IMAT Physics · Formula sheet

Heat and Thermodynamics formulas

9 formulas, 3 reference tables and 17 common traps for IMAT Physics Heat and Thermodynamics, grouped by subtopic.

Temperature, Heat and How Heat Moves

Temperature, heat and the Kelvin scale

Celsius to kelvin

T\,(\text{K}) = \theta\,(^\circ\text{C}) + 273

$T$ absolute temperature, in K
$\theta$ temperature in degrees Celsius

Thermal expansion of solids and liquids

Linear expansion

\Delta L = \alpha L_0 \Delta T

$\Delta L$ increase in length, in m
$\alpha$ coefficient of linear expansion, in K⁻¹
$L_0$ original length, in m
$\Delta T$ temperature rise, in K (or °C)

Heat transfer: conduction, convection and radiation

Mechanism	How the energy moves	Needs a medium?	Everyday example
Conduction	Vibrating particles and free electrons pass energy to neighbours	Yes; works best in solids, especially metals	The handle of a metal pan getting hot
Convection	Warm, less dense fluid rises and cooler fluid sinks, forming a current	Yes; only liquids and gases	Warm air rising above a radiator
Radiation	Electromagnetic waves, mainly infrared	No; it crosses a vacuum	Sunlight warming the Earth through space

Common traps

A temperature change is not converted by adding 273

Adding 273 converts a temperature reading. A difference of 50 °C is a difference of 50 K, because the two scales have the same step size. Options that add 273 to a temperature rise are wrong.

Heat and temperature are different quantities

Temperature is an average per particle; heat is energy that moves. A large cool lake holds far more internal energy than a small hot spark. Statements treating a body as containing heat, or temperature as an amount of energy, are wrong.

Feeling colder does not mean a lower temperature

Objects left in the same room reach the same temperature. Metal feels colder than wood because it conducts energy out of your skin faster, not because it is cooler.

Water does not always expand when heated

Between 0 °C and 4 °C water contracts as it warms, and it is densest at about 4 °C. Options claiming every liquid expands on heating at every temperature, or that water is densest at 0 °C, are wrong.

Specific Heat, Latent Heat and Mixing

Learn this subtopic in the notes

Specific heat capacity: the energy to warm a mass

Heating without a change of state

Q = m c \Delta T \qquad Q = P t

$Q$ energy supplied, in J
$m$ mass, in kg
$c$ specific heat capacity, in J kg⁻¹ K⁻¹
$\Delta T$ temperature change, in K
$P,\ t$ heater power in W, and time in s

Latent heat and changes of state

Energy for a change of state

Q = m L

$Q$ energy absorbed or released, in J
$m$ mass that changes state, in kg
$L$ specific latent heat, in J kg⁻¹

Calorimetry: heat lost equals heat gained

Mixing in an insulated container

m_1 c_1 (T_1 - T) = m_2 c_2 (T - T_2)

$m_1, c_1, T_1$ mass, specific heat and starting temperature of the hotter body
$m_2, c_2, T_2$ the same for the colder body
$T$ the shared final temperature

Common traps

Power times time needs seconds

A watt is a joule per second, so

Q = Pt

gives joules only with

t

in seconds. Leaving minutes in gives an answer 60 times too small, and that wrong value is usually one of the options.

Thermal conductivity is not specific heat capacity

Specific heat capacity says how much energy warms a kilogram; thermal conductivity says how fast heat passes through a material. A calorimetry calculation never needs the conductivity.

Melting breaks forces between molecules, not bonds inside them

When wax or ice melts, the energy overcomes the attractions between molecules. The covalent bonds within each molecule are untouched; breaking those would be a chemical change. Statements that melting breaks the bonds inside the molecules are wrong.

The temperature does not rise during a change of state

While a pure substance melts or boils, the energy supplied goes into the change of state and the temperature stays constant. A heating curve shows a flat section there, not a slope.

Melted ice must still warm up

When ice melts and the mixture ends above 0 °C, the meltwater also warms from 0 °C to the final temperature. The energy balance needs

m L_f + m c T

on the gaining side, not

m L_f

alone.

Ideal Gases and Kinetic Theory

Learn this subtopic in the notes

The ideal gas equation and the three gas laws

Ideal gas equation

pV = nRT \qquad \frac{p_1 V_1}{T_1} = \frac{p_2 V_2}{T_2}

$p$ pressure, in Pa
$V$ volume, in m³
$n$ amount of gas, in mol
$R$ gas constant, 8.31 J mol⁻¹ K⁻¹
$T$ absolute temperature, in K

Kinetic theory: temperature and molecular speed

Mean kinetic energy of a molecule

\overline{E_k} = \tfrac{1}{2} m \langle v^2 \rangle = \tfrac{3}{2} k_B T

$m$ mass of one molecule, in kg
$\langle v^2 \rangle$ mean of the squared speeds
$k_B$ Boltzmann constant, 1.38 × 10⁻²³ J K⁻¹
$T$ absolute temperature, in K

Common traps

Gas laws need kelvin, never Celsius

Proportions such as

p \propto T

hold only for absolute temperature. Heating from 10 °C to 20 °C does not double the pressure; it raises it by about 3.5%, from 283 K to 293 K.

Boyle's law is not a straight line on a p against V graph

At constant temperature,

p = \text{constant}/V

, a curve that falls ever less steeply. A straight line through the origin appears only when

p

is plotted against

1/V

Heavier molecules do not have more kinetic energy

At the same temperature all gas molecules have the same mean kinetic energy, whatever their mass. The heavier ones simply move more slowly, with a smaller mean square speed.

The First Law and Gas Processes

Learn this subtopic in the notes

The first law of thermodynamics and the work done by a gas

First law and work at constant pressure

\Delta U = Q - W \qquad W = p\,\Delta V

$\Delta U$ change in internal energy of the gas, in J
$Q$ heat given to the gas, in J
$W$ work done by the gas, in J
$p,\ \Delta V$ pressure in Pa, change in volume in m³

Isothermal, isobaric, isochoric and adiabatic processes

Process	What stays constant	First law becomes	Line on a p against V graph
Isothermal	Temperature, so the internal energy of an ideal gas	$Q = W$	A hyperbola, $pV$ constant
Isobaric	Pressure	$Q = \Delta U + p\Delta V$	A horizontal line
Isochoric	Volume, so no work is done	$Q = \Delta U$	A vertical line
Adiabatic	No heat in or out	$\Delta U = -W$	A curve steeper than the isotherm through the same point

Common traps

Work done by the gas and work done on it have opposite signs

Compression means work is done on the gas, so the work done by the gas is negative. Putting 600 J of compression into

\Delta U = Q - W

+600

gives the internal energy change with the wrong sign.

Adiabatic does not mean constant temperature

Adiabatic means no heat flows. The temperature changes, because work changes the internal energy. Isothermal is the opposite case: the temperature is constant, and heat does flow in or out.

The Second Law, Entropy and Heat Engines

Learn this subtopic in the notes

Heat engines, efficiency and the Carnot limit

Efficiency and the Carnot limit

\eta = \frac{W}{Q_h} = 1 - \frac{Q_c}{Q_h} \qquad \eta_{\max} = 1 - \frac{T_c}{T_h}

$Q_h,\ Q_c$ heat taken from the hot reservoir, heat given to the cold one, in J
$W$ useful work done, in J
$T_h,\ T_c$ reservoir temperatures, in K

The second law and entropy

Process	Entropy change	Why
Ice melting	Increases	An ordered crystal becomes a disordered liquid
Liquid evaporating	Increases	Molecules spread out into a much larger volume as gas
Friction slowing a sliding block	Increases	Ordered motion becomes random thermal motion
Heat flowing from hot to cold	Increases overall	The cold body gains more entropy than the hot body loses
Frictionless swing of a pendulum in a vacuum	Close to zero	No energy is turned into heat, so the motion is reversible

Common traps

The Carnot formula needs kelvin

Putting Celsius temperatures into

1 - T_c/T_h

gives a far too large efficiency. Convert both temperatures to kelvin first.

No real engine reaches the Carnot efficiency

The Carnot value is a ceiling for an ideal, reversible engine. Any claim that an engine beats it, or reaches 100% with a cold side above absolute zero, breaks the second law.

Entropy can fall locally

The second law is about the total entropy of an isolated system. A living cell, a freezer or a growing crystal can lower its own entropy, as long as it raises the entropy of its surroundings by more.

Heat and Thermodynamics formulas

Temperature, Heat and How Heat Moves

Temperature, heat and the Kelvin scale

Thermal expansion of solids and liquids

Heat transfer: conduction, convection and radiation

Common traps

Specific Heat, Latent Heat and Mixing

Specific heat capacity: the energy to warm a mass

Latent heat and changes of state

Calorimetry: heat lost equals heat gained

Common traps

Ideal Gases and Kinetic Theory

The ideal gas equation and the three gas laws

Kinetic theory: temperature and molecular speed

Common traps

The First Law and Gas Processes

The first law of thermodynamics and the work done by a gas

Isothermal, isobaric, isochoric and adiabatic processes

Common traps

The Second Law, Entropy and Heat Engines

Heat engines, efficiency and the Carnot limit

The second law and entropy

Common traps

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