IMAT Chemistry · Formula sheet

States of Matter and Gas Laws formulas

8 formulas, 9 reference tables and 20 common traps for IMAT Chemistry States of Matter and Gas Laws, grouped by subtopic.

Full notes with worked examples

Solids, Liquids, Gases and the Kinetic Model

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Absolute temperature and the average kinetic energy of particles

Kelvin from Celsius, and kinetic energy

T\,(\text{K}) = \theta\,({}^{\circ}\text{C}) + 273.15 \qquad \bar{E}_k \propto T

$T$ absolute temperature, in K
$\theta$ temperature in degrees Celsius (IMAT often uses 273 instead of 273.15)
$\bar{E}_k$ average kinetic energy of the particles

The three states of matter in the particle model

State	Arrangement and spacing	Motion of particles	Shape and volume	Compressible?
Solid	Close together, usually in a regular pattern	Vibrate about fixed positions	Fixed shape and fixed volume	Almost not at all
Liquid	Close together, no regular pattern	Move around and slide past one another	Takes the shape of the container, fixed volume	Almost not at all
Gas	Far apart, random	Fast, random, straight lines between collisions	Fills the whole container	Very easily

A gas at room conditions is mostly empty space: the gaps between molecules are about ten times the size of the molecules.

Water is unusual: ice is less dense than liquid water

Property	Ice at 0 °C	Liquid water at 0 °C
Density	About 0.92 g/cm³	About 1.00 g/cm³
Volume of 1 kg	About 1.09 L	About 1.00 L
Arrangement	Open lattice held by hydrogen bonds; molecules further apart	Some hydrogen bonds broken; molecules packed closer
Average kinetic energy	Same as the liquid at 0 °C	Same as the ice at 0 °C

Water is one of very few substances whose solid floats on its own liquid.

Common traps

Particles in a solid still move

Particles in a solid are not still. They vibrate about fixed positions, and they vibrate more as the solid gets hotter. Options saying the particles of a solid have no kinetic energy are wrong.

Only kelvin temperatures can be used in ratios

Going from 20 °C to 40 °C does not double the kinetic energy: in kelvin it is 293 K to 313 K, a rise of under 7%. Any calculation where the answer depends on a ratio of temperatures must use kelvin.

At the same temperature, heavier molecules are slower, not more energetic

All gases at the same temperature have the same average kinetic energy per molecule. Since

E_k = \tfrac{1}{2}mv^2

, the heavier molecules move more slowly.

The solid is not always the densest state

For most substances the solid is denser than the liquid, but water is the exception: ice is less dense than liquid water because its molecules are further apart. The explanation is the spacing of the molecules, not their kinetic energy.

Changes of State, Boiling and Phase Diagrams

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Solutes lower the freezing point and raise the boiling point

Freezing point depression

\Delta T_f = i\,K_f\,m \qquad \Delta T_b = i\,K_b\,m

$\Delta T_f,\ \Delta T_b$ fall in freezing point, rise in boiling point, in K
$K_f,\ K_b$ constants for the solvent; for water 1.86 and 0.512 K kg/mol
$m$ molality, in mol of solute per kg of solvent
$i$ number of particles each formula unit gives in solution

The six changes of state and their energy

Change	Name	Energy
Solid to liquid	Melting (fusion)	Absorbed
Liquid to solid	Freezing (solidification)	Released
Liquid to gas	Evaporation or boiling (vaporisation)	Absorbed
Gas to liquid	Condensation	Released
Solid to gas	Sublimation	Absorbed
Gas to solid	Deposition	Released

Vapour pressure, evaporation and the boiling point

Feature	Evaporation	Boiling
Where	At the surface only	Throughout the liquid, with bubbles
Temperature	Any temperature	Only where vapour pressure equals the external pressure
Speeded up by	Higher temperature, larger surface, wind	Supplying heat faster (the temperature stays the same)
Effect on the liquid	Cools the liquid left behind	Temperature stays at the boiling point

Phase diagrams in outline: triple point and critical point

Feature	Meaning	Example values
Triple point	Solid, liquid and gas coexist	Water: 0.01 °C and about 0.006 atm
Critical point	End of the liquid to gas line	Water: about 374 °C and 218 atm
Normal melting and boiling points	Where the 1 atm line crosses the two lines	Water: 0 °C and 100 °C
Triple point above 1 atm	No liquid can exist at 1 atm; the solid sublimes	Carbon dioxide: triple point 5.1 atm, so dry ice sublimes at about $-78\ ^\circ\text{C}$ at 1 atm

Common traps

Freezing means liquid to solid

Freezing is the change from liquid to solid. A gas turning directly into a solid is deposition, and a solid turning directly into a gas is sublimation. Options that call gas to solid freezing are wrong.

The temperature does not rise during melting or boiling

While a pure substance melts or boils, the heat supplied breaks attractions between particles and the temperature stays constant. Only when the change is complete does the temperature start rising again.

More pressure on a liquid means a higher boiling point

Raising the pressure above water from 100 kPa to more than that raises its boiling point; it does not lower it. That is how a pressure cooker cooks faster. Heating more strongly only makes the water boil faster, at the same temperature.

The triple point of water is not 0 °C at 1 atm

0 °C at 1 atm is the normal melting point of ice. The triple point of water is at 0.01 °C and a very low pressure, about 0.006 atm.

A solute lowers the freezing point; it never raises it

Adding salt or sugar to water lowers the temperature at which it freezes. Options saying a solute raises the freezing point, or raises the melting point of ice, are wrong.

The solute does not supply the energy to melt ice

When salt or sugar makes ice melt below 0 °C, the energy for melting comes from the surroundings and the mixture gets colder. The solute only changes the melting point; it gives the water molecules no energy.

Pressure Units and the Gas Laws

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Boyle's law: pressure and volume at constant temperature

Boyle's law

P_1 V_1 = P_2 V_2

$P_1, V_1$ pressure and volume before the change
$P_2, V_2$ pressure and volume after the change (same temperature, same amount of gas)

Charles's law and Gay-Lussac's law: the effect of temperature

Charles, Gay-Lussac and the combined gas law

\frac{V_1}{T_1} = \frac{V_2}{T_2} \qquad \frac{P_1}{T_1} = \frac{P_2}{T_2} \qquad \frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}

$T_1, T_2$ absolute temperatures, in K
$P, V$ pressure and volume, in any units used consistently

Avogadro's law and the molar volume of a gas

Amount of gas from its volume

n = \frac{V}{V_m}

$n$ amount of gas, in mol
$V$ volume of the gas
$V_m$ molar volume at the stated conditions (22.4 L/mol at STP)

Units of pressure and their values for one atmosphere

Unit	Symbol	Value of 1 atm
Pascal	Pa	101 325 Pa
Kilopascal	kPa	101.325 kPa
Bar	bar	1.01325 bar
Millibar (hectopascal)	mbar or hPa	1013.25 mbar
Millimetre of mercury	mmHg	760 mmHg
Torr	Torr	760 torr

Learn 101.3 kPa and 760 mmHg; the others follow from the prefixes.

Common traps

1 atm is 101.3 kPa, not 1013 kPa

The number 1013 belongs to millibars (or hectopascals): 1 atm is 1013 mbar = 101.3 kPa. A value of 1013 kPa is ten atmospheres. Watch which prefix goes with which number.

Inverse proportion, not a direct decrease

When the volume increases at constant temperature, the pressure decreases in inverse proportion: double the volume, half the pressure, so

PV

is constant. Decreasing in direct proportion would mean

P/V

is constant, which is not true. The temperature does not change in an isothermal process.

Convert to kelvin before using any gas law ratio

Doubling the Celsius temperature does not double the pressure. To double the pressure of a gas at 50 °C in a rigid container, the kelvin temperature must double, from 323 K to 646 K, which is 373 °C. Answers built on Celsius ratios are always among the options.

Equal volumes mean equal moles, not equal masses

At the same temperature and pressure, 1 L of hydrogen and 1 L of carbon dioxide contain the same number of molecules, but the carbon dioxide is 22 times heavier. Avogadro's law is about the number of particles.

The Ideal Gas Equation and Mixtures of Gases

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The ideal gas equation PV = nRT

Ideal gas equation

PV = nRT

$P$ pressure, in Pa (or atm with the other R)
$V$ volume, in m³ (or L with the other R)
$n$ amount of gas, in mol
$R$ gas constant, 8.314 J/(mol K) or 0.0821 L atm/(mol K)
$T$ absolute temperature, in K

Dalton's law of partial pressures

Partial pressure

p_A = x_A\,P_{\text{total}} = \frac{n_A}{n_{\text{total}}}\,P_{\text{total}}

$p_A$ partial pressure of gas A
$x_A$ mole fraction of A
$n_A, n_\text{total}$ moles of A and total moles of gas

Graham's law: lighter gases diffuse and effuse faster

Graham's law

\frac{\text{rate}_1}{\text{rate}_2} = \sqrt{\frac{M_2}{M_1}}

$\text{rate}_1, \text{rate}_2$ rates of effusion of gases 1 and 2
$M_1, M_2$ their molar masses

Real gases and the assumptions of the ideal gas model

Idea	Ideal gas model	Real gas
Volume of the molecules	Negligible	Small but real; matters at high pressure
Forces between molecules	None	Weak attractions; matter at low temperature
Can it be liquefied?	Never	Yes, by cooling and compressing
Closest to ideal	Any gas, at any conditions	Helium and hydrogen, at high temperature and low pressure

Common traps

Litres with R = 8.314 gives an answer 1000 times wrong

R = 8.314

needs pressure in Pa and volume in m³. Putting in litres gives a pressure 1000 times too large. With litres and atmospheres, use

R = 0.0821

Adding gas at constant pressure and volume cools the gas

From

T = PV/(nR)

, if more moles go into the container while

P

and

V

stay fixed, the temperature must fall, and so does the kinetic energy of the molecules that were there at the start. At fixed

n

, raising either

P

V

raises

T

and the kinetic energy.

Partial pressures follow moles, not masses

In a mixture of 1 g of hydrogen and 1 g of helium, hydrogen exerts two thirds of the pressure, because it has twice as many moles. Always convert masses to moles before sharing out the pressure.

Rate goes with the square root of the molar mass

A gas 4 times heavier is 2 times slower, not 4 times slower. Leaving out the square root is the commonest wrong answer.

Pure Substances, Mixtures and Separation Techniques

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Elements, compounds and mixtures

Type	What it is	Examples	How it can be separated
Element	One kind of atom	Iron, oxygen gas, $\mathrm{S_8}$	Cannot be broken down chemically
Compound	Elements bonded in a fixed ratio	Water, sodium chloride, carbon dioxide	Only by a chemical reaction
Homogeneous mixture	Uniform throughout, one phase	Salt water, air, brass	Physical methods such as distillation
Heterogeneous mixture	Not uniform, visible parts	Sand in water, oil and water, granite	Physical methods such as filtration

Separation techniques: filtration, distillation, chromatography and others

Technique	Separates	Based on	Example
Filtration	Insoluble solid from a liquid	Particle size	Sand from water
Evaporation or crystallisation	Dissolved solid from its solution	The solvent is volatile, the solid is not	Salt from sea water
Simple distillation	Solvent from a solution	Very different boiling points	Pure water from salt water
Fractional distillation	Miscible liquids	Boiling points closer together	Ethanol from water; fractions of crude oil
Chromatography	Dissolved substances in small amounts	Attraction to the stationary and mobile phases	Dyes in an ink; amino acids
Separating funnel	Immiscible liquids	Density; liquids do not mix	Oil from water
Centrifugation	Suspended solid from a liquid	Density	Blood cells from plasma

Common traps

Looking uniform does not make a substance pure

Salt water and air look the same throughout, but they are homogeneous mixtures: their composition can change and their parts can be separated physically. A pure substance has a fixed composition and sharp melting and boiling points.

Filtration cannot remove a dissolved substance

A dissolved solid passes through filter paper with the solvent. To get salt out of salt water you must evaporate the water, or distil it off. Filtration only removes undissolved solids.

States of Matter and Gas Laws formulas

Solids, Liquids, Gases and the Kinetic Model

Absolute temperature and the average kinetic energy of particles

The three states of matter in the particle model

Water is unusual: ice is less dense than liquid water

Common traps

Changes of State, Boiling and Phase Diagrams

Solutes lower the freezing point and raise the boiling point

The six changes of state and their energy

Vapour pressure, evaporation and the boiling point

Phase diagrams in outline: triple point and critical point

Common traps

Pressure Units and the Gas Laws

Boyle's law: pressure and volume at constant temperature

Charles's law and Gay-Lussac's law: the effect of temperature

Avogadro's law and the molar volume of a gas

Units of pressure and their values for one atmosphere

Common traps

The Ideal Gas Equation and Mixtures of Gases

The ideal gas equation PV = nRT

Dalton's law of partial pressures

Graham's law: lighter gases diffuse and effuse faster

Real gases and the assumptions of the ideal gas model

Common traps

Pure Substances, Mixtures and Separation Techniques

Elements, compounds and mixtures

Separation techniques: filtration, distillation, chromatography and others

Common traps

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