IB PHYSICSSUBSIDIARY LEVEL

THERMAL PHYSICS

NOTES

3.1

Thermal Concepts3.1.1

Molecular theory of solids, liquids and gases

Whether a substance is a solid, liquid or gas at room temperature, depends on the forces between its particles, the distance between the particles and how fast they are moving.

The strength of the forces between particles depends on:

*whether they are ions, molecules or atoms. For molecules, the forces depend on which atoms are present and the structure of the molecule.

*the distance between the particles the greater the distance, the weaker the force.

*the speed of the particles the faster the speed, the greater the distance between them and the weaker the force.

The table below summarises the properties of the states of matter in terms of molecular motion.

StatePropertyParticle behavior

solidFixed shape and volume.

Specific melting point when pure.

Most expand when heated, contract

when cooled.

Can't be compressed.Particles held close together by strong forces and are stacked neatly in a crystal lattice. Particles vibrate on the spot with average kinetic energy much less than the binding energy - energy that must be absorbed for a particle to break free.

liquidShape varies to fit lowest part of the container.

Specific boiling point when pure.

Most expand when heated, contract

when cooled.

Specific melting point when pure.

Can't be compressed.Particles close but slightly further apart than in a solid. Forces between particles are weak allowing movement of particles around each other. Average kinetic energy of the particles is similar in size to the binding energy.

gasNo shape, will expand to completely fill any container.

Exert a pressure on the walls of the container that depends on the number of particles, the volume and the temp.

Can be compressed considerably.Particles are far apart compared to their size and exert virtually no force on each other except during collisions. Average kinetic energy much greater than the energy binding them together when in liquid state.

3.1.2

Temperature and absolute temperature

Temperature is a measure of the average kinetic energy of the vibrations called thermal motion of the particles in a substance. The higher the temperature, the faster the thermal motion. When moving particles collide, energy is transferred from the one with more KE to the one with less KE. This is why a warm object cools down and a cool object warms up when they are in contact until the two are at the same temperature. This is called thermal equilibrium. Thermal energy is transferred from the object at the higher temperature to the object at the lower temperature until the two are in thermal equilibrium.

A temperature scale is constructed by choosing two reproducible events such as the melting of ice and the boiling of water. The temperature values of these two events are then chosen. For the Celsius scale these are 0oC and 100oC. For the Fahrenheit scale these are 32oF and 212oF. A thermometer is constructed that contains a liquid with a freezing point below that of ice and a boiling point above that of water. Mercury is suitable. It is placed in a reservoir with thin glass walls that enable rapid thermal energy transfer from an object to the mercury. Up from the reservoir is a thin tube that will cause a large change in the level of the mercury over the temperature range. The thermometer is put in melting ice and the level of mercury marked. The thermometer is put in boiling water and the level marked again.

2.

These marks are named 0oC and 100oC or 32oF and 212oF depending on the scale chosen. The distance between the marks is divided up into 100 equal divisions for the Celsius thermometer and 180 divisions for the Fahrenheit thermometer.

The Celsius scale is based on the temperature of melting ice being assigned the value zero. This is an arbitrary zero. At 0oC, the H2O molecules still have thermal motion. Try dropping an ice cube into liquid air. It causes the liquid air to boil! Recall that temperature is related to the thermal motion of particles so it follows that absolute zero of temperature is when thermal motion of particles ceases. The Kelvin temperature scale starts at absolute zero and each degree is the same as the Celsius scale. Zero Kelvin is -273.14 oC but this is conveniently rounded off to -273 oC. The word degree is omitted when using the Kelvin scale.

The following equivalences apply:

0K = -273 oC and 273K = 0oC

Thus Kelvin = Celsius + 2733.1.3

Internal energy

The particles of a substance move and rotate. They have kinetic energy of motion and rotation. Forces between particles means potential energy is stored in them. The total of the potential and kinetic energies of the particles is called the internal energy of the substance.

Previously it was stated that temperature of a substance is related to the motion of its particles More specifically, temperature is a measure of the average kinetic energy of the particles of a substance and will be proven to be true later in this study. The temperature of a substance determines if internal energy will be transferred to or from its surroundings.

Internal energy is the total of the kinetic and potential energies of the particles of the substance.

Conduction transfers internal energy within a substance and from one substance to another in contact when a temperature difference exists.

During collisions, the particles with the greater kinetic energy slow down and the ones with smaller kinetic energy speed up. The higher temperature substance cools and the lower temperature substance warms up until the temperatures are the same. Solids are the best conductors because the particles are stacked close together and collisions occur readily between them. Gases are the poorest conductors because the particles are far apart and collisions between them occur less often than in solids.

Heat is infra-red electromagnetic waves that travel at the speed of light. Thermal energy is the energy transferred to or from a substance when heat is absorbed or radiated away. When heat is absorbed, the internal energy of a substance increases. When heat is radiated away, its internal energy decreases. 3.1.4

Specific heat capacityThermal Capacity is the internal energy increase needed to cause a 1oC increase in the temperature of an object. The value depends on the mass of the object and the substance that makes up the object.

The bigger the mass, the more particles it contains and the more internal energy needed to increase their average kinetic energy. It takes longer to boil the water in the electric jug when it is full!

The forces between the particles of a substance are different for different substances. This means the ease of increasing the average kinetic energy of the particles varies. Less internal energy increase occurs when 100 g of oil has its temp raised 1 oC than for 100 g of water.

The unit of Heat Capacity is Joule per oC, written as J oC-1. The change in the internal energy of an object is equal to its Heat Capacity times its change in temperature.

i.e. H = HC x T

3.

The Specific Heat Capacity is the Heat Capacity of a specific mass of sample. This is either the gram or the kilogram. Strictly speaking, SI units should involve the kilogram but the gram is in common usage with Specific Heat Capacities. Thus the Specific Heat Capacity (SHC) is the change in internal energy when 1 kg of a substance has its temperature changed by 1 oC. The unit of SHC is Joule per kilogram per oC or Joule per gram per oC, written as J kg-1 oC-1 and J g-1 oC-1.The change in the internal energy of an object is equal to its specific heat capacity times its mass times its change in temperature.

i.e. H = SHC x m x T

Some examples of substances and their Specific Heat Capacities are:

water4200 J kg-1 oC-1

ice 2100 J kg-1 oC-1 lead130 J kg-1 oC-1

The diagram opposite shows a direct way of determining the Specific Heat Capacity of water. An electric current raises the temperature of the heating coil that causes an increase in the internal energy of the water. The electrical energy transferred to heat energy can be determined from the readings on the meters and the heating time. The mass of the water and its rise in temperature are measured. Assuming complete transfer of electrical energy to internal energy of the water, a value for the Specific Heat Capacity of water can be calculated.

The steel can and the parts of the calorimeter dipping in the water also have their internal energy raised. Corrections for these must be introduced if an accurate value for the SHC of water is to be obtained.

e.g. The arrangement above caused a 6.4oC rise in temp. when 100 g of water was heated for 2.0 min. Calculate the SHC of water.

Electrical energy transfer = 7.0 J C-1 x 3.5 C s-1 x 120 sec

= 2900 J

Internal energy increase = SHC x m x T

2900 = SHC x 0.10 x 6.4

SHC = 4600 J kg-1 oC-1The accepted value for the SHC of water is 4186 J kg-1 oC-1. A larger value was obtained because the internal energy of the calorimeter was increased. Consequently, the temperature rise of the water was less than if only the water's internal energy was increased. This produced the larger value for the SHC.

The error can be reduced, by including the increase in the internal energy of the calorimeter. The mass of the steel in the calorimeter was 80 g and its SHC is 450 J kg-1 oC-1.

Internal energy increase = SHCwater x m x T + SHCsteel x m x T

2900 = SHCwater x 0.10 x 6.4 + 450 x 0.080 x 6.4

SHCwater = 4200 J kg-1 oC-1

4.The diagram opposite shows an indirect way of finding the SHC of copper. The cube has been heated and its temperature measured. The cube was placed in a calorimeter containing a known mass of water at a known temperature. The temperature at equilibrium can be used to calculate the SHC of copper.

e.g. The calorimeter used contained 80 mL of water at 15.0oC. A 150 g

cube of copper was placed in a beaker of boiling water for several minutes.

The cube was removed and without delay, placed in the calorimeter. The

temperature at equilibrium was 26.5oC.

Internal energy decrease of Cu = SHCCu x 0.15 x (100.0 - 26.5)

IE incr. of H2O + IE incr. of steel = SHCH2O x 0.080 x (26.5-15.0) + 450 x 0.080 x (26.5-15.0)

= 4200 x 0.080 x 11.5 + 450 x 0.080 x 11.5

= 4.3 x 103 J

Assuming the internal energy decrease of Cu is equal to the internal energy increase of the H2O and the steel:

SHCCu x 0.15 x (100.0 - 26.5) = 4.3 x 103

SHCCu = 390 J kg-1 oC-1

The thermal capacity of the 150 g cube of copper = 390 x 0.150

= 58.5 J oC-1

3.1.5

Phase changeAfter it has been raining, the water on the ground evaporates. Evaporation of a liquid can occur even though it is not boiling. A few of its particles have kinetic energy greater than the binding energy and escape into the air. The temperature of the liquid can increase as it evaporates.

When the liquid is boiling, a large number of its particles have kinetic energy greater than the binding energy and escape. Evaporation occurs at the fastest rate but the temperature remains constant,

In a liquid, the average kinetic energy of the particles is similar in size to the binding energy.

Some of the particles have more kinetic energy than the average and some have less than the average. Particles with more KE than the binding energy, move to the top of the liquid and escape from its surface. This will occur even if the liquid is not boiling. When the particles with more KE than average leave the liquid, the average KE of the particles remaining becomes lower. Temperature is a measure of the average KE so evaporation of a liquid causes its temperature to drop. This is why evaporation of perspiration cools your body.

ChangeSubstance behaviorParticle behavior

meltingAs temperature increases, the solid

expands and when a certain temperature is reached, the solid loses its fixed shape and the material becomes runny.As temp increases, particles vibrate faster. Distance between particles increases slightly and forces between particles weaken. Binding energy and av. KE become similar.

evaporatingAs temp. increases, the volume of liquid gradually decreases. The amount of the gaseous state increases in the area above the liquid.As temp. increases, particles vibrate faster. When the KE of a particle exceeds the binding energy, it moves off by itself.

sublimingIf the pressure of the air above a solid is sufficiently low it changes straight to a gas.At low air pressure, the KE of particles exceeds their binding energy at low temps.

When a substance is changing state, its temperature remains the same. The internal energy increases during the process, but this is hidden by the temperature remaining constant. This hidden internal energy increase is called latent heat. The table on the next page summarises the processes of internal energy increases for the melting of ice and the evaporation of water.

5.

ChangeSubstance behaviorParticle behavior

meltingWhen ice is melting, the temperature of the ice/water mixture remains at 0oC until all the ice has meltedHeat energy must be absorbed if ice is to melt. The internal energy of the molecules increases during the melting process. The internal energy absorbed does not increase their average KE. It increases the potential energy stored between the molecules.

evaporatingWhen water is boiling, its temperature remains at 100oC. If the evaporated steam is trapped in a sealed container and heating continues, the temperature will rise above 100oC. A pressure cooker does this.Heat energy must be absorbed if water is to evaporate. The internal energy of the molecules increases during the evaporating process. The internal energy absorbed does not increase their average KE. It increases the potential energy stored between the molecules.

For a typical substance surrounded by normal air pressure, the graph of its temperature versus

heat energy added is shown below.

The state of a substance depends on both its temperature and the pressure of the air around it.

The graph of pressure versus temperature shows the conditions when it will be a solid a liquid

or a gas.

The triple point marks the only temperature and pressure that the solid, liquid and gas states can

exist together. The critical point marks the temperature that if exceeded, increasing the pressure

will not cause the gas to become a liquid. The molecules might be very close but no horizontal

surface forms. Deposition is when a gas changes straight to a solid. It occurs in clouds

when water vapour turns into snowflakes when the temperature and pressure are low enough.

6.

For carbon dioxide, the triple point is at -56 OC and 5.1 atmosphere. Therefore when placed on

the table and warming up at 1 atmosphere of pressure, it will sublime.

For water, the triple point is at 0 OC and 0.006 atmosphere. Therefore when ice is placed on

the table and warming up at 1 atmosphere of pressure, it will melt.

Freeze dried food is created by freezing the food so the water content turns into solid ice. Then,

the air pressure around the food is dropped sufficiently and warmed slightly until the ice

sublimes and leaves the food as water vapour. This makes the food light for hikers to carry but

free from bacteria that would multiply if the water content were removed by strongly heating it.3.1.6

Specific latent heat.

Specific latent heat is the amount of internal energy increase needed to cause 1 kg of a substance to change its state without changing its temperature. Specific latent heat of Fusion is for melting and Specific latent heat of Vaporization is for evaporating. The unit is J kg-1.

LH of Fusion of ice = 3.33 x 105 J kg-1 LH of Vaporization of water = 2.26 x 106 J kg-1The LH of V is greater than the LH of F due to a much greater increase in the distance between the molecules during evaporation and hence a larger increase in the potential energy between the molecules.

From the definition of Latent Heat it follows that:

Change in IE during a change of state J = Latent Heat of substance J kg-1 x mass in kg

A saucepan contains 2.0 kg of water at 20oC. When heated on the stove, the internal energy of the water increases by 9.0 x105 J. How much of the water evaporated?

The LHV of water is 2.3 x 106 J kg-1.

Increase in IE = Increase for water 20 to 100oC + Increase for water 100oC to steam 100oC

= SHCwater x mwater x T + LHVwater x mwater evaporated

9.0 x105 = 4186 x 2.0 x 80 + 2.3 x 106 x m

9.0 x105 = 6.7 x 105 + 2.3 x 106 x m

m = 0.1 kg

3.2

Modelling a gas3.2.1

Pressure.The pressure on a surface is equal to the force on the surface divided by its area. P = F/A. The unit of pressure is N m-2. A pressure of 1 N m-2 is called 1 Pascal Pa.

Imagine a very thin sheet of plywood with a mass of 100 g and an area of 1 m2. The wood weighs about 1 N. When the wood lies flat on the floor it applies a pressure of 1 Pa on the floor. The Pascal is a small unit. Air applies a force of about 10 N on each cm2. This amounts to about105 N m-2 or 105 Pa. The unit kiloPascal kPa is used frequently. Normal air pressure is about 101 kPa.

The walls of a container are continually bombarded by fast moving gas molecules. When a molecule bounces off the wall, it pushes outward against the wall and the wall pushes inwards on the molecule. Each molecule is very tiny and many collisions occur each second. It seems that the container is receiving a constant outward force. The force per square metre of area is the pressure of the gas.

3.2.2

Kinetic model of an ideal gas.

An ideal gas consists of tiny particles that are small compared to their separation. The particles

only apply forces on each other during collisions. Most gases are close to being ideal.

7.

The kinetic model of an ideal gas assumes:

*that a gas consists of a large number of very small particles that are far apart and are moving randomly in all directions with a range of speeds. The average kinetic energy of the particles is a measure of the gas's temperature.

*the particles obey the laws of mechanics.

*The particles exert no force on each other except during collisions.

*When a particle collides with the container wall or another particle it bounces off making an elastic collision. During impact the particle applies a force to the container and the container applies a force back on the particle. The net outward force of all the particles on the container wall gives rise to the pressure of the gas.

Real gas exert molecules small forces on each other between collisions. When a gas is compressed or its temperature is low the gas is is close to being a liquid, it does not show ideal behavior. 3.2.3

Mole, molar mass and the Avogadro constantOne mole of a substance is the mass of the substance that contains the same number of particles as there are in 12.00 grams of Carbon 12. Carbon 12 is the isotope containing 6 protons and 6 neutrons in the nuclei. The mass of a carbon atom is 1.99 x 10-23 g so in 12.0 g there are 12.0/1.99 x 10-23 or 6.02 x 1023 atoms of carbon. Thus a mole of a substance is the mass of 6.02 x 1023 particles of the substance. This number is called Avogadro's constant NA. The word particle can stand for a single He atom of helium or a single CO2 molecule of carbon dioxide.

Consider a molecule of CO2. It consists of a carbon atom and two oxygen atoms. A carbon atom has 12 particles in its nucleus and an oxygen atom has 16 particles in its nucleus. In a CO2 molecule there are 12 + 16 x 2 or 44 particles in its nucleus. There are 12 particles in the nucleus of carbon atoms and 12 g of carbon is 1 mole and contains 6.02 x 1023 carbon atoms. For CO2 there are a total of 44 particles in the 3 nuclei making up the molecule. To a good approximation, 44 g of CO2 is 1 mole of CO2 and contains 6.02 x 1023 molecules of carbon dioxide.

The numbers 12 and 16 are the mass numbers of carbon and oxygen. The number 44 is the molecular mass of carbon dioxide. More accurate mass numbers of elements can be found in periodic tables e.g. C 12.00, O 15.99 and Cl 35.45. These values can be used to calculate accurate molecular masses. e.g. 1 mole of chlorine Cl2 is 35.45 x 2 or 70.9 g.

To sum up the mass in gram of one mole of a substance is approximately equal to the total number of protons and neutrons in a particle of the substance and accurately equal to the sum of the mass numbers of the atoms in a particle of the substance.

The mole concept was discussed in this section because you will see soon there is an advantage to measure the mass of a gas in mole.

3.2.4

Equation of state for an ideal gas.If a gas is in a sealed container with a fixed volume, the pressure of the gas increases when its temperature increases. When the temperature increases, the average KE of the particles increases. More collisions occur with the walls each second and the collisions are more forceful. The outward force on the container increases.

When the pressure on a gas is increased, the volume of the gas decreases. When the volume decreases, the molecules travel less distance before colliding with the walls. More collisions occur with the walls each second. The outward pressure on the container increases to match the pressure outside the container.

8.

The pressure of a gas is also affected by the mass of the gas. The greater the mass of gas, the more particles present and the more collisions that occur with the walls of the container each second. If the volume remains the same as in pumping up a tyre, the pressure increases. If the pressure remains the same, the volume increases.

A bike pump is a cylinder of gas with a piston. Putting your finger over the hole and pushing in the piston increases the pressure on the air in the pump. When the pressure on the air is increased, the volume of air decreases. When the volume decreases, the molecules travel less distance before colliding with the walls. More collisions occur with the walls each second. The outward force on the container increases and the pressure of the air inside the pump increases.

The movement of the piston down on the gas does work on the gas. The gas molecules bounce off the moving piston with a faster speed just like a ball that is hit by a moving tennis racquet. The Temperature of the gas increases as the average kinetic energy of the gas molecules has increased. Conduction of heat energy to the surroundings results in a decrease in internal energy of the gas. When the piston is pulled back (finger still on the hole) the temperature of the gas is lower than its initial value. This is the principle that fridges use to operate.

All the discussions above indicate there is a connection between the pressure, temperature, volume and mass of a gas.

The walls of a container are continually bombarded by fast moving gas molecules. When a

molecule bounces off the wall, it pushes outward against the wall and the wall pushes inwards

on the molecule. Each molecule is very tiny and many collisions occur each second. It seems

that the container is receiving a constant outward force. The force per square metre of area is the

pressure of the gas.

The pressure on a surface is equal to the force on the surface divided by its area. P = F/A. The

unit of pressure is N m-2. A pressure of 1 N m-2 is called 1 Pascal Pa. The Pascal is a small unit.

Air applies a force of about 10 N on each cm2. This amounts to about105 N m-2 or 105 Pa. The

unit kiloPascal kPa is used frequently. Normal air pressure is about 101 kPa.

If the temperature of a gas remains constant

the volume of the gas is inversely

proportional to its pressure. V ( 1/P or

PV = constant. This relationship is called

Boyle's Law, as it was discovered by Robert

Boyle in the mid 1600's.

The diagram shows a gas inside a container that has a

piston free to slide. The piston will stop in the

position where the gas pressure inside the container

is the same as atmospheric pressure. When the gas

is heated, its pressure will remain constant and its

volume increases as the piston slides.

Jacques Charles discovered around 1800 that the volume of the gas increased in equal amounts

for equal rises in temperature. This is now described as volume is directly proportional to its

absolute Temperature (measured in K) V (T when P constant. This is called Charles' Law.

If a gas is in a sealed container with a fixed volume, the pressure of the gas increases when its

temperature increases. When the temperature increases, the average KE of the particles

increases. More collisions occur with the walls each second and the collisions are more forceful.

The outward force on the container increases. The air pressure in a cold car tyre is lower than it

pressure after the car has been driven and the tyre has increased temperature. Recommended

tyre pressures apply when the tyre is cold.

9.

Joseph Gay-Loussac found that for a gas with constant volume, its pressure is directly

proportional to its absolute temperature i.e. P ( T when V constant. This is Gay-Loussac's Law.

The three Gas Laws just described are not true laws in that they are only true for ideal gases.

They are good approximations when applied to a real gas that is not close to becoming a liquid

when pressure is high or temperature is low.

Combining the relationships discussed so far gives:

PV T

PV = KT

Adding more molecules to a sample of gas means more collisions with the walls and pressure

would increase. For the pressure and temperature to remain the same, the volume increase to

reduce the frequency of molecules hitting the walls of the container. In the last equation, if P

and T remain constant and the number of molecules increases, the value of K must increase

also. Thus K is proportional to the number of molecules N.

K = kN

PV = kNT

where N/V is the number of molecules per m3.

The table shows the number of molecules per m3 when at STP - Standard Temperature and

Pressure - At STP, pressure is 1 atmosphere, 1.01 x 105 N m-2 and temperature is 0oC, 273 K.

Gas

No. of molecules per m3

Hydrogen

2.68 x 1025

Helium

2.68 x 1025

Oxygen

2.68 x 1025

When at the same temperature and pressure, the number of molecules per m3 is the same for all

gases. Therefore the constant k is the same for all gases i.e. it is a universal constant.

k is called Boltzmann's constant

Imagine N molecules of an ideal gas are in a cubic container with

volume V and area of sides A.

A molecule with mass m is moving directly at one wall with velocity v.

Assuming the molecule bounces off the wall at the same speed:

v = 2 v

The change in momentum of the molecule is given by:

p = 2m v

10.

In the time t it takes the molecule to reach the wall, if n molecules hit the wall, the average

impulse applied to the wall is:

F t = 2m v x n

The volume of the gas between the dotted line and the right hand wall is vt x A

This volume is the fraction vtA/V of the total volume. Therefore the number of molecules in

this volume is N x vtA/V.

However half the molecules in this volume will be moving away from the wall. Therefore the

number of molecules n that collide with the wall in time t is 0.5NvtA/V. Substituting the value

of n into the impulse equation gives:

F t = 2mv x 0.5NvtA/V

F = Amv2 x N/V

The pressure P of the gas on the wall is force F divided by area A. Therefore:

P = mv2 x N/V

This last line was calculated assuming all the molecules were travelling directly towards the

wall. Call this the x direction. The last equation becomes:

P = mvx2 x N/V

However, the molecules will be moving in random directions. Molecules will have velocity

components in the x, y and z directions.

For a molecule with velocity v, its components have magnitudes vx, vy and vz. Applying

Pythagoras in 3-D gives:

v2 = vx2 + vy2 + vz2

Given the randomness of the motion of a large number of molecules, the average values of vx2,

vy2 and vz2 should be equal. Therefore:

v2av = 3 vx2av

hence vx2av = 1/3 v2av

The expression for pressure now becomes:

P = 1/3 mv2av x N/V

= 2/3 x mv2av x N/V

= 2/3 x KEav x N/V

hence PV = 2/3 x KEav x N

Earlier it was established that PV = kNT. Therefore:

2/3 x KEav x N = kNT

hence KEav = 3/2kT

The average kinetic energy of the gas molecules is directly proportional to its absolute

temperature. The opposite viewpoint is the temperature of a gas is a measure of the average

Kinetic Energy of its molecules.

11.

What is the average speed of an air molecule at 20oC? Most of the air is nitrogen. The mass

of a nitrogen molecule N2 is 4.65 x 10-26 kg. The absolute temperature is 293 K.

KEav = 3/2kT

mvav2 = 3/2kT

x 4.65 x 10-26 x vav2 = 1.5 x 1.38 x 10-23 x 293

vav = 511 m s-1

Ever wondered why the hair of people sitting behind the windscreen of an opened topped car is

blown forward!!!? The air molecules are rushing into the space behind the windscreen faster

than the car and its passengers are travelling. Wear a hat or you will have a sore scalp.

The pressure of a gas is affected by the mass of the gas. When V ( 1/P, V ( T and V ( m are

combined, then PV ( mT. As an equation PV = mT where (kappa) is a constant.

Consider the following information about gases at STP - 0oC and 1 atmosphere:

Gas

mass kg

Volume m3

no. of mole

oxygen

0.032

0.0224

1

hydrogen

0.002

0.0224

1

carbon dioxide0.044

0.0224

1

Each gas has the same volume 0.0224 m3, the same temperature 273 K, the same pressure

1.01 x 10-5 N m-2 but a different mass in kg. Hence if the information for each gas is substituted

into the equation PV = mT, a different value for would be obtained for each gas.

However, the number of mole is the same for each gas. If the mass is measured in number of

mole n, a new constant R will apply. i.e. PV = RnT. All gases in the table have the same values

for P, V, n and T so R is a universal constant.

PV = RnT

1.01 x 105 x 0.0224 = R x 1 x 273

R = 8.3 J mol-1 K-1 called the universal gas constant

By tradition the ideal gas equation is written as PV = nRT.

In a closed container, the mass of gas remains constant. Therefore:

PV/T = nR

= constant for the same no. of mole

Previously it was established that PV = 2/3 KEav x N where KEav is the average kinetic energy of

a single molecule and N is the total number of molecules.

Comparing this to the ideal gas equation PV = RnT means:

2/3 KEav x N = nRT

rearranging KEav = 3/2 x n/N x RT

The total kinetic energy of all the molecules is N times the last equation i.e.

KEtotal = 3/2nRT

One mole is the mass of 6.023 x 1023 molecules of the gas. The number 6.023 x 1023 is called

Avogadro's constant NA. If 3.012 x 1023 molecules are present, the mass is 0.5 mol. Hence the

number of mole is equal to the number of molecules present divided by Avogadro's constant

i.e. n = N/NA. Rearranging, n/N = 1/NA. Substituting into the second last equation gives:

KEav = 3/2 RT/NA

12.

What is the average kinetic energy of a gas molecule at 20oC?

Using KEav = 3/2kT

Using KEav = 3/2 RT/NA

KEav = 3/2 x 1.38 x 10-23 x 293

KEav = 3/2 x 8.3 x 293/6.023 x 1023

= 6.1 x 10-21 J

= 6.1 x 10-21 J

In a sealed container, the mass of the gas remains constant. This means PV/T = nR i.e. PV/T is a

constant. For two different sets of P, V and T values, this is usually written as:

P1V1/T1 = P2V2/T2

You will be expected to solve problems involving the ideal gas equation. Pressure gauges

measure the pressure difference between the gas pressure inside a container and the air outside.

If a gauge reads 1atmosphere, the pressure inside the container is one atmosphere higher than

the air outside. The pressure of the gas inside the container is 2 atmospheres. This is called the

absolute pressure. Thus absolute pressure equals gauge pressure plus 1 atmosphere.

3.2.5

Differences between real and ideal gases

The table below describes differences between real and ideal gases.

Real

Ideal

Smallish number of particles.

Very large number of particles.

Molecules have size and shape.

Molecules are point particles.

Molecules can be close or far apart.

Molecules are far apart compared with size.

The volume of the molecules issignificant

The volume of the molecules is insignificant

compared with the container.

compared with the container.

Molecules exert forces on each other

Molecules exert no forces on each other

between collisions.

between collisions.

Molecules have energy of rotation and

Molecules have no energy of rotation or

vibration within the molecule.

vibration.

During collisions some of the kinetic

Collisions are elastic. Total kinetic energy

energy is transferred to energy of rotation

before and after the collision is the same.

and vibration.

Do not obey the ideal gas laws when close

Always obey the ideal gas laws.

to turning into a liquid.

PV = nRT not always accurate.

PV = nRT always accurate.

Exchange energy with surroundings.

Do not exchange energy with surroundings.

Copyright B & G Scientific 2014. May be copied for student use. Thermal Physics.