IB Physics Notes For Thermal Physics
Topics Covered:
 Temperature And Heat Flow
 Kelvin And Celsius
 Gases
 Internal Energy Of Molecules
 Specific Heat Capacity
 Methods Of Measuring Heat Capacities And Specific Heat Capacities
 The States Of Matter And Latent Heat
 Methods Of Measuring Latent Heats
 Gas Laws 1
 Gas Laws 2
 Definitions
 Molecular Model Of An Ideal Gas
 Pressure Law
 Charles' Law
 Boyle's Law
 Sample Questions
Temperature And Heat Flow

The direction of the natural flow of thermal energy between two objects is determined by the 'hotness' of each object.

Thermal energy naturally flows from hot to cold. The temperature of an object is a measure of how hot it is. In other words, if two objects are placed in thermal contact, then the temperature difference between the two objects will determine the direction of the natural transfer of thermal energy. Thermal energy is naturally transferred 'down' the temperature difference  from high temperature to low temperature.

Eventually, the two objects would be expected to reach the same temperature. When this happens, they are said to be in thermal equilibrium.

Heat is not a substance that flows from one object to another. What has happened is that thermal energy has been transferred. Thermal energy (heat) refers to the nonmechanical transfer of energy between a system and its surroundings.
Kelvin And Celsius

In order to use Kelvin and Celsius, you do not need to understand the details of how either of these scales has been defined, but you do need to know the relation between them. Most everyday thermometers are marked with the celsius scale and temperature Is quoted in degrees Celsius (°C).

There is an easy relationship between a temperature 'Tas' measured on the Kelvin scale and the corresponding temperature 't' as measured on the Celsius scale. The approximate relationship is T (K) = (°C) + 273. This means that the 'size' of the units used on each scale is identical, but they have different zero points.

The Kelvin scale is an absolute thermodynamic temperature scale and a measurement on this scale is also called the absolute temperature. Zero Kelvin is called absolute zero.
Gases

For a given sample of gas, the pressure, the volume and the temperature are all related to one another.

The SI units of pressure are $N{m}^{2}$ or Pa (Pascals). 1*Pa = 1*N

The temperature, t, of the gas is measured in °C or K

In order to investigate how these quantities are interrelated, we choose:

one quantity to be the independent variable (the thing we alter and measure)

another quantity to be the dependent variable (the second thing we measure).

The third quantity needs to be controlled (i.e. kept constant ).

The specific values that will be recorded also depend on the mass of gas being investigated and the type of gas being used so these need to be controlled as well.

Internal Energy Of Molecules
 The molecules have kinetic energy. They are moving. To be absolutely precise. a molecule has either translational kinetic energy
 The molecule also has potential energy because of the intermolecular forces.
 Molecules with large mass move with a lower average speed
 Molecules with a small mass, move with a higher average speed
Specific Heat Capacity
 Two different blocks with the same mass and same energy input will have a different temperature change. We define the thermal capacity C of an object as the energy required to raise its temperature by 1 K.
 Different objects (even different samples of the same substance) will have different values of heat capacity.
 Specific heat capacity is the energy required to raise a unit mass of a substance by 1 K. 'Specific' here just means 'per unit mass'.
 A particular gas can have many different values of specific heat capacity—it depends on the conditions.
 It generally takes the same amount of energy to raise the temperature of an object from 25 °C to 35 °C as it does for the same object to go from 402 °C to 412 °C. This is only true so long as energy is not lost from the object.
 If an object is raised above room temperature, it starts to lose energy. The hotter it becomes, the greater the rate at which it loses energy.
Methods Of Measuring Heat Capacities And Specific Heat Capacities

Electrical Method
 In this method, the following are the sources of experimental error come from the loss of thermal energy from the apparatus.
 The container for the substance and the heater will also be warmed up.
 It will take some time for the energy to be shared uniformly through the substance.
 In this method, the following are the sources of experimental error come from the loss of thermal energy from the apparatus.

Method of Mixtures
 The known specific heat capacity of one substance can be used to find the specific heat capacity of another substance.
 Procedure:
 Measure the masses of the liquids
 Measure the two starting temperatures
 Mix the two liquids together.
 Record the maximum temperature of the mixture
 If no energy is lost from the system then, energy lost by hot substance cooling down = energy gained by cold substance heating up
 The main source of experimental error is
 The loss of thermal energy from the apparatus; particularly while the liquids are being transferred.
 The changes in the temperature of the container also need to be taken into consideration for a more accurate result.
The States Of Matter And Latent Heat

When a substance changes phase, the temperature remains constant even though thermal energy is still being transferred.

Cooling Curve For Molten Lead (Idealized)

The amount of energy associated with the phase change is called the latent heat. The technical term for the change of phase from solid to liquid is fusion and the term for the change from liquid to gas is vaporization.

The energy given to the molecules does not increase their kinetic energy so it must be increasing their potential energy. Intermolecular bonds are being broken and this takes energy. When the substance freezes bonds are created and this process releases energy.

It is a very common mistake to think that the molecules must speed up during a phase change. The molecules in water vapour at 100 °C must be moving with the same average speed as the molecules in liquid water at 100 °C.

The specific latent heat of a substance is defined as the amount of energy per unit mass absorbed or released during a change of phase. In symbols,

Specific latent heat $L=\frac{Q}{M}(Jk{g}^{1});Q=ML$

In the idealized situation of no energy loss, a constant rate of energy transfer into a solid substance would result in a constant rate of increase in temperature until the melting point is reached:

The specific heat capacity of the solid as the gradient of the line that corresponds to the liquid phase is greater than the gradient of the line that corresponds to the solid phase. A given amount of energy will cause a greater increase in temperature for the liquid when compared with the solid.

Methods Of Measuring Latent Heats

The two possible methods for measuring latent heats are very similar in principle to the methods for measuring specific heat capacities. A method for measuring the

Method 1: Electrical Circuit.

The amount of thermal energy provided to water at its boiling point is calculated using electrical energy = I t V. The mass vaporized needs to be recorded.

The specific latent heat $L=\frac{ItV}{({m}_{1}{m}_{2})}$

Sources of experimental error

Loss of thermal energy from the apparatus.

Some water vapour will be lost before and after timing.



Method 2: Fusion Of Water

Providing we know the specific heat capacity of water, we can calculate the specific latent heat of fusion for water. For example, ice (at 0 °C) is added to warm water and the temperature of the resulting mix is measured.

If no energy is lost from the system then, energy lost by water cooling down = energy gained by the ice

Sources of experimental error

Loss (or gain) of thermal energy from the apparatus.

If the ice had not started at exactly zero, then there would be an additional term in the equation in order to account for the energy needed to warm the ice up to 0 °C.

The water clinging to the ice before the transfer.



Gas Laws 1
 Although pressure and volume both vary linearly with Celsius temperature, neither pressure nor volume is proportional to Celsius temperature.
 A different sample of gas would produce a different straightline variation for pressure (or volume) against temperature but both graphs would extrapolate back to the same low temperature, 273 °C. This temperature is known as absolute zero.
 As pressure increases, the volume decreases. In fact, they are inversely proportional.
 At constant $V,p\propto Tor\frac{p}{T}=$ constant (the pressure law)
 At constant $p,V\propto Tor\frac{V}{T}=$constant (Charles's law)
 At constant $T,p\propto \frac{1}{V}orpV=$ constant (Boyle's law)
 These relationships are known as Ideal gas laws.
 The temperature is always expressed in Kelvin
 . These laws do not always apply to experiments done with real gases.
 A real gas is said to 'deviate' from ideal behaviour under certain conditions (e.g. high pressure).
Gas Laws 2

The three ideal gas laws can be combined together to produce one mathematical relationship: $\frac{pV}{T}=$constant

This constant will depend on the mass and type of gas.

If we compare the value of this constant for different masses of different gases, it turns out to depend on the number of molecules that are in the gas  not their type.

In this case, we use the definition of the mole to state that for n moles of an ideal gas: $\frac{pV}{nT}=$a universal constant

The universal constant is called the molar gas constant R. The SI unit for R is J $mo{l}^{1}{K}^{1}$

R = 8.314 J $mo{l}^{1}$ ${K}^{1}$
Definitions
The concepts of the mole, molar mass and the Avogadro constant are all introduced so as to be able to relate the mass of gas (an easily measurable quantity) to the number of molecules that are present in the gas.
 Ideal gas
 An ideal gas Is one that follows them, laws for all values of P, Vand T

Mole

It is the basic SI unit for 'amount of substance'. One mole of any substance is equal to the amount of that substance that contains the same number of particles as 0.012 kg of carbon12. When writing the unit it is (slightly) shortened to the mol.


Molar mass

The mass of one mole of a substance is called the molar mass. A simple rule applies. If an element has a certain mass number, A, then the molar mass will be A grams.


Avogadro constant, NA

This is the number of atoms in 0.012 kg of carbon12. It is 6.02 x ${10}^{23}$.


Ideal Gases And Real Gases

An ideal gas is a one that follows the gas laws for all values of p, V and T and thus ideal gases cannot be liquefied. Real gases, however, can approximate to ideal behaviour providing that the intermolecular forces are small enough to be ignored. For this to apply, the pressure/density of the gas must be low and the temperature must be moderate.

Molecular Model Of An Ideal Gas
 Kinetic Model Of An Ideal Gas
 Assumptions:
 Newton's laws apply to molecular behaviour
 There are no intermolecular forces except dining a collision
 The molecules are treated as points
 The molecules are in random motion
 The collisions between the molecules arc elastic (no energy is lost)
 There is no time spent in these collisions.
 Assumptions:
 The pressure of a gas is a result of collisions between the molecules and the walls of the container in which that gas has been stored.
 When a molecule bounces off the walls of a container its momentum changes (due to the change in direction  momentum is a vector).
 There must have been a force on the molecule from the wall (Newton II).
 There must have been an equal and opposite force on the wall from the molecule (Newton III).
 Each time there is a collision between a molecule and the wall, a force is exerted on the wall.
 The average of all the microscopic forces on the wall over a period of time means that there is effectively a constant force on the wall from the gas.
 This force per unit area of the wall is what we call pressure. $P=\frac{F}{A}$.
 Since the temperature of a gas is a measure of the average kinetic energy of the molecules, as we lower the temperature of a gas the molecules will move slower.
 At absolute zero, we imagine the molecules to have zero kinetic energy. We cannot go any lower because we cannot reduce their kinetic energy any further!
Pressure Law
 Macroscopically, at a constant volume, the pressure of a gas is proportional to its temperature in kelvin.
 Microscopically this can be analysed as follows
 If the temperature of gas goes up, the molecules have more average kinetic energy  they are moving faster on average.
 Fastmoving molecules will have a greater chance of momentum when they hit the walls of the container. Thus the microscopic force from each molecule will be greater.
 The molecules are moving faster so they hit the walls more often.
 For both these reasons, the total force on the wall goes up.
 Thus the pressure goes up. lowtemperature high temperature
Charles' Law

Macroscopically, at constant pressure, the volume of a gas is proportional to its temperature in kelvin.

Microscopically this can be analysed as follows

A higher temperature means faster moving molecules.

Faster moving molecules hit the walls with a greater microscopic force

If the volume of the gas increases, then the rate at which these collisions take place on a unit area of the wall must go down.

The average force on a unit area of the wall can thus be the same. Thus the pressure remains the same.

Boyle's Law
 Macroscopically, at a constant temperature, the pressure of a gas is inversely proportional to its volume.
 Microscopically this can be seen to be correct.
 The constant temperature of gas means that the molecules have a constant average speed.
 The microscopic force that each molecule exerts on the wall will remain constant.
 Increasing the volume of the container decreases the rate with which the molecules hit the wall the average total force decreases.
 If the average total force decreases the pressure decreases.
Sample Questions
 When the volume of a gas is isothermally compressed to a smaller volume, the pressure exerted by the gas on the container walls increases. The best microscopic explanation for this pressure increase is that at the smaller volume
 A. the individual gas molecules are compressed
 B. the gas molecules repel each other more strongly
 C. the average velocity of gas molecules hitting the wall is greater
 D. the frequency of collisions with gas molecules with the walls is greater
 A lead bullet is fired into an iron plate, where it deforms and stops. M a result, the temperature of the lead increases by an amount AT. For an identical bullet hitting the plate with twice the speed, what is the best estimate of the temperature increase?
 A. $\u2206$T
 B. 2 $\u2206$T
 C. 4 $\u2206$T
 In winter, in some countries, the water in a swimming pool needs to be heated.
 Estimate the cost of heating the water in a typical swimming pool from 5 °C to a suitable temperature for swimming. You may choose to consider any reasonable size of the pool. Clearly show any estimated values. The following information will be useful:
 Specific heat capacity of water 4186 J $k{g}^{1}{K}^{1}$
 The density of water 1000 kg ${m}^{3}$
 Cost per kW h of electrical energy $0.10
 (i) Estimated values
 (ii) Calculations
 Estimate the cost of heating the water in a typical swimming pool from 5 °C to a suitable temperature for swimming. You may choose to consider any reasonable size of the pool. Clearly show any estimated values. The following information will be useful:
 This question is about determining the specific latent heat of fusion of ice. A student determines the specific latent heat of fusion of ice at home. She takes some ice from the freezer, measures its mass and mixes it with a known mass of water in an insulating jug. She stirs until all the ice has melted and measures the final temperature of the mixture. She also measured the temperature in the freezer and the initial temperature of the water. She records her measurements as follows:
 Mass of ice used: ${m}_{1}$ = 0.12 kg
 The initial temperature of ice: Ti —12 °C
 The initial mass of water: ${m}_{w}$ = 0.40 kg
 The initial temperature of water: ${T}_{w}$ = 22 °C
 The final temperature of the mixture: ${T}_{f}$= 15 °C
 The specific heat capacities of water and ice are
 ${c}_{w}$= 4.2 kJ $k{g}^{1}$ °${C}^{1}$ and ${c}_{i}$ = 2.1 kJ $k{g}^{1}$°${C}^{1}$
 a) Set up the appropriate equation, representing energy transfers during the process of coming to thermal equilibrium, that will enable her to solve for the specific latent heat Li of ice. Insert values into the equation from the data above, but do not solve the equation.
 b) Explain the physical meaning of each energy transfer term in your equation (but not each symbol).
 State an assumption you have made about the experiment, in setting up your equation in (a).
 d) Why should she take the temperature of the mixture immediately after all the ice has melted?
 e) Explain from the microscopic point of view, in terms of molecular behaviour, why the temperature of the ice does not increase while it is melting.