Physics

HEAT AND THERMO-DYNAMICS    Thermal effects

Key unit Competence

By the of this unit, the learner should be able to evaluate applications of first and second laws of thermodynamics in real life.

Unit goals

By the end of this unit, I will be able to:

differentiate between Internal energy and total energy of a system.

* explain the work done by the expanding gas.

* state the first law of thermodynamics.

* state the second law of thermodynamics.

* explain thermodynamic processes in heat engines.

Introduction

Before, you learnt that:

• Heat is a form of energy.

• Heat can be changed / transformed from one form to another.So, if in a system heat changes from one form to another, its called thermal dynamic system.

The systems to discuss in this unit include refrigerators, heat pumps, car engines. Remember that heat is the measure of total internal energy of a body. This means that particles of a body vibrate because of energy they have.

Thermal energy and internal energy

Activity 1

Have you ever boiled water on a sauce pan with a cover?

Describe what happens to the cover when water boils?

When water boils, the vapour pushes the cover off the sauce pan. You have already seen in your early secondary that heat is a form of energy. Therefore, when this saucepan is heated, the heat gained is used to boil off the water and extra work is done to push the sauce pan cover. This total heat energy supplied is called thermal energy.

Science in action! Discover

• In groups of five, explain why an inflated bicycle tube bursts when it is left on sunshine for a very long time.

• Similarly explain why a balloon full of air bursts as it rises in the atmosphere.

• Note down your observation in your exercise books.You already know the characteristics of the three states of matter that is; solids, liquids and gases. In this unit, we shall be interested in studying the behaviour of molecules in matter.

When the bicycle tube is left exposed to sunshine, it gets heated and the molecules in the gas gain energy and hence its kinetic energy increases. As a result, they collide frequently with the walls of the tube and therefore exert high pressure on the walls and the tube bursts.

The same thing happens with the balloon in air.

The energy possessed by the molecules of the gas is called internal energy of the gas. This energy depends on the temperature of the gas. When a gas is heated its temperature increases and hence the average speed of molecules also increases increasing the internal energy of the gas. Further increase of heat supplied means that extra energy is absorbed by the molecules of the gas, hence expanding and pushing the tyre. As a result the tyre bursts.

Activity 2

List down three utensils used for cooking food in your homes.Describe how these utensils are used to cook the food. Are they always left open while cooking?

In all the above, there exists energy exchanges and such things are called systems. Systems can either be closed or open. When water is being boiled in an open sauce pan, vapour is allowed to escape. It is an example of an open system. When someone cooks meat using a closed container, no gas is allowed to escape. Its an example of a closed system.

Whenever heat flows to or from a system, or work is done on or by a system, there is a change in the energy of this system. The study of the processes that cause these energy changes is termed thermodynamics.

Thermodynamic systems

Activity 3: Discover

In groups of five, discuss how heat is transferred in the three states of matter. Do you think heat can be transferred from one state of matter to another? For example from gas to a solid or from a solid to a liquid or from a liquid to a gas and vice versa?

Heat is the energy that flows by conduction, convection or radiation from one body to another because of a temperature difference between them. These bodies where exchange of heat to other forms of energy occurs are called thermodynamic systems.

A thermodynamic system consists of a fixed mass of matter, often a gas, separated from its surroundings, perhaps by a cylinder and a piston. For example heat engines such as a petrol engine, a steam turbine and jet engine all contain thermodynamic systems designed to convert heat into mechanical work. Head pumps and refrigerators are thermodynamic devices for transferring heat from a cold body to a hotter one.

In such devices, energy is transferred from one system to another by a force moving its point of application in its own direction.

The energy of a system, whether transferred to it as heat or work is termed as the internal energy of the system.

When there is no heat transfer between two systems, that is, the two are at the same temperature, they are said to be in thermal equilibrium.

Activity 4

Have you ever observed smoke moving in the atmosphere.Move outside class and go towards the kitchen and observe how smoke is moving. Describe briefly how it moves.Why does it move like that?

You have already seen in your early secondary that molecules in a gas are more further apart and are always in constant random motion while moving at high speed colliding with one another and the walls of the container, and when the gas is heated their speed increases.

Smoke particles are always in random motion and when they are moving in air, they collide with air molecules and a zigzag pattern is seen.

Similarly, when smoke is put in a container and then closed, the particles are seen to be in a random motion. Smoke is an example of a real gas.

In thermodynamics, we are mainly interested in ideal gas. At higher temperatures, a real gas behaves like an ideal gas.

Activity 5

Have you ever heard of an ideal gas?What are the differences between a real gas and an ideal gas?

When a gas is heated, molecules move further apart and the forces of attractions between them become negligible and the gas becomes ideal.

When the molecules become further apart, the gas expands and the volume of the individual molecule becomes so small compared to the entire volume of the gas. It therefore becomes negligible compared to the volume of the gas and the gas becomes ideal.

When the molecules are colliding with one another, collisions are assumed to be perfectly elastic. In this case, the gas becomes ideal because for a real gas we expect to have time between approach and separation during collision.

Exercise

What is a perfectly elastic collision?

Work done by an expanding gas

Activity 6: Discover

Explain why a pump gets hot when one pumps air into a tyre.

When you compress air in a bicycle pump, your muscles transfer energy to the handle, which in turn transfers energy to the molecules of air in the pump. This additional energy makes the molecules move faster. As they are compressed into a smaller space, they also collide more often with the wall of the pump, so they transfer more energy to the metal wall and it becomes hot.

We have already seen how heat can be transferred, so you probably have a good idea what Q means. Work is simply a force Multiplied by distance in the direction of force.

A gas can be heated by compressing it, for example with a bicycle pump. Hence the temperature of the gas can be raised either by doing work in compressing it or by heating it. Likewise the temperature can be lowered by either making the gas do work in expanding or by extracting heat from it.

Consider a mass of gas enclosed in a cylinder by a frictionless piston of cross-section area A which is in equilibrium under the action of an external force, F, acting downwards (i.e pressing the gas). Let this force be infinitely reduced so that the piston is pushed up by the gas a distance Δx, which is so small that the pressure of the gas is virtually unchanged by the expansion

If the gas is heated, it will expand and push the piston thereby doing work on the piston. If the piston is pushed down, on the other hand, it does work on the gas. This is an example on how work is done by a thermodynamic system.

The piston must be held in position by force PA (by definition of pressure).

The external work done by the gas,

ΔW against F will be;ΔW = FΔx = PAΔx= PΔV,

ΔV = FΔx is the increase in volume of the gas.

Suppose that the pressure is kept constant during the expansion, and the gas expands from V1 to V2, then the total work done by the gas is given by calculus as; W = ʃdW= ʃPdW

It follows that; W = P (V2 – V1)

Specific heat capacities of gases

(i) Weigh a given quantity of a gas.

(ii) Confine the gas in a closed container.

(iii) Place a thermometer in the container.

(iv) Using a given source of heat, supply heat to a gas by keeping its volume constant.

(v) Record the change in temperature using the thermometer.

(vi) You can do this by closing the container.(vii) Calculate the quantity of heat supplied.

(viii) Repeat the above procedures by keeping the pressure constant.

(ix) Calculate the heat supplied for the same temperature change as in above.

(x) Compare the two quantities of heat supplied.

Do you notice that the heat needed at constant pressure is higher than that at constant volume?

Why do you think it is so?

Gases are considered to have a number of specific heat capacities. A change in temperature of a gas is likely to cause large changes in pressure and volume of the gas but for solids or liquids, the change in pressure is neglected.

In your early secondary, you have already seen that heat energy is calculated by measuring the mass of liquids and solids. However, in gases, we replace the mass with the number of moles of a gas.

When the specific heat capacity of a gas is measured in terms of its moles, it is known as principal specific heat capacity. There are two important heat capacities: the molar heat capacity and constant volume (Cv ) and molar heat capacity at constant pressure (CP).

The principal molar heat capacity at constant volume (Cv) is defined as the heat required to increase the temperature of one mole of a gas at constant volume by one Kelvin.

The principal molar heat capacity at constant pressure (CP) is the amount of heat required to increase the temperature of one mole of a gas at constant pressure by one Kelvin.

The molar heat capacities have units Jmol—1K-1

Since at constant volume, the work done by a gas is zero from (W=P∆V, ∆V, =0), then it is evident that the principal K molar heat capacity at constant pressure, CP is greater than that at constant volume, Cv. Why?

The reason as to why this is so can be done by considering cylinders in (a) and (b) each initially containing one mole of gas at temperature T and pressure, P. The piston in (a) is fixed and that in (b) is frictionless and can move freely but has a constant force applied to it. If heat is supplied to each until thetemperature has risen by one Kelvin, the increase of internal energy must be the same in each case (Since the temperature rise is the same).

All the heat supplied in case (a) is used to increase the internal energy of the gas. In (b), however, the gas expands and work is done by it on the piston; the heat supplied in this case equals the increase of internal energy plus the work done in the expansion of the gas.

The first law of the thermodynamics

We have already seen that when a quantity of heat ∆Q is supplied to a gas, two things happen: (i) the heat supplied may increase the internal energy, U of the gas and (ii) the gas may expand and do some work, W in moving the piston.

The magnitude of internal energy depends on; the temperature of the gas i.e. the internal energy is high at a high temperature and low at low temperature.

The amount of heat supplied is equal to the change in internal energy of the gas plus the work done by the gas.i.e. ∆Q = ∆U + ∆W (i)

The above is the statement of the first law of thermodynamics.

since ∆W = P∆V,

It follows that ∆Q = ∆U + P∆W

When n moles of a gas are considered, the amount of heat supplied at constant pressure is nCPΔT, whereas the amount of heat supplied at constant volume would be nCvΔT.

Relationship between Cp and Cv

.Consider one mole of a gas at a pressure P, temperature T and volume V, heated to cause the same temperature change, ∆T first at constant volume and secondly, at constant pressure.

Applications of first law of thermodynamics in particular gas changes

Isovolumetric process (Isochoric process)

Activity 7

(i) Have you ever heard of an isovolumetric or isochoric process?Study the Figure and answer questions that follow;

(ii) What substance is likely to be getting cooked using the can on the left? Give a reason for your answer.

(iii) Why is the can covered and not open?

(iv) If one tried to open it while its on fire, what do you think would happen?

The above can be used to boil liquids for example milk effectively. As the liquid inside the can heats up, its pressure increases, but its volume stays the same (unless, of course, the can explodes). It therefore means that all the processes inside the can take place at constant volume. They are called isovolumetric or isochoric processes.

When the pressure in a system changes but the volume is constant, you have what is called an isochoric process. An example of this would be a simple closed container, which can’t change its volume as seen above.

Activity 8

How much work does the fire do on the can? In this case, the volume is constant, from the law of thermodynamics, no

work is being done since ∆V = 0

This process takes place at constant volume and since

∆V = 0, ∆W = sP∆W = 0

∆Q = ∆U + CV ∆T

In this process, the energy supplied is used to increase the internal energy since the internal energy is independent of the volume.

Isobaric process

Activity 9Study the Figure here showing a woman preparing sauce.

(i) What name of the utensil is she using?

(ii) Why is the utensil open?

(iii) Do you think it is good to use an open utensil to boil liquids?

Boiling liquids in open containers is very safe for example if the container is closed, pressure may build up in the container and force it to burst. Boiling in open containers imply that the pressure of the substance is kept constant. This process is called an isobaric process. An isobaric process is the one that occurs at constant pressure.

Heating of water in an open vessel and the expansion of a gas in a cylinder with a freely moving piston are typical examples of isobaric processes. In both cases, the pressure is equal to atmospheric pressure. For example when water

is being heated, its volume increases and the pressure inside the container is constant since the number of collisions between water molecules and the walls of the container is constant.

The same process occurs when a gas enclosed in a cylinder with a frictionless piston is heated such that at any time, the gas pressure equals the external pressure.

Work done by the gas in the isobaric process

When the gas expands from volume V1 to V2,

ΔW = PΔV = P(V2 – V1)

From a PV graph, the work equals the area under the graph.

In this process, the energy supplied is used to increase the internal energy since the internal energy is independent of the volume.

∆Q = ∆U + ∆W

∆Q = CV∆T + P(V2- V1)

Isothermal change (constant temperature)

Activity 10

(i) Get a polythene bag and fill it with air.

(ii) Insert a thermometre in the bag and place it in the ice-water mixture.

(iii) Note what happens.Do you notice that the gas condenses and the volume decreases?What happens to the temperature recorded by the thermometer?

You can notice that the temperature remains constant. This change is called Condensation and is an example of isothermal process.

Do you think this process is reversible?

An isothermal change can be reversible. An isothermal change is the change that occurs at constant temperature. It is either a compression or expansion of a gas at a constant temperature.

If the volume increases, the pressure must decrease and if the volume decreases, the pressure must increase

Conditions necessary for an isothermal process to occur

Activity 11: Discover

(i) On a cold day, how do you keep yourself warm?

(ii) In groups of five, describe how you can keep the temperature of the system constant.

For an isothermal process to take place, the gas must be contained in a thin –walled heat conducting vessel/container in good thermal contact with a constant temperature.

The process must be carried out slowly to allow time for heat exchange to take place

Work done in Isothermal Change

Activity 12: Science at work

(i) Have you ever tried to boil water in a closed sauce pan?

(ii) What happens to the cover when the vapour starts to come off the water?

(iii) Notice that this vapour pushes the cover off the pan

We say that the vapour does work on the cover.

From the first law of thermodynamics, ∆Q = ∆U + ∆W.

When the volume of gas changes by ∆V at constant temperature then the

pressure has also to change so that the ideal gas equation is satisfied.

From the above equation, the following can be drawn;

(i) When the gas expands (i.e V2 > V1), then W is positive.

(ii) When the gas is compressed (i.e V1 >V2), thus W is negative, meaning that work is done on the gas in compressing it.

Adiabatic change

Activity 13

(i) Pump a bicycle tyre using a pump until it is full.

(ii) Open the tube slowly while placing your other hand in its path.

(iii) Do you notice that the the air coming out of the tyre is hotter than the surrounding air?

As one pumps, the air molecules are compressed into a smaller space. They also collide more often with the wall of the pump, so they transfer more energy to one another and become hot. No heat has been supplied to the system. It is called an adiabatic compression.

Activity:14

(i) Now pump the tyre and leave it standing for sometime.

(ii) Make sure you don’t expose it to sun shine.

(iii) Open the valve after two hours while your hand is placed in the path of air from it.

Do you notice that the air is colder than its surrounding?

Heat has been lost but not to the surroundings. When the air is left standing, expansion occurs. This is associated with a decrease in temperature. It is called an adiabatic expansion.

An adiabatic change is process in which no heat enters or leaves the gas system. It is either an expansion or a compression.

Since ∆Q = ∆U + P∆V and ∆Q = 0

∆Q = CV∆T + P∆V Or ∆U = P∆V

If the gas expands, it does work, its internal energy is reduced and hence the temperature is lowered.

If the gas is compressed, work is done on the gas, its internal energy will increase and therefore its temperature rises.

Conditions that are necessary for an adiabatic change to occur

Activity 15

How do you always protect yourself from a bad weather?

On a cold day, we always wear woolen jackets to protect ourselves from coldness. Therefore no heat is either lost to the surrounding and or gained. In this case, an adiabatic process is achieved.

For an adiabatic process to be achieved, the gas must be contained in a thick –walled and perfectly insulated isolated container.

The process must be carried out rapidly to avoid any possible heat exchanges between the gas system and the surroundings

Equations for an adiabatic change

From the first law of thermodynamics, ∆Q = ∆U + P∆V

For an adiabatic process, ∆Q = 0, and for 1 mole, ∆U = CV∆T1, CV∆T = 0 (i)

For infinitesimal small changes, but from the ideal gas equation, for one mole,

Activity 16

Derive the expression for temperature and pressure for adiabatic change i.e T γ P1– γ = a constant

Example I

A gas has a volume of 0.02 m3 at a pressure of 2 x 105 Pa and a temperature of 27oC. It is heated at constant pressure until its volume increases to 0.03 m3.

Calculate the:

(i) External work done.

(ii) New temperature of the gas.

(iii) Increase in internal energy of the gas if its mass is 16g, its molar heat capacity at constant volume is 0.8Jmol-1K-1 and the molar mass is 32g.

Example 2

An ideal gas at 17o C has a pressure of 760mm Hg is compressed (i) isothermally (ii) a diabatically, until its volume is halved.

Calculate in each case the final temperature and pressure of the gas. Assume that Cp= 2100Jmol-1K-1 and CV = 1500Jmol-1K-1

Exercise

1. Distinguish between isothermal and adiabatic changes, clearly stating the conditions under which they occur in practice.

2. Define the two principal molar heat capacities of a gas and derive an expression relating the two. Explain the difference between these two principal molar heat capacities.

3. A quantity of oxygen is compressed isothermally until its pressure is doubled, it is then allowed to expand adiabatically until its original volume is restored. Find the final pressure in terms of its original pressure. Draw a PV diagram for the above processes.

4. 0.45m3 of a gas at a temperature of 15o C expands adiabatically and its temperature falls to 4o C.a) What is the new volume if γ = 1.40b) The gas is then compressed isothermally until the pressure returns to its original value. Calculate the final volume of the gas.

5. A vessel containing 2m3 of air initially at a temperature 25o C and pressure 760mmHg, is heated at constant pressure until its volume is doubled. Find

(a) the final temperature

(b) the external work done by the air in expanding,

(c) the quantity of heat supplied.

6. (Assume that the density of air at s.t.p is 1.293kgm-3 and that the principal molar heat capacity of air at constant volume is 20.4Jmol-1K-1.An ideal gas at a temperature 45o C and pressure 1.0 x 105Nm-2occupies a volume of 2.0 x 10-3 m3. It expands adiabatically to twice its volume. Find the final temperature and pressure. Represent this process on PV- diagram.(Take γ = 1.40)

Second Law of thermodynamics

The second law of thermodynamics can be stated in many equivalent ways, each expressing a different facet of its meaning. There are so many different forms of it because it is of such significance. William Thomson (Lord Kelvin) in 1851 stated it in this form;

“no heat engine can perform a cyclic operation whose only result is to convert internal energy into mechanical energy”

The second law was stated by Rodolph Clausus in 1850 in this form;

“no refrigerator (or heat pump) can transfer internal energy from a cold reservoir to a hot reservoir without some external agent doing work.”

Applications of the second law of thermodynamics

Heat engines

Activity 17

* Have you ever heard of an engine?

* Where exactly do we find engines?

* What do you think an engine is?

* How do you think the engine operates?

Any device which will convert heat cyclically into mechanical work is called a heat engine.The material which, on being supplied with heat, performs mechanical work is called the working substance. It is a machine, which changes heat energy, obtained by burning a fuel, to kinetic energy. In an internal combustion engine, e.g petrol. Diesel, jet engine, the fuel is burnt in the cylinder chamber where the energy change occurs. This is not so in other engines e.g steam turbines. All practical engines use one of the two working substances either water (in the reciprocating steam engine and the turbine) or air (in the internal combustion engine). The working of an engine is a thermodynamic process in which at one point no heat enters or leaves a system during expansion or compression of the fluid composing the system.

The Carnot Cycle

The cycle of operations through which the working substance has been taken is calledcarnot’s cycle

There are two main ways in which the carnot cycle differs from that of any practical engine.

First, the heat absorbed is all taken in at one constant temperature and all the heat rejected to the sink is given out at another constant temperature. In this manner, it is very much simpler than any practical engine.

Secondly, as no work is done at any stage in overcoming frictions, and no heat is lost to the surrounding, the cycle is completely reversible. This means that if we had carried out the whole sequency of changes in the reverse order, every operation would have been exactly reversed. This is called an ideal heat engine because in all practical engines work, is done in overcoming friction and heat is lost to the surroundings.

Otto Cycle and Diesel Cycle

Otto Cycle

An Otto cycle is an idealized thermodynamic cycle which describes the functioning of a typical spark ignition reciprocating piston engine, the thermodynamic cycle most commonly found in automobile engine

The Pressure Volume diagram above represents the Otto cycle which has the following strokes; the intake (A) stroke is performed by an isobaric expansion, followed by an adiabatic compression (B) stroke (along 1-2). Through the combustion of fuel, heat is added in an isovolumetric process (2-3), followed by an adiabatic expansion process ( 3-4), characterising the power (C) stroke. The cycle is closed by the exhaust (D) stroke, characterized by isovolumetric cooling and isobaric compression processes.

The processes are described by:

Process 1-2 is an isentropic compression of the air as the piston moves from bottom dead centre (BDC) to top dead centre (TDC)

.Process 2-3 is a constant –volume heat transfer to the air from an external source while the piston is at top dead centre. This process is intended to represent the ignition of the fuel –air mixture and the subsequent rapid burning.

Process 3-4 is an isentropic expansion (power stroke).

Process 4-1 completes the cycle by a constant-volume process in which heat is rejected from the air while the piston is a bottom dead centre.

The Otto cycle consists of adiabatic compression, heat addition at constant volume, adiabatic expansion, and rejection of heat at constant volume. In the case of a four-stroke Otto cycle, technically there are two additional processes; one for the exhaust of waste heat and combustion products (by isobaric compression), and one for the intake of cool oxygen –rich) air (by isobaric expansion); however, these are often omitted in a simplified analysis. Even though these two processes are critical to the functioning of a real engine, wherein the details of heat transfer and combustion chemistry are relevant, for the simplified analysis of the thermodynamic cycle, it is simpler and more convenient to assume that all of the waste-heat is removed during a single volume change.

Diesel Cycle

The diesel cycle is the thermodynamic cycle, which approximates the pressure and volume of the combustion chamber of the Diesel engine, invented by Rudolph Diesel in 1897. It is assumed to have constant pressure during the first part of the “combustion” phase V2to V2 in the diagram, below). This is an idealised mathematical model; real physical diesels do have an increase in pressure during this period, but it is less pronounced than in the Otto cycle. The idealized Otto cycle of a gasoline engine approximates constant volume during that phase, generating more of a spike in a P-V diagram.

The Idealised Diesel Cycle

From P-V diagram for the Ideal Diesel cycle, the cycle follows the numbers 1-4 in clockwise direction.