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Ideal gases is where thermal physics meets Newtonian mechanics. A handful of equations (pV=nRT=NkT, pV=31Nm(crms)2 and 21m(crms)2) carry a lot of marks. Expect 1-mark MCQs on ratios and assumptions, 2–4 mark calculations that hinge on kelvin and kilogram conversions, a 4–6 mark derivation or “explain using the kinetic model” question, and a gas-law practical with a graph to extrapolate.
By the end you’ll be able to
State Boyle’s, Charles’s and the pressure law and describe the Boyle’s/Charles’s law required practical
Use pV = nRT and pV = NkT, and convert between moles, molecules and mass
List the kinetic-theory assumptions and derive pV = ⅓Nm(c_rms)²
Use ½m(c_rms)² = (3/2)kT = 3RT/2N_A and calculate rms speeds
Explain Brownian motion and how the gas laws were developed empirically before the molecular model
What the exam asks
Gas laws and absolute zero: Boyle’s law, Charles’s law and the pressure law, what each graph looks like, and how extrapolation gives absolute zero.
The equation of state:pV=nRT and pV=NkT, and converting between mass, moles and molecules. This is where most marks are lost, usually to °C instead of K or grams instead of kilograms.
The kinetic model: the assumptions, a molecular explanation of pressure and of each gas law, and the derivation of .
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3 questions on ideal gases and kinetic theory. Every option is explained once you answer.
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PromptCard 1 of 3
List the assumptions of the kinetic theory model of an ideal gas.
=
23kT
pV=31Nm(crms)2
Molecular energy: mean kinetic energy 23kT, rms speed, and the internal energy of an ideal gas.
Evidence and history: Brownian motion, and the fact that the gas laws are empirical (found by experiment) while kinetic theory is theoretical (derived from a model).
Practical work: AQA Required Practical 8 (Boyle’s and Charles’s laws), Edexcel Core Practical 14 (pressure and volume at fixed temperature), OCR PAG8 (gas laws and estimating absolute zero) and the Eduqas/WJEC specified practical (estimating absolute zero).
Core ideas
Units first: kelvin, moles and kilograms
T/K=θ/∘C+273n=NAN=Mmassmmolecule=NAM
The molar massM must be in kg mol−1. Nitrogen (Mr=28) has M=0.028kg mol−1.
The Avogadro constantNA=6.02×1023mol− is the number of particles in one mole.
Every gas-law equation uses absolute temperature. A temperature change is the same in K and °C, but a temperature ratio is not.
The gas laws (empirical)
For a fixed mass of gas:
Law
Held constant
Relationship
Straight-line graph
Boyle’s law
T
pV=constant
p against V1, through the origin
Charles’s law
p
TV=constant
V against , through the origin at 0 K
Pressure law
V
Tp=constant
p against , through the origin at 0 K
Combined: TpV=constant. If you plot V or p against temperature in °C, you get a straight line that meets the temperature axis at about −273∘C. That extrapolation is how absolute zero is estimated in the practical.
Absolute zero is the temperature at which the particles have the minimum possible internal energy. For an ideal gas this means zero kinetic energy and, from the laws above, zero pressure.
The equation of state
pV=nRT=NkTk=NAR
R=8.31J mol−1 K−1 is per mole; k=1.38×10−23J K−1 is per molecule. Use R with moles and k with molecules. An ideal gas obeys pV∝T exactly. Real gases get close at low pressure and at temperatures well above their boiling point, where the molecules are far apart and intermolecular forces are negligible.
The kinetic theory model: assumptions
A gas contains a very large number of identical molecules moving randomly.
The volume of the molecules is negligible compared with the volume of the container.
All collisions (with each other and with the walls) are perfectly elastic.
There are no forces between molecules except during collisions.
The duration of a collision is negligible compared with the time between collisions.
Newton’s laws apply to the molecules.
The model does not assume that all the molecules have the same speed. They have a spread of speeds; only averages appear in the results.
Explaining pressure with the model
A molecule hitting a wall reverses its momentum component perpendicular to the wall, so the wall exerts a force on it and, by Newton’s third law, it exerts an equal and opposite force on the wall. Pressure is the total force from very many collisions per second, divided by the area.
Boyle’s law: halve V at constant T. The speeds are unchanged, but each molecule hits the walls more often, so the rate of change of momentum per unit area rises and the pressure doubles.
Pressure law: raise T at constant V. Molecules move faster, so each collision gives a larger change in momentum and collisions are more frequent. Both increase p.
Charles’s law: raise T at constant p. Faster molecules would raise p, so the gas expands until the lower collision rate brings p back down.
Deriving pV=31Nm(crms)2
Take a cube of side L containing N molecules, each of mass m. Consider one molecule with velocity component u1 towards one wall.
Each collision with that wall changes its momentum by −2mu1, so the wall receives momentum 2mu1.
It returns to the same wall after travelling 2L, so the time between collisions is u12L.
Force on the wall = rate of change of momentum: F1=2L/u1.
Pressure from this molecule: p1=L2.
Add all N molecules: p=Vm.
Motion is random, so u2=v2=. Since , it follows that .
Therefore pV=31Nmc.
Collisions between molecules are ignored because, with elastic collisions between identical molecules, they only exchange velocities and do not change the total momentum delivered to the walls.
Temperature is mean kinetic energy
Equate 31Nm(crms)2 with NkT:
21m(crms)2=23kT=2NA3RTcrms=m3kT=M3RT
At the same temperature, every ideal gas has the same mean kinetic energy per molecule, but lighter molecules move faster: crms∝M1.
crms∝T. To double the rms speed you must quadruple the kelvin temperature.
The root-mean-square speed is the square root of the mean of the squared speeds. It is slightly larger than the mean speed.
An ideal gas has no intermolecular forces, so its internal energy is all kinetic: for a monatomic gas, U=N×23kT=.
The spread of speeds
At any temperature the molecular speeds follow the Maxwell–Boltzmann distribution: a curve that starts at zero, rises to a peak and has a long tail at high speed. At a higher temperature the peak moves to a higher speed and becomes lower and broader. The area under the curve (the total number of molecules) stays the same. The high-speed tail explains why a few molecules can escape from a liquid (evaporation) or from a planet’s atmosphere even when the rms speed is well below the escape velocity.
Brownian motion and the history
Smoke particles in air (or pollen grains in water) seen through a microscope jiggle randomly. They are being struck unevenly by far smaller, invisible, fast-moving molecules. Einstein’s 1905 analysis turned this into quantitative evidence that atoms and molecules exist. The gas laws were found first, by experiment (Boyle in the 1660s, Charles and Gay-Lussac around 1800). Kinetic theory came later as a model that explains them. The laws are empirical; the model is theoretical.
Worked examples
Exam technique
Write the conversions first: °C to K, g to kg, cm3 to m3 (×10), litres to (). A one-line list earns method marks even if the arithmetic slips.
The total amount of gas is fixed, and the pressure equalises: RTpV+RTp(3V)=RTp′V+R(2T)p′(3V).
So 4p=2.5p′ and p′=1.6p. Gas flows from the hot flask into the cold one, so you cannot treat the whole sample as being at one temperature.
−
6
m3
×10−3
Pick R or k from the question: moles or mass → R; number of molecules or one molecule → k.
“Explain using the kinetic model” (3–4 marks): always go through collisions → change in momentum → force (rate of change of momentum) → pressure = force/area, and say which of speed or collision frequency changes.
Derivation (5–6 marks): number each step, define every symbol, and say explicitly where “random motion” gives the factor 31.
Ratio MCQs:crms∝T/M and p∝(crms)2 at constant N and V. Check whether the answer needs a square root.
Practical graphs:p against V1 for Boyle; V or p against θ for absolute zero, where the intercept is found by calculation (θ0=−gradientc), not by drawing an axis to −300 °C.
3RT/M
crms
m s−1
M
T
crms
2
crms
T
p
A
R
T
M
kg mol−1
Boyle: pV constant; Charles: V/T constant; pressure law: p/T constant. Extrapolating V or p against θ to zero gives absolute zero ≈ −273 °C.
Kinetic assumptions: random motion, negligible molecular volume, elastic collisions, no forces except in collisions, negligible collision time.
Pressure = rate of change of momentum at the walls per unit area.
pV=31Nm(crms)2; the 31 comes from random motion in three dimensions.
21m(crms)2=23kT, so crms=3RT/M and ideal-gas internal energy is 23nRT (monatomic).
Brownian motion is evidence for molecules; the gas laws are empirical, the kinetic model is theoretical.