understand that the root-mean-square speed cr.m.s. is given by c<>

Topic‑Map Checklist – Cambridge International AS & A Level Physics 9702 (2025‑2027)

Syllabus Item	Covered in these notes?	Comments / Further work
15.1 Physical quantities & units	No	See “Prerequisite Knowledge” below.
15.2 Scalars & vectors	No	See “Prerequisite Knowledge”.
15.3 Ideal‑gas equation & kinetic‑theory derivation	Yes	Derivation of \(c_{\text{rms}}\) and link to \(pV = Nk_{\!B}T\).
15.4 Mean translational kinetic energy	Yes	Result \(\langle KE\rangle = \tfrac32k_{\!B}T\).
15.5 Errors & uncertainties (AO3)	Partial	Brief AO3 note on experimental verification (e.g., effusion).
15.6 Scalars, vectors & resolution	No	Covered in prerequisite section.
15.7 Practical skills (AO3)	Partial	Suggested experiment box.

Prerequisite Knowledge (What you need before this)

• SI units, especially joule (J), kilogram (kg), mole (mol) and kelvin (K).
• Understanding of vectors – components, magnitude, and the idea of isotropy (properties are the same in all directions).
• Basic error analysis: quoting results to the correct number of significant figures.

Learning Objective

Show that the root‑mean‑square (r.m.s.) speed of the molecules in an ideal gas is

$$c_{\text{rms}}=\sqrt{\frac{3k_{\mathrm B}T}{m}}=\sqrt{\frac{3RT}{M}}$$

and that the average translational kinetic energy of a molecule is

$$\langle KE\rangle=\frac32k_{\mathrm B}T.$$

Key Assumptions of the Kinetic Theory

• The gas consists of a very large number of tiny particles (atoms or molecules) moving in random directions.
• Each particle is a point mass; its own volume is negligible compared with the volume of the container.
• No intermolecular forces act except during perfectly elastic collisions.
• Collisions between particles and with the walls are perfectly elastic.
• The time between successive collisions is much longer than the duration of a collision.

Derivation of the r.m.s. Speed

1. Set‑up – A cubic container of side length \(L\) holds \(N\) molecules, each of mass \(m\).
2. Force from a single molecule – A molecule with velocity components \((v_x,v_y,v_z)\) strikes the wall perpendicular to the \(x\)-axis.
   • Change in momentum on the collision: \(\Delta p = 2mv_x\).
   • Time between two successive impacts on the same wall: \(\displaystyle \Delta t=\frac{2L}{|v_x|}\).
   • Average force exerted by this molecule on the wall: $$F=\frac{\Delta p}{\Delta t}= \frac{2mv_x}{2L/|v_x|}= \frac{mv_x^{2}}{L}.$$
3. Total pressure – Summing over all \(N\) molecules and using isotropy \(\bigl(\langle v_x^{2}\rangle=\langle v_y^{2}\rangle=\langle v_z^{2}\rangle\bigr)\): $$p=\frac{F_{\text{total}}}{A}= \frac{1}{3}\,\frac{Nm\langle v^{2}\rangle}{V},\qquad V=L^{3}$$ where \(\displaystyle \langle v^{2}\rangle=\langle v_x^{2}+v_y^{2}+v_z^{2}\rangle\).
4. Link with the ideal‑gas equation – The ideal‑gas law for \(N\) molecules is $$pV = Nk_{\!B}T.$$ Equating the two expressions for \(pV\): $$\frac13 Nm\langle v^{2}\rangle = Nk_{\!B}T$$ gives $$\langle v^{2}\rangle = \frac{3k_{\!B}T}{m}.$$
5. Definition of r.m.s. speed – By definition, $$c_{\text{rms}}=\sqrt{\langle v^{2}\rangle}= \sqrt{\frac{3k_{\!B}T}{m}}.$$
6. Macroscopic form – Using \(R=N_{\!A}k_{\!B}\) and \(M=N_{\!A}m\): $$c_{\text{rms}}=\sqrt{\frac{3RT}{M}}.$$

Limitations of the Model (AO2)

• Assumes an ideal gas – no intermolecular forces and point‑like particles. Real gases deviate at high pressure or low temperature (e.g., condensation).
• Neglects quantum effects; the derivation breaks down for gases at temperatures comparable with the molecular rotational or vibrational energy spacings.
• Only translational kinetic energy is considered – rotational and vibrational contributions become important for polyatomic gases at higher temperatures.

Physical Significance

• The r.m.s. speed is a statistical measure of the “typical” speed of molecules at a given temperature.
• It varies as \(\sqrt{T}\) and inversely as \(\sqrt{M}\) (or \(\sqrt{m}\)).
• Because \(\langle KE\rangle=\tfrac32k_{\!B}T\), temperature is a direct measure of the average translational kinetic energy of the gas.
• The r.m.s. speed appears in the kinetic‑theory expression for pressure and underlies the Maxwell–Boltzmann speed distribution (useful for extension questions).

Practical (AO3) Note

Experimentally the r.m.s. speed can be inferred from effusion rates (Graham’s law) or from the speed of sound in a gas. When reporting results, quote to the appropriate number of significant figures (typically 3 sf for textbook problems).

Worked Example (with Significant‑Figure Reminder)

Find the r.m.s. speed of nitrogen (\(\mathrm{N_2}\)) molecules at \(300\ \text{K}\).

1. Molar mass: \(M = 28.02\ \text{g mol}^{-1}=2.802\times10^{-2}\ \text{kg mol}^{-1}\) (3 sf).
2. Universal gas constant: \(R = 8.314\ \text{J mol}^{-1}\text{K}^{-1}\) (4 sf).
3. Apply the formula: $$c_{\text{rms}}=\sqrt{\frac{3RT}{M}} =\sqrt{\frac{3(8.314)(300)}{2.802\times10^{-2}}} \approx 5.17\times10^{2}\ \text{m s}^{-1}.$$ Rounded to 3 sf, \(c_{\text{rms}} = 5.20\times10^{2}\ \text{m s}^{-1}\).

Typical r.m.s. Speeds of Common Gases at 300 K

Gas	Molar Mass \(M\) (kg mol⁻¹)	r.m.s. Speed \(c_{\text{rms}}\) (m s⁻¹)
Helium (He)	4.00 × 10⁻³	1 300
Hydrogen (H₂)	2.02 × 10⁻³	1 950
Nitrogen (N₂)	2.80 × 10⁻²	517
Oxygen (O₂)	3.20 × 10⁻²	480
Carbon Dioxide (CO₂)	4.44 × 10⁻²	426

Optional Extension – Maxwell‑Boltzmann Speed Distribution

Although not required for the core syllabus, the r.m.s. speed is one of three characteristic speeds in the Maxwell‑Boltzmann distribution (most probable, mean, and r.m.s.). Understanding the shape of the distribution helps answer higher‑order AO2 questions about diffusion and effusion.

Suggested Diagram

Key Points to Remember

• The r.m.s. speed depends only on temperature and molecular (or molar) mass; pressure and volume cancel out during the derivation.
• Higher temperature → higher average kinetic energy → higher \(c_{\text{rms}}\).
• Lighter molecules move faster than heavier ones at the same temperature.
• \(\langle KE\rangle=\tfrac32k_{\!B}T\) is a cornerstone of kinetic‑theory and is required for AO2 questions.
• r.m.s. speed is useful for estimating rates of diffusion, effusion, and for interpreting the Maxwell–Boltzmann speed distribution.
• When applying the formula, keep track of units (use SI) and quote results to the appropriate number of significant figures.