we describe concepts such as pressure and temperature?

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Did you ever do this? Close your mouth and blow out your cheeks. What holds out your cheeks? If you said pressure then basically you have said nothing since pressure is just a word we use to describe a more fundamental process. Your cheeks moved outwards because something pushed them. Keep in mind that the air in your mouth is made of atoms which are hitting the insides of your cheeks many times a second. So some how we must define pressure in terms of the rate at which atoms are hitting the inside of your cheeks.

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Let's examine what we mean by pressure. An atom heads towards a wall with momentum mv. It strikes the wall and returns in the opposite direction with momentum -mv. This change in momentum, mv - (-mv) = 2mv, divided by the collision time gives the force applied to the wall. Now imagine billions and billions of these collisions per second over a certain area of the wall. This is what we define as pressure i.e the force exerted by the change in momentum of the atoms divided by the area. It can be shown that the ideal gas equation can be rewritten in terms of more fundamental parameters

where n is the number of moles, M is the mass of one mole, and <v2> is the average of the speed squared of the gas particles. So for the first time we see that the ideal gas equation can be written in a form that depends on the microscopic parameters of atoms i.e their mass and velocity. But really, nothing has changed, the above equation must still give the same results as the more familiar ideal gas equation, which deals with macroscopic parameters such as temperature. So if we equate the two we have

PV(macroscopic) = PV(microscopic)

rearranging this we get

where vrms is known as the root mean square velocity, which is another way of defining the average velocity.

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How fast do gas atoms go?

What atomic speeds are we talking about: Find the root mean square speed of 1 mole of nitrogen molecules at a pressures of 101 kPa and temperature of 300 K. Note that 1 mole of nitrogen has a mass of 28 grams = 0.028 kg.

Solution:

This is faster than the speed of sound!!

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Now we are in a position to get the average translational kinetic energy of gas particles.

where m is the mass of one particle (atom or molecule). The molar mass M is defined as the mass of one particle multiplied by Avogadro's number = NA = 6.023 x 1023. That is

M = mNA

substituting this in the expression for kinetic energy we get

for convenience we write  as one constant i.e

k is known as Boltzman's constant.

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Boltzman's constant is named in honour of Ludwig Boltzman who began the branch of physics know as statistical mechanics. He combined classical mechanics with the statistical description of the motion of atoms as in Brownian motion. He also had theories about the irreversibility of many processes. A large section of the scientific community did not accept them mostly because they could not understand them. Finally, ill and depressed, took his own life on September 5 1906. Since then, all of his theories have been fully justified.

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From the above equation for kinetic energy, we can see that what we call temperature is simply the kinetic energy divided by a constant. So temperature is a direct measure of the kinetic energy of atoms and molecules.

Although we have talked about the average kinetic energy, let's look at the actual speeds of the atoms / molecules in a gas, which are given by the following

This is known as a maxwellian distribution of velocities. Note that there is a continuous distribution of velocities from zero to very high values. So some particles are not moving while others are well and truly supersonic. However, there are not too many particles at the extremes of the distribution. As we have seen, the average in the square of the speeds defines the temperature. Note that the peak moves to higher velocities as the temperature is increased.

This is the end of the thermal section of the course.