Ideal gases

• Kelvin, cubic metres, and N is molecules while n is moles. Pressure is momentum given to the walls; the ⅓ is random motion shared among three directions; \(\tfrac{3}{2}kT\) is the same for every gas at the same temperature.

• 15.1.1 I can understand — understand that amount of substance is an SI base quantity with the base unit mol
• 15.1.2 I can use — use molar quantities where one mole of any substance is the amount containing a number of particles of that substance equal to the Avogadro constant NA
• 15.2.1 I can understand — understand that a gas obeying pV ∝ T, where T is the thermodynamic temperature, is known as an ideal gas
• 15.2.2 I can recall — recall and use the equation of state for an ideal gas expressed as pV = nRT, where n = amount of substance (number of moles) and as pV = NkT, where N = number of molecules
• 15.2.3 I can recall — recall that the Boltzmann constant k is given by k = R/N_A
• 15.3.1 I can state — state the basic assumptions of the kinetic theory of gases
• 15.3.2 I can explain — explain how molecular movement causes the pressure exerted by a gas and derive and use the relationship pV = ⅓Nm⟨c²⟩, where ⟨c²⟩ is the mean-square speed (a simple model considering one-dimensional collisions and then extending to three dimensions using ⅓⟨c²⟩ = ⟨c_x²⟩ is sufficient)
• 15.3.3 I can understand — understand that the root-mean-square speed c_r.m.s. is given by √⟨c²⟩
• 15.3.4 I can compare — compare pV = ⅓Nm⟨c²⟩ with pV = NkT to deduce that the average translational kinetic energy of a molecule is (3/2)kT, and recall and use this expression

Units, significant figures and working are part of the physics. Give a calculated answer to the same number of significant figures as the least precise data, or one more; keep full precision in the working and round only at the end; write the unit with every final answer. A fifth of the qualification is experimental: Papers 3 and 5 test AO3 only, and their questions may be set in contexts outside the syllabus content, so the practical work in this chapter is set out as method, recording, graphs and uncertainties rather than as theory.

• “Put the numbers straight into pV = nRT.” Correct T in kelvin, V in m3, and N is molecules while n is moles. Convert first: \(\theta + 273.15\), cm3 × 10−6, kPa × 103, g mol−1 ÷ 1000. Using °C in \(p_1/T_1 = p_2/T_2\) turned a modest warming into a tripled pressure in worked example 2.
• “Gas pressure is caused by molecules colliding with each other.” Correct It is caused by molecules colliding with the walls: each collision reverses the momentum component perpendicular to the wall, the wall exerts a force on the molecule, and by Newton's third law the molecule exerts an equal and opposite force on the wall. Pressure is the total of these forces per unit area.
• “The momentum change at the wall is mcx, and the time between hits is l/cx.” Correct Both are doubled: \(+mc_x\) to \(-mc_x\) is a change of magnitude \(2mc_x\), and the molecule goes there and back, \(2l\), before it hits the same wall again. The two 2s cancel, which is why the force is \(mc_x^2/l\).
• “The r.m.s. speed is the average speed.” Correct \(c_\mathrm{r.m.s.} = \sqrt{\langle c^2\rangle}\): square, then mean, then root. It is larger than the mean speed unless every speed is equal, and \(\langle c^2\rangle\) is not (mean speed)2.
• “Heavier molecules have more kinetic energy at the same temperature.” Correct The average translational kinetic energy is \(\tfrac{3}{2}kT\) for every gas at temperature T. Heavier molecules have the same average kinetic energy and a lower r.m.s. speed, \(c_\mathrm{r.m.s.} \propto 1/\sqrt{m}\).
• “pV = nRT is on the formulas sheet.” Correct Only \(p = \tfrac{1}{3}\dfrac{Nm}{V}\langle c^2\rangle\) is given. \(pV = nRT\), \(pV = NkT\), \(k = R/N_\mathrm{A}\) and \(\tfrac{1}{2}m\langle c^2\rangle = \tfrac{3}{2}kT\) must be recalled. The constants \(N_\mathrm{A}\), R, k and u are on the Data sheet.
• “A mole is 6.02 × 1023 grams.” Repair A mole is an amount of substance: the amount containing NA particles. NA = 6.02 × 1023 mol−1 is a number per mole, not a mass.
• “M for oxygen is 32 kg mol−1.” Repair 32 g mol−1 = 0.032 kg mol−1. A molecular mass that comes out near 10−23 kg has missed this factor of 1000.
• “In pV = nRT, T can be in °C.” Repair T is always the thermodynamic temperature in kelvin; p ∝ T only on that scale. Add 273.15 (or 273).
• “V = 500 cm3 goes straight into pV = nRT.” Repair Convert to m3 (× 10−6): 5.00 × 10−4 m3. Convert kPa to Pa as well. Only the ratio form p1V1/T1 = p2V2/T2 lets p and V stay in matching non-SI units.
• “In pV = NkT, N is the number of moles.” Repair N is the number of molecules; n is the number of moles. N = nNA, and nR = Nk.
• “p1V1/T1 = p2V2/T2, so the pressure in the cylinder after some gas was used is…” Repair The two-state form needs a fixed amount of gas. If gas escapes or is added, use pV = nRT separately for each state.
• “Gas pressure is caused by molecules colliding with each other.” Repair It is caused by molecules colliding with the walls: the momentum change at each collision means a force on the wall (Newton's third law), and pressure is the total force per unit area.
• “The momentum change in a collision with the wall is mcx.” Repair From +mcx to −mcx is a change of −2mcx for the molecule, +2mcx for the wall.
• “The time between collisions with a wall is l/cx.” Repair It is 2l/cx: the molecule must go to the opposite wall and back.
• “The ⅓ comes from the cube having three pairs of walls.” Repair It comes from ⟨cx2⟩ = ⅓⟨c2⟩: c2 = cx2 + cy2 + cz2, and random motion makes the three mean squares equal. The shape of the container does not matter.
• “⟨c2⟩ is the square of the mean speed.” Repair It is the mean of the squared speeds, which is larger. For 200, 400, 500, 600, 800 m s−1: ⟨c2⟩ = 2.90 × 105 m2 s−2 but (mean)2 = 2.50 × 105 m2 s−2.
• “cr.m.s. is the average speed.” Repair It is the square root of the mean of the squared speeds, larger than the mean speed unless all the speeds are equal.
• “Heavier molecules have more kinetic energy at the same temperature.” Repair The average translational kinetic energy is &frac32;kT for every gas; heavier molecules move more slowly, cr.m.s. ∝ 1/√m.
• “Doubling the Celsius temperature doubles the molecules' kinetic energy.” Repair The kinetic energy is proportional to T in kelvin: 20 °C to 40 °C is only 293 K to 313 K, a 7% increase.

• Read the command word before you decide how much to write. This syllabus has fifteen of them: calculate, comment, compare, define, describe, determine, explain, give, identify, justify, predict, show (that), sketch, state and suggest. Define wants a precise meaning — for a physical quantity, usually an equation in words with every quantity named. State and give want a fact and nothing more. Describe wants the points or the features. Explain wants the reasons and the relationships — a describe-level answer to an explain question is incomplete however well written it is. Show (that) gives you the result and asks for the structured evidence that leads to it, so every step must appear — and a final value worked to one more significant figure than the one printed makes it plain that you calculated it rather than copied it. Sketch wants a freehand graph with its key features — intercepts, asymptotes, the shape — correct, but no plotted scale.
• Two words to keep apart. Amount of substance is the quantity; mole is its unit. “The number of moles” is everyday shorthand for the amount of substance in mol.
• Rules this section used (from the syllabus's practical requirements): raw readings of a quantity to the same precision; a false-origin intercept found by substituting a point into y = mx + c; a gradient triangle using points more than half the line apart; the worst acceptable line through every error bar; uncertainty in a gradient = |best-fit gradient − worst-acceptable gradient|; percentage uncertainties added, never combined in quadrature. “Human error” is never an acceptable source of error: name the physical cause, as the unheated tube above.
• Interleave with the chapters that use this one. When you reach topic 16, re-answer: why is the internal energy of a monatomic ideal gas \(\tfrac{3}{2}nRT\), and why does it depend on temperature alone? When you reach topic 23, re-answer: how do the molar mass and NA turn the mass of a radioactive sample into the number of undecayed nuclei N in A = λN? Recalling a topic inside a new context is worth more than another pass over this chapter on its own; at A Level, Paper 4 assumes the whole of the AS content, so nothing here is ever finished with.

• Cambridge International AS & A Level Physics 9702 has five components. Topic 15 is A Level content, so it is examined in Papers 4 and 5. A Level content: examined in Paper 4 (A Level structured, which also requires the AS content) and, as practical context, Paper 5. AS Level candidates take Papers 1, 2 and 3; A Level candidates take all five, either staged over two years (Papers 1–3 in year one, Papers 4 and 5 in year two) or together in one series. Examinations are available in the June and November series, and in March in India.
• Across both the AS Level and the A Level the assessment objectives are weighted AO1 40% (knowledge and understanding), AO2 40% (handling, applying and evaluating information) and AO3 20% (experimental skills and investigations). AS candidates are graded a–e; A Level candidates A*–E. The Data and formulas sheet is printed as page 2 of Papers 1 and 2 and as pages 2 and 3 of Paper 4: it gives the constants and a short list of formulas. Every other equation in this chapter is one the syllabus says you must recall, and this chapter says which is which.
• There is no multiple-choice paper on A Level content, so Topic 15 appears in Paper 4 structured questions. Expect to state what an ideal gas is and the assumptions of the kinetic theory, to explain how molecular movement causes pressure, and to show that pV = ⅓Nm⟨c2⟩ follows from the model, step by step, or that ½m⟨c2⟩ = &frac32;kT follows from comparing it with pV = NkT.
• Calculations of n, N and mass from pV = nRT or pV = NkT, two-state changes for a fixed amount of gas, r.m.s. speeds from p = ⅓ρ⟨c2⟩ or from &frac32;kT, and ratios of speeds for different gases. Only p = ⅓(Nm/V)⟨c2⟩ is given on the Data and formulas sheet; pV = nRT, pV = NkT, k = R/NA and ½m⟨c2⟩ = &frac32;kT are recall. NA, R, k and u are given.
• The syllabus names no experiment for this topic, but the gas laws are natural Paper 5 contexts: vary the temperature of a fixed volume of air and measure its pressure with a gauge, plot p against θ and extrapolate to p = 0; or vary the pressure on air in a syringe and plot V against 1/p, whose gradient is nRT. The largest uncertainty comes from the long extrapolation to absolute zero; air that is not at the bath's temperature causes a systematic error, which repeating the readings cannot remove.
• Read the command word before you decide how much to write. This syllabus has fifteen of them: calculate, comment, compare, define, describe, determine, explain, give, identify, justify, predict, show (that), sketch, state and suggest. Define wants a precise meaning — for a physical quantity, usually an equation in words with every quantity named. State and give want a fact and nothing more. Describe wants the points or the features. Explain wants the reasons and the relationships — a describe-level answer to an explain question is incomplete however well written it is. Show (that) gives you the result and asks for the structured evidence that leads to it, so every step must appear — and a final value worked to one more significant figure than the one printed makes it plain that you calculated it rather than copied it. Sketch wants a freehand graph with its key features — intercepts, asymptotes, the shape — correct, but no plotted scale.