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Theme B — The Particulate Nature of Matter

B.3.7 Average Kinetic Energy of a Molecule — IB Physics HL Revision Notes & Topical Questions

5 min read 10 parts Lesson 31 of 53 1 quiz

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B.3.7 Average Kinetic Energy of a Molecule — IB PHYSICS SL notes: Show that the mean translational kinetic energy of a molecule is (3/2)kBT and use it to find rms speeds and the internal energy of a monatomic ideal gas.Kinetic theory — pV = ⅓Nm⟨c2⟩.What the symbols mean: p = pres…

  • Multiplying by 3/2
  • Temperature
  • Elastic collisions
  • No forces between collisions
  • Per molecule
  • Dependence

Part 1 of 8Two equations for pV

Kinetic theory
  • Kinetic theory pV = ⅓Nm⟨c2⟩.
    What the symbols mean: p = pressure (Pa) · V = volume (m3) · N = number of molecules · m = mass of one molecule (kg) · <c2> = mean square speed (m2 s−2)
Ideal gas
  • Ideal gas pV = NkBT.
Equating
  • ⅓m⟨c2⟩ = kBT.
Multiplying by 3/2
  • ⟨Ek⟩ = ½m⟨c2⟩ = (3/2)kBT.
Meaning
  • Meaning the mean translational kinetic energy of a molecule is proportional to the absolute temperature.
  • It does not depend on the type of gas, the pressure or the volume.
  • A helium atom and a nitrogen molecule at the same temperature have the same average kinetic energy but different speeds.
Temperature
  • Temperature it measures the average random translational kinetic energy.

Part 2 of 8Assumptions of the ideal gas model

Assumptions of the ideal gas model
Elastic collisions
  • Elastic collisions momentum and kinetic energy are conserved when molecules collide.
No forces between collisions
  • No forces between collisions so molecules have no potential energy.
Internal energy
  • Internal energy the internal energy of an ideal monatomic gas equals the total kinetic energy of its molecules.

Part 3 of 8Internal energy

Internal energy
Per molecule
  • (3/2)kBT, valid for one molecule only, so multiply by N for the whole gas.
Total
  • U = (3/2)NkBT = (3/2)nRT.
Per mole
  • (3/2)RT, using R = NAkB.
Dependence
  • U depends only on temperature and the amount of gas.
Proportional
  • U ∝ T, so heating a gas at constant volume raises U and T.
Change
  • ΔU = (3/2)nRΔT.
Monatomic gases
  • Monatomic gases helium, neon and argon, with only translational energy.
Diatomic molecules
  • Diatomic molecules they also have rotational energy.
Use
  • Use this is the basis of the first law of thermodynamics for monatomic gases.

Part 4 of 8Root-mean-square speed

From
  • ½m⟨c2⟩ = (3/2)kBT.
Result
  • Result crms = √(3kBT/m) = √(3RT/M).
  • m is the mass of one molecule in kg and M the molar mass in kg mol−1.
Dependence
  • Dependence crms grows as the square root of T and falls as the square root of the mass.
Doubling T
  • Doubling T it increases the speed only by √2.
At 300 K
  • At 300 K oxygen about 480 m s−1 and hydrogen about 1900 m s−1.

Part 5 of 8Absolute zero and phase changes

1Absolute zero
  • Absolute zero as T approaches 0 K the translational kinetic energy of ideal gas molecules falls to zero,
    • although real particles keep a small zero-point energy.
2Graph
  • Graph mean kinetic energy against absolute temperature is a straight line through the origin with gradient (3/2)kB.
3Celsius axis
  • Celsius axis the line is the same but cuts the axis at −273 °C.
4Boiling and melting
  • Boiling and melting the mean kinetic energy stays constant while the supplied energy raises the potential energy.
5Real gas
  • Real gas its internal energy includes potential energy as well as kinetic energy;
    • only for an ideal gas is U purely kinetic.

Part 6 of 8Typical values

Room temperature (300 K)
  • Room temperature (300 K) the mean kinetic energy is about 6 × 10−21 J, or 0.039 eV.
Bond energies
  • Bond energies they are several eV, so few collisions at room temperature break bonds.
Reference value
  • Reference value kBT is a useful energy scale throughout physics.

Part 7 of 8Degrees of freedom

1Factor 3/2
  • Factor 3/2 it comes from three independent translations, each contributing ½kBT.
2Diatomic gas
  • Diatomic gas at room temperature two rotations add kBT, giving U = (5/2)nRT.
3IB assumption
  • IB assumption unless told otherwise the gas is monatomic with U = (3/2)nRT.
4Isothermal process
  • U is unchanged because it depends only on temperature.

Part 8 of 8More from the specification

More from the specification
Microscopic molecular kinetic energy
  • Microscopic molecular kinetic energy relating mean translational kinetic energy to absolute temperature provides a statistical means to define thermal state.
Kinetic Interpretation of Absolute Temperature
  • The average translational kinetic energy of an ideal gas particle in a closed container is directly proportional to its thermodynamic temperature: Ek = 3/2 * kB * T,
  • meaning temperature is defined by random atomic and molecular motions.
Mole Calculations and Avogadro's Constant
  • The amount of substance n in moles is related to total number of molecules or atoms N by N = n * NA,
  • where NA = 6.02 x 1023 mol-1; mass m relates to molar mass M via n = m / M.
Ideal Gas State Equations
  • Combining the gas laws gives pV = nRT = N * kB * T;
    • to solve for unknown quantities such as final pressure or initial temperature, you rearrange the equation based on which state variables remain constant.
Internal Energy of a Monatomic Ideal Gas
  • For a monatomic gas, negligible intermolecular forces mean internal energy U consists solely of particle kinetic energies: U = 3/2 * n * R * T = 3/2 * N * kB * T.
Molecular kinetic energy
  • Molecular kinetic energy average translational kinetic energy of gas particles in a container is directly proportional to absolute thermodynamic temperature.
Avogadro constant and moles
  • Avogadro constant and moles one mole of any gaseous substance contains 6.02 × 1023 constituent atoms or molecules.
Root-mean-square speed
  • Root-mean-square speed calculated by taking the square root of the mean squared speeds of gas molecules inside a closed container.
Boltzmann constant
  • K = R/NA, the gas constant per molecule
Avogadro constant
  • The number of particles per mole, 6.02 × 10²³ mol⁻¹
N = nNA
What the symbols mean: N = number of particles · n = amount (mol) · NA = Avogadro constant (6.02 × 1023 mol−1)

Key terms

TermMeaning
Mean translational kinetic energyAverage energy of translation per molecule, (3/2)kBT
Internal energy of an ideal gasTotal kinetic energy of its molecules, (3/2)nRT for a monatomic gas
R.m.s. speedRoot of the mean square molecular speed, √(3kBT/m)
Monatomic gasGas whose particles are single atoms
Zero-point energySmall energy that real particles keep at absolute zero
Isothermal processProcess at constant temperature

Avoid these mistakes

1
Applying (3/2)kBT to the whole gas without multiplying by N — it is per molecule.
2
Using °C — use kelvin.
3
Thinking heavier molecules have more kinetic energy at the same T — the mean kinetic energy is equal for all gases.
4
Applying U = (3/2)nRT to a gas that is not monatomic or ideal — rotational or potential energy adds to U.
5
Using the molar mass in g mol−1 in crms — use kg mol−1.

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