Structure and Properties of Materials

1

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 L 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, T 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

conduction, convection, and radiation

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-1andkCu = 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.

Radiation

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.