Heat is a measure of thermal energy stored in an object, and is measured in Joules with symbol Q. is the rate of change of heat, which is measured in Watts.

Temperature is a measure of how hot objects are and is measured in Kelvin or degrees centigrade. If in °C, the symbol is q
and if it is in K, the symbol is T. The unit for temperature is ** never** °K. Dq is the temperature change, and this has an effect on - . Heat always flows from hot objects to cold ones.

Thermal equilibrium – balanced rates of heat flow.

If the heat flow per second into an object equals the heat flow per second out then the object is in thermal equilibrium. Temperature is a bulk property, and is related to the average kinetic energy of the molecules. When the molecules have no kinetic energy, then they are stationary. At this point the temperature is 0K, or absolute zero. This is equal to –273.5 °C.

is the temperature gradient – the change in temperature per unit length.

Thermometers rely on the physical properties of matter to produce measurements of the temperature. Examples of this are:

- Thermal expansion of liquids – in a liquid in glass thermometer (there is a vacuum in the top)
- Electrical resistance in platinum resistance thermometers (long, thin wire) and thermistors (semiconductor material)
- Pressure of an ideal gas (constant volume gas thermometer)
- The thermo-electric effect (a junction between two dissimilar metals generates an EMF, which varies with temperature. This causes a small current to flow.

A few mV are produced across the 1m wire because of the driver cell. The temperature probe produces an EMF which can be balanced by adjusting the length, l, of wire. Thus the 1m wire can be calibrated to read temperatures directly.

A constant volume gas thermometer has a constant volume of gas in a large container, the pressure of which can be read using a manometer.

Type of thermometer |
Thermometric property |
Advantages |
Disadvantages |
Range |

Mercury in glass |
Length (as cross-section is constant) |
Convenient. Portable. |
Fragile. Limited range. |
234 - 723 K -39 – 450 °C |

Resistance |
Electrical resistance |
Wide range. Accurate. Good for small temperature differences. |
Slow to respond due to oil surround. |
90 – 903 K -183 – 630 °C |

Constant volume gas |
Pressure of a fixed mass at a constant volume. |
Wide range. Very accurate. |
Bulky. Slow to respond (high mass \ high heat capacity. |
-270 – 1500 °C |

Thermocouple |
EMF across the junction of two dissimilar metals. |
Wide range. Quick to respond. |
Needs a high resistance voltmeter. Problems with non-linearity. |
-250 – 1500 °C. |

All temperature scales need 2 fixed points. For the Celsius scale these are pure boiling water at atmospheric pressure (100°C) and pure melting ice at atmospheric pressure (0°C). The formula for calculating the scale for any device where temperature is proportional to the quantity being measured is , where X is the quantity being measured.

The fixed points for this scale are absolute zero and the triple point of water. This is the point at which water can exist in all three states at once. The equipment for the determination of the point is shown. This temperature is 0.01°C.

1K is equal in magnitude to 1°C. 0K = -273.15°C. Therefore q = T – 273.15

The internal energy is the amount of energy that the particles in a substance have. The symbol for internal energy is U. Internal energy in a solid is due to the kinetic and potential energy of the particles. As they move in a simple harmonic fashion, the energy transfers back and forth between these two types. Therefore,

Internal Energy = Kinetic Energy + Potential energy

This is constant at constant temperature. In a gas, U is entirely kinetic (as in an ideal gas there are no forces between the particles).

There are two methods of increasing the internal energy of an object. You can either:

- Change the temperature (leading to a heat flow in or out of the object)
- Do mechanical work on the object.

The term heat is used to describe the process of heat flow. The amount of internal energy in a solid or a gas will be used instead of the heat stored in it.

Heat and work describe energy in the process of transfer. Heat flow is by conduction, convection or radiation. These all take place from one object to another because of temperature difference. Work is transferred by a force moving through a distance.

Internal energy changes accompany either a change of state or a change of temperature.

The specific latent heat of fusion of a substance is the heat required to change 1kg of the substance from solid to liquid at its melting point.

DU=*l*D*m*

This is a method of measuring the specific latent heat of fusion of ice. Some water is added into a calorimeter at 5°C above room temperature, and dry melting ice is added until the temperature is 5°C below room temperature. The mass of the calorimeter, water and of the ice added need to be taken. Then:

Energy lost by calorimeter + Energy lost by water in calorimeter = Energy gained changing state + Energy gained heating water

mcDT + mcDT = lDm + mcDT

The real value = 334 400 J kg^{-1}

The temperature rise produced by the addition of any given amount of heat to a body is determined by the mass of the body and the substances of which it is composed.

The heat capacity, C, of a body is defined as being the heat required to produce unit temperature rise. This is sometimes called thermal capacity. It follows that the temperature of a body whose heat capacity is C rises by D q when an amount of heat is added to it, and so

D Q = CD q

The unit of heat capacity is J K^{-1}.

The term specific heat capacity refers to the heat capacity of unit mass of a substance. The specific heat capacity (c) of a substance is the heat required to produce unit temperature rise in unit mass of a substance.

It follows that if the temperature of a body of mass m and specific heat capacity c rises by D
q
when an amount of heat D
Q is added to it then D
Q = mcD
q
. The unit of specific heat capacity is J Kg^{-1} K^{-1}.

By using a copper calorimeter with a known mass of water in it is possible to calculate the specific heat capacity of brass.

Method

- Weigh the calorimeter.
- Weigh the calorimeter with approximately 70 ml of water in.
- Heat a mass of brass in the water until it boils.
- Quickly transfer the brass to the calorimeter and note the temperature change.

Calculation

Energy received by water and calorimeter = mcD q

= D q (mc + mc)

= ([57.49 ´ 0.39] + [72.98 ´ 4.18])

= 2619.82

Energy given out by the brass mass = mcD q

Therefore, c = E / mD q

so c=2619.82 / (100.15 ´ 72) = 0.363 J / g°C.

Consider an ideal gas being heated. 2 things may happen to it.

- Its internal energy may increase.
- Work may be done on the surroundings (i.e. the gas expands)

These are linked by the equation , where DU is the internal energy of the gas, DQ is the heat supplied and DW is the work done on the gas. There are three ways of doing work on something:

- Applying a force over a distance - DW=FD
*x* - Changing the volume of a gas at constant pressure - DW = pDV
- Doing work on an object electrically - DW = VIDt

A system that is isothermal exists at constant temperature. Thus an isothermal change is a change at constant temperature. DU = DQ + DW, but as the temperature is constant, DU = 0. Therefore, DQ = -DW, or the heat flow out of the object is equal to the work done on it. An example of a change like this is compressing a gas in the Boyle’s Law apparatus - the heat flow out of the gas is equal to the work done on the gas mechanically.

A system that is adiabatic is one where no heat is gained or lost. DU = DQ + DW, but there is no heat flow, DQ = 0. Therefore DU = DW, and so the change in internal energy is equal to the work done on the object in question.

When a lightbulb is turned off, it is at room temperature, the same temperature as the air around it, and therefore is in thermal equilibrium. Thus, when it is first turned on, there is no heat flow out of the lightbulb. Thus, it is an adiabatic system. DU = DW, as the work done by the electricity on the lightbulb (DW = VIDt) goes entirely into heating the wire in the bulb.

When a lightbulb has been on for a while, the wire in it reaches a constant temperature. At this stage DU = 0, and therefore the lightbulb is now an isothermal system. Therefore, DQ = -DW, as the heat flowing out of the lightbulb is equal to the work being done upon it by the electric current flowing.

Consider a uniform conducting rod, made of metal.

As the rod is uniform, A is constant. T_{2} > T_{1} and DT = temperature difference = T_{2} - T_{1}. Assuming no heat losses, the graph of temperature against length looks as follows:

It is clear that the temperature gradient is constant = . The rate of heat flowing through the rod is proportional to the temperature gradient and also the area of the rod, and therefore . The constant of proportionality in this equation is the thermal conductivity of the material that the rod is made of, k, and therefore . k is a constant, units W m^{-1} K^{-1}. This equation can be used to calculate how quickly heat will transfer along a substance. A lower value of k means that the value of DQ/DT is lower, and therefore that the substance is a better insulator.

The U value for a substance is the heat flow per metre squared per unit temperature difference, and is given by the equation . This is for a finished item, and takes the size of the item into account.

Sheets of rubber and cardboard, both 2mm thick with area 100cm^{2}, are pressed together, and the outer faces kept at 0°C and 25°C. Find the quantity of heat that flows in 2 hours, given that k_{rubber} = 0.13 W m^{-1} K^{-1} and k_{carboarrd} = 0.05 W m^{-1} K^{-1}.

1. Calculate the temperature at the junction.

The rate of heat flow through each of the substances is the same, and therefore:

As A and D*x* are constant, they cancel. Putting in values . As the total temperature difference is 25°C, . Thus q = 25-(18.1) = 6.94 K.

The key point is that for a composite material with a constant DT, the heat flow through each part will be the same and equal to the total heat flow.

Thermal Conductivities - some examples and typical values:

Copper 385

Aluminium 238

Cardboard 0.2

Water 0.6

Air 0.03

The value for air varies with pressure and humidity. Metals are 10^{3} times better as thermal conductors that non metals and liquids, because of free electrons. Gasses have even lower thermal conductivities. On heating a metal bar, the free electrons gain thermal energy and distribute this by collisions with the fixed positive metal ions in the lattice. In poor conductors, heat is passed on by wave motion - the hot parts become more and more disturbed and pass the disturbance on like a wave.

- The attraction between the gas molecules is negligible.
- The volume of the molecules is negligible compared with the volume of the gas.
- The molecules are perfectly elastic spheres.
- The duration of a collision is negligible compared with the time between collisions.

Consider a gas molecule in a box.

Side of length l

Number of molecules in box = N

u = speed of a molecule in the x direction

m = mass of a molecule

Change of momentum in the x direction = 2mu {=mu-(-mu)}

Time between 2 collisions with the right hand wall =

Force = rate of change of momentum

Pressure = Force / Area

Generalising for N molecules:

Let be the mean value of all the squares of all the x speeds.

i.e.

Because motion is random, x-direction motion = y-direction motion = z-direction motion.

and by Pythagoras theorem

Therefore pressure =

Where is the mean square speed, and therefore is the root mean square speed or r.m.s. speed. This is a type of average speed, and is measured in ms^{-1}. Given data on a set of molecules and their speeds, square the speeds, multiply by the frequencies, sum and take the square root. This value is always slightly greater than the mean speed.

Increased temperature affects the distribution of the speeds of molecules in the following ways:

- Most probable speed, mean speed and r.m.s. speed all increase (as the peak on the graph shifts to the right).
- The width of peak increases and the peak broadens out as the range of speeds increases.
- The peak falls in height (as the area under the curve must remain the same) and therefore the number of molecules with the most probable speed decreases.
- The Maxwellian tail is higher – there are more molecules with higher speeds.

Equal volumes of gas under the same conditions (pressure and temperature) contain equal numbers of molecules.

This can be proved by using the kinetic theory.

Consider equal volumes of 2 gasses, A and B, at the same pressure and temperature. Since pV is constant and , .

As the gasses are at the same temperature, and ,

. Dividing one equation by the other:

Using the Kinetic Theory to look at temperature

Using the ideal gas law, pV = nRT, and assuming one mole of gas (and therefore pV = RT), where R = the gas constant = 8.31.

If V = volume, N = total number of molecules and n = number of molecules per unit volume, then . Since ,

The mean kinetic energy of one molecule = , therefore , where R is the molar gas constant and N is the number of molecules, which for 1 mole of gas is equal to Avagadro’s Constant. R/N = k, which is Boltzman’s constant, where k = 1.38 ´
10^{-23} JK^{-1}. Thus KE = 3/2kT.

Rearranging the formula , we get r.m.s. speed = , where M is the molar mass of the gas.. This allows us to calculate ratios of speeds at constant temperatures.

Boyle’s Law states that the pressure of a fixed mass of gas at constant temperature is inversely proportional to its volume. It can be demonstrated using apparatus such as that shown:

By the kinetic theory of gasses, . Thus, from a graph of r against p, which can be produced from the graph of V against p produced by the Boyle’s Law apparatus, the r.m.s. speed can be calculated.

This states that for a fixed mass of gas at constant pressure, absolute temperature is proportional to volume. It can be demonstrated using mercury and capillary tubes.

This states that for a fixed mass of gas at constant volume, pressure is proportional to absolute temperature.

Combining the three laws, we conclude that pV/T is a constant. This can be written as pV = nRT, where n is the number of moles of gas and R is the gas constant, equal to 8.31 J K^{-1} mol^{-1}. This is the ideal gas law, and can be used to do calculations about gasses.

The Boltzman constant is defined as k= = 1.38 ´ 10^{-23}. Combining this with the kinetic theory:

Using the relation , . Substituting in k= gives , the ideal gas equation.

(Fluids are liquids and gasses)

There are 2 equations relating to pressure in a fluid:

P=F/A (For a force applied to an area, e.g. piston systems)

DP=rgDh (For changes in height of a liquid)

One example of the second is in a manometer. This is a U shaped tube with a liquid in the bottom. One side is open to the atmosphere, and the other side has a pressure applied to it. This forms a situation as shown in the diagram beside. Dh is due to the extra pressure applied to the U tube at one end, and therefore Dp=rgDh.

In a fluid that is not moving there is static pressure. This has 2 important features:

- The pressure acts equally in all directions.
- The pressure increases with depth.

Archimedes' principle states that a body in a fluid experiences an upthrust equal to the weight of fluid displaces. Thus an object less dense than the fluid it is immersed in will float at the point where the weight of fluid displaced is equal to the weight of the body.

Oil does not flow as easily as water or air but air becomes turbulent very easily.

A shape like (a) has a stable flow pattern (the flow lines remain the same). The lines shown are called streamlines. If the streamlines are parallel then the flow is laminar. (b) is illustrating turbulence, caused by early separation of the air flow. A vacuum is caused behind the body which sucks the air flow in. On a golf ball (c), there are dimples on the surface, which make it irregular. This causes turbulence, which causes the ball to move further.

The viscosity of a fluid is its resistance to flow. The coefficient of viscosity is given the Greek symbol h (eta, units N m s^{-2}). The variation of speed of flow of a liquid in a tube is as follows:

Viscosity is a direct result of laminar flow. It is caused by particles transferring between the lamina and thus slowing the liquid’s flow speed down.

Consider a volume of liquid flowing from one container to another through a narrow pipe, as shown in the diagram below:

The factors that are going to affect the rate of flow of liquid are:

- r, the radius of the pipe
- h, the viscosity of the fluid
- P/L, the pressure difference per unit length or pressure gradient.

Therefore, . Expressing these in base quantities of mass, length and time:

L^{3}T^{-1} = (ML^{-1}T^{-1})^{x} L^{y} (ML^{-2}T^{-2})^{2}

Looking at the indices of these base quantities:

Length: 3 = -x+y-2z (1)

Time: -1 = -x-2z (2)

Mass: 0 = x+z (3)

From (2) and (3): -1 = -x+2x = x \x = -1

From (1) and (2): 3 = -x+y+2x = x+y \ y = 4

From (3) 0 = -1 + z \ z = 1

Therefore - the formula for velocity of liquid flow down a tube.

Stoke’s Law gives the link between the viscous drag force and the viscosity for a sphere falling through a fluid as F = 6phrv, where r is the radius of the sphere and v its velocity.

At terminal velocity, there is no acceleration on the falling body, and therefore it is in dynamic equilibrium. This means that the weight of the body is balanced by the viscous drag force acting. Thus, mg = F + u, where F is the viscous drag force and u is the upthrust. It is possible to use this to measure the coefficient of viscosity for a substance.

A fuel is a material from which energy can be obtained. A primary fuel is a fuel that occurs naturally (i.e. not petrol, electricity, etc. – these are produced from primary fuels.)

Primary energy is the total amount of energy needed (in the original energy source) to do a job. Primary energy = useful energy to do a job + energy wasted (in transporting energy from generation to utilisation).

Primary energy sources include coal, oil, gas and nuclear. Between them these three energy sources account for 93% of world energy usage. They are finite sources – i.e. there is a fixed amount of them available.

Energy from oil is used primarily for heating and transport. However, oil is also used in the manufacture of fertilisers and plastics. Thus it can be thought that the usage of oil as an energy source is wasteful. The world’s known oil reserves are being used up rapidly, but new discoveries and better methods of extraction are continually increasing the amount of oil available. Oil is expected to run out as a resource in the middle of the next century, whereas coal is expected to last longer.

Town gas is an alternative to natural gas, and is produced from coal by heating it in the absence of air.

The dash for gas describes the move to convert from producing electricity from coal and oil to gas. This is short term as the world’s gas reserves are very limited.

There are huge reserves of coal available – they are expected to last much longer than those of oil and gas. These values are based on a 10% yield, as many reserves are inaccessible. There are problems associated with re-opening coal mines, as they tend to flood with water. Coal as a fuel also has problems – is contains sulphur dioxide, which causes respiratory problems and acid rain.

Nuclear power works on the basis of the fission of fissile material. The principal nuclear fuel is uranium – 235, of which there are only very limited resources, about 50 years worth. The majority of uranium found on the Earth is uranium – 238, which is of no value in an ordinary nuclear reactor. In a fast breeder reactor, plutonium is used as a fuel. Plutonium is produced when uranium – 238 absorbs fast electrons. However, plutonium can also be used in nuclear weapons.

Uranium is a concentrated energy source – 10^{6} kg of coal are needed to produce as much energy as 1kg of uranium. However, there are problems with the byproducts of the nuclear reactions. The two major by-products are barium and krypton, both of which are radioactive in the forms that are produced.

A chain reaction takes place in a nuclear reactor, . Other by-products are possible as well as Barium and Krypton, but these are the two most likely possibilities. The neutrons that are produced are fast moving, and therefore need to be slowed down before they can be captured by uranium atoms. To do this a moderator is used. Q is the quantity of heat that is produced. This is used to boil water to make steam. The water that circulates within the reactor needs to be isolated from any other water supplies because it is radioactive, and therefore a heat exchanger is used, where the water is in thermal but not physical contact.

Other daughter products are possible, and some combinations produce three neutrons instead of two (the mean number is 2.3). The heat change is a result of a change in binding energy – the products of fission are atomically more stable than uranium. Each kilogram of U-235 releases about 80 terrajoules of energy.

To prevent the reaction becoming uncontrollable, control rods are used to absorb some of the neutrons. These are made of boron or cadmium. A moderator is used to slow down the neutrons so that they can be absorbed by the uranium atoms. This is generally graphite or water. The moderator nucleus needs to have a similar mass to the neutron, so that the collision of the two is sufficiently elastic to transfer some of the energy to the atom. This heats up the moderator and slows down the neutrons, so that the energy of the neutrons is roughly equal to the thermal energy of the lattice – thus they are called thermal neutrons.

It is not necessary for the neutrons to actually impact a nucleus to interact with it. There is an area round the nucleus where electrons can be scattered and absorbed – these quantities are called s
_{scattering} and s
_{abs} . For control rods larger values of s
_{abs} are needed, the opposite for moderators.

The critical mass is the mass of fuel where enough neutrons are kept within the nucleus to cause a chain reaction. This is a function of the mass/surface area ratio of the shape and mass of the fuel used.

Spent fuel is placed in water to cool down and for the activity to decrease. It is then used for sterilisation, and any fuel within the waste reprocessed. Some waste then remains, this must be sealed and then either buried or dumped at sea. The container used must be very permanent to prevent leakage.

Thermal reactors need a moderator and control rods, and use uranium-235. They look as follows:

[no diagram...]These use plutonium as a fuel, which can be split by fast neutrons. A blanket of U-238 round the outside captures neutrons and forms more plutonium, which is then used for fission.

For geothermal power it is necessary to have hot rocks, often due to radioactive decay, which are porous or have cracks in. Cold water is pumped down into the rocks, and how water or steam is piped off. In some places such hot water gushes to the surface naturally, for example in Iceland, in excess of 200°C.

The average heat flow in the Earth is 0.1 Wm^{-2}.

There are fuels produced from living material. Typically they are produced from the anaerobic decay of dead plants and animals, which produces methane. This is often piped off landfill sites. Another common use of biofuels are in the third world, especially cow dung.

The power output of the sun is 4 ´ 10^{26} Watts. From this it can be calculated that the flux at the Earth’s surface is 1400 W m^{-2}. The solar constant in England is closer to 860 W m^{-2}, as England is at a latitude of 52°. Only the equator receives the full 1400 W m^{-2}.

These are used to heat water, and thus to reduce domestic power consumption. A blackened surface absorbs infra-red solar radiation, and water running over the surface is warmed. A more efficient system has a high surface area and a good thermal contact between the panel and the water. A high proportion of the time that the water spends in the system should be spent in the panel. The system is generally closed and a heat exchange mechanism is used.

These are also called solar cells, but should not be referred to as solar panels, which properly are only the water filled type. They are often used on pocket calculators and satellites, where they are useful because there is no atmosphere. They are often made with a gold back, which means that they can be folded, etc. In remote places they are also used because it is easier than supplying electricity.

In any substance there are bands of energy levels that electrons can be present in. The three possible arrangements are shown in the diagram below:

[..]In the valance band, the electrons have very low energy and cannot move. They are being used to bond the substance together. If the forbidden band is large, the substance is an insulator. If the conduction and valance bands overlap, then the substance is a conductor (e.g. a metal). In a semiconductor, the forbidden band is very small. If a photon strikes the wafer, and hf > forbidden band, then an eolectron jumps from the valance band to the conductance band, and therefore a small current is produced. The problem with solar power is the large area of panels needed – a 100MW station needs at least 71000 m^{2} of panels.

The generators used are called wind turbines, or more correctly aero-generators.

They are useful for private use (e.g. remote farms) and any extra electricity generated can be fed into the National Grid. The problems with wind power are that it is very dilute as a source (a huge area is needed to produce sufficient power) and the source is very variable – when the wind don’t blow… The cogs in the gearboxes of the generators are also noisy and the turbines themselves are an eyesore. They tend to be situated on moorland and coastal areas, where it is windy and people do not mind the eyesore quite so much. To generate a lot of electricity it is necessary to have a large sail area and a high wind speed.

An array of "ducks" are set up in series. The ducks nod, moving with the waves, and reduce the amplitude of the waves. The relative motion of 2 ducks is used to generate electricity. The engineering needed to overcome problems with changing wave size and unpredictability is significant is sophisticated, but can be developed. Ducks can produce 70 kw per m of ducks laid. The principal reason for the non-development of ducks has been political.

Hydroelectric power is driven by the sun via the hydrological cycle. A dam is built across the mouth of a river, and a reservoir builds up behind the dam. This is let out through a turbine, which generates electricity. The problems with hydroelectric power stem largely from the flooding of the valley behind the dam, to form the reservoir. A hydroelectric plant is also of little value if it does not rain. There are also problems with affecting the migration of fish, etc. down the river. The plants are useful as a top up, because they can be switched on and off at very little expense. Although the volume used is much less than that in a tidal system, the drop is much larger, and therefore the energy generated is similar.

The sun drives the following energy sources:

- Solar power
- Hydroelectric power, via the hydrological cycle
- Fossil fuels, via photosynthesis
- Wind, by differential heating of the tropics and poles, and of the land and sea.

In a heat engine, we are converting thermal energy into mechanical work. Examples of this are steam engines and petrol engines. This process is not 100% efficient.

Consider a piston in a cylinder of gas which expands. Work is done against atmospheric pressure. For a useful engine the process must be a cyclical one, and so some form of heat sink is necessary to remove heat from the gasses inside the cylinder and thus reduce the volume of the gasses. Therefore 100% efficiency is not possible.

% efficiency = h =

Carnot ‘s heat engine – 1840s.

]Example - A power station. Superheated steam goes in at 700K and leaves at 300K. Efficiency = 400/700 ´ 100 = 57% efficient.

To increase efficiency it is necessary to minimise T_{2} and maximise T_{1}. The system cannot be 100% efficient as it is not possible to have T_{1} = 0K.

In a heat engine, one tries to convert random thermal motion into uniform kinetic motion. The second law of thermodynamics states that it is not possible to convert heat continually into work without transferring some heat from a hotter body to a cooler one. This means that it is not possible to have a heat engine that is 100% efficient.

CHPs use the hot water produced by cooling steam to heat buildings, for example the buildings in the power station, nearby commercial buildings, etc. The buildings must be in the locality otherwise the heat will escape. This increases the effective efficiency of the power station by using some of the "waste" heat energy. Water is pumped out at about 70°C and returns at about 30°C.

A steam engine is about 20% efficient, and a car engine about 25%. The maximum theoretical possible efficiency of a car engine is 67%, but this is impossible to achieve practically.