IB Physics HL B.3 Gas laws Questions

SLSL HLHL

Practise IB Physics HL B.3 by combining ideal-gas calculations with molecular momentum, internal-energy and model-limit analysis.

143 questions in this syllabus scope, with question parts counted together.

Exam points & syllabus

Exam points

For pressure loading across a surface, use the perpendicular force F = AΔp and justify whether the assumed pressure difference is realistic.
Choose the actual contact area: a solid cylinder uses its base, while a thin open cylinder can rest on an annular rim of area about 2πRx.
For a stopper supporting an oil column, add the column’s weight and the stopper’s own weight to obtain the required lifting force.
Check pressure dimensions by showing that J/m³ = N m/m³ = N/m² = Pa.
Distinguish amount n from particle count N; use N = (m/M)NA to compare samples and M/NA to estimate a single particle’s mass.
For equal masses, compare molecular molar masses inversely to identify which gas sample contains the most molecules.
State distinct ideal-gas assumptions and contrast the model with finite particle size, intermolecular forces or liquefaction in real gases.
Assess the negligible-particle-volume assumption by dividing N times the volume of one particle by the total gas volume.
Distinguish an elastic collision’s conserved kinetic energy from the molecular momentum change that transfers force to a wall.
Explain pressure through the average rate of molecular momentum transfer at the walls and the resulting force per unit area.
Compare isothermal expansion, which lowers wall-collision rate at unchanged mean speed, with fixed-volume heating, where faster particles raise pressure.
For a fixed gas amount, compare states with pV/T = constant, using kelvin and the correct fixed-pressure, fixed-volume or fixed-temperature constraint.
Read gas-cycle coordinates at the requested states and use their pV ratio for temperature without importing an unnecessary law for another leg.
Use numerical pV or p/T values at two graph points to test ideal behaviour over the stated temperature or volume range.
Read axis roles and Celsius offsets carefully: constant-volume p versus kelvin T passes through the origin, while a Celsius scale is shifted.
Choose PV = nRT for molar amount or PV = NkBT for particle number, converting pressure and volume to compatible SI units.
Compare gas samples through p ∝ nT/V, and for a fixed gas species use p = (R/M)ρT to compare pressure, density and temperature.
Use the physical container geometry before applying the gas equation, and show extra precision when checking a stated approximate pressure or volume.
For vessels joined at a common maintained temperature, conserve total gas amount and find common pressure from pfΣV = Σ(pV).
Distinguish final values from changes: subtract the original pressure or gas amount when the question asks for an increase after joining or filling.
For connected containers kept at different temperatures, use common pressure and n ∝ V/T rather than assuming equal temperatures.
For open gas at fixed pressure and volume, use nT = constant to explain escape on heating or calculate particle gain on cooling.
Link graph gradients to the gas equation: constant-pressure V–T has gradient nR/p, and constant-volume p–T has gradient nR/V.
For a constant-area gas column, use V = AH so p versus 1/H is linear and its gradient b gives n = bA/(RT).
When a mercury plug is inverted, combine the change in its weight-pressure contribution with the gas-column length and the fixed-temperature gas law.
Determine gas amount from a p–V state or p–T graph, then convert mol to mass when the question asks for the whole sample’s mass.
Define R through PV = nRT and identify the quantities represented, keeping amount in mol distinct from particle number.
Calculate monatomic ideal-gas internal energy with U = 3NkBT/2 = 3nRT/2, relating the total to N times mean translational particle energy.
State the ideal-approximation conditions required: low pressure with high temperature, or low pressure with low density.
Judge gas-model extrapolation physically: extremely large 1/H means small volume and high pressure, where ideal assumptions can fail.

Covered syllabus points

B.3.1—Pressure B.3.2—Amount of substance B.3.3—Ideal gas model B.3.4—Empirical gas laws B.3.5—Ideal gas equations B.3.6—Molecular pressure model B.3.7—Ideal gas internal energy B.3.8—Ideal gas approximation limits

Exam points

For a surface with pressure on both sides, use the pressure difference in F = AΔp rather than one side’s absolute pressure.
Evaluate an unusually large pressure-force estimate by linking its size to an unrealistic assumption or a mechanism that reduces the pressure difference.
Derive a solid cylinder’s contact pressure by writing weight = ρAhg and dividing by the base area to obtain p = ρgh.
For a thin-walled open cylinder, use the annular rim’s contact area A ≈ 2πRx, so pressure is W/(2πRx).
For a stopper under oil, include the weight of the oil column above its circular area and add the stopper’s own weight to find the lifting force.
Show that energy per unit volume has pressure units: J/m³ = (N m)/m³ = N/m² = Pa.

[Maximum number: 5]

A boat is moved from land to water by rolling it across a set of cylindrical airbags.

(a)

When fully inflated, an unloaded airbag has a diameter of 1.80 m and a length of 24.0 m . At a temperature of $15^{\circ} \mathrm{C}$ , an airbag can hold 4200 mol of gas.

[ 5 ]

(i)

Show that the pressure in an airbag is about 0.2 MPa .

When the boat is placed on the airbags, the airbags are compressed so that the effective contact area is as shown in the diagram.

In this arrangement, the maximum safe pressure before an airbag bursts is four times the value calculated in (a)(i). The boat is supported by airbags on a slope that is at an angle of $4.0^{\circ}$ to the horizontal. There are fifteen airbags supporting the boat at all times.

[ 2 ]

$\begin{aligned} V=\pi r^{2} h=\pi 0.9^{2} \times 24=61.07 \mathrm{~m}^{3} P=\frac{(4200)(8.31)(273+15)}{61.07}=0.16 \mathrm{MPa} \end{aligned}$

Allow MP1 to be seen in MP2.

(ii)

Estimate the maximum safe mass that this arrangement can hold.

[ 3 ]

$\begin{aligned} \mathrm{F}=4 P A=4\left(1.6 \times 10^{5}\right)(15 \times 24 \times 1.8)=4.14 \times 10^{8} \mathrm{~N} \mathrm{F}=m g \cos \theta=\mathrm{m}(9.8)(\cos 4)=9.78 \mathrm{~m} \mathrm{~N} m=\frac{4 P A}{g \cos \theta}=\frac{4.14 \times 10^{8}}{9.78}=4.3 \times 10^{7} \mathrm{~kg} \end{aligned}$

Do not award MP2 if the cos θterm is omitted.

Allow ECF for MP3.

Exam points

Define a mole by its Avogadro number of specified entities, and distinguish amount n in mol from the number of particles N.
Use N = (m/M)NA to estimate particle numbers from sample mass, or compare counts using mass divided by molar mass so NA cancels.
For equal sample masses, identify the smallest molecular molar mass as giving the greatest number of molecules.
Find a single atom or molecule’s mass from M/NA, converting molar mass from g/mol to kg/mol before calculating.

[Maximum number: 2]

Magnesium-27 nuclei $\left({ }_{12}^{27} \mathrm{Mg}\right)$ decay by beta-minus $\left(\beta^{-}\right)$ decay to form nuclei of aluminium-27 (Al).

Small amounts of magnesium in a material can be detected by firing neutrons at magnesium-26 nuclei. This process is known as irradiation.

Magnesium-27 is formed because of irradiation. The products of the beta-particle emission are observed as the magnesium-27 decays to aluminium-27.

The smallest mass of magnesium that can be detected with this technique is $1.1 \times 10^{-8} \mathrm{~kg}$ .

Show that the smallest number of magnesium atoms that can be detected with this technique is about $10^{17}$ .

So $1.1 \times 10^{-8} \mathrm{~kg} \equiv \frac{1.1}{0.027} \times 10^{-8}$ «mol»

OR

Mass of atom $=27 \times 1.66 \times 10^{-27}$ «kg»

2.4-2.5 × $10^{17}$ atoms

Exam points

State distinct kinetic-model assumptions: negligible particle volume, no forces between collisions, random motion, elastic collisions or short collision time.
Define an ideal gas through its idealized particle interactions or obedience to the ideal-gas equation throughout the model’s range.
Contrast real gases with the ideal model using finite molecular size, intermolecular forces or the possibility of liquefaction.
Estimate the occupied-volume fraction as N times a particle’s volume divided by the gas volume, and use its small size to assess the point-particle model.
During isothermal expansion, keep average molecular KE constant while explaining a lower wall-collision rate and greater distance between collisions.
Distinguish elastic kinetic-energy conservation from momentum change: a molecule reflecting from a wall transfers momentum even in the ideal model.
Test ideal behaviour during a constant-temperature change by calculating pV at two graph points and showing agreement before drawing a conclusion.

[Maximum number: 1]

An ideal monatomic gas is kept in a container of volume $2.1 \times 10^{-4} \mathrm{~m}^{3}$ , temperature 310 K and pressure $5.3 \times 10^{5} \mathrm{~Pa}$ .

State what is meant by an ideal gas.

Exam points

For a fixed amount of gas, use p₁V₁/T₁ = p₂V₂/T₂ with temperatures in kelvin, reading the required endpoints rather than the path between them.
For an isobaric change, use V/T = constant or the equivalent ideal-gas equation to find the new volume or temperature.
For an isothermal change, use p₁V₁ = p₂V₂ and retain the pressure ratios in the correct order.
For a constant-area gas column, use V ∝ length and reverse the sign of the mercury weight-pressure contribution when the tube is inverted.
At constant volume, draw p against kelvin T as a straight line through the origin; recognize the Celsius offset when identifying graph axes.
Use two numerical p/T ratios to assess ideal behaviour over a stated constant-volume temperature range.
In a constant-area tube at fixed temperature, explain why plotting p against 1/H gives a straight line through the origin because V ∝ H.
Recognize pV/T = constant for a real gas as an empirical relationship that can be tested against experimental data.
For a show-that gas-law result, display full substitution or enough significant figures beyond the given approximate target.

[Maximum number: 2]

The equipment shown in the diagram was used by a student to investigate the variation with volume, of the pressure p of air, at constant temperature. The air was trapped in a tube of constant cross-sectional area above a column of oil.

The pump forces oil to move up the tube decreasing the volume of the trapped air.

Outline how the results of this experiment are consistent with the ideal gas law at constant temperature.

$V\propto H$ , thus the ideal gas law gives $p\propto\frac{1}{H}$ .

So the graph should be a straight line through the origin, as observed.

Exam points

Choose PV = nRT for amount in mol or PV = NkBT for particle number, and convert absolute temperatures to kelvin.
Rearrange the gas equation for pressure, volume or temperature, converting kPa or MPa to Pa for SI calculations.
Compare different gas samples using p ∝ nT/V or T ∝ pV/n, retaining every amount, volume and temperature ratio.
For a fixed sample, combine pV/T between states; use pV = constant isothermally and p/T = constant at fixed volume.
Read the required states on a cycle and use their pV ratio for temperature; use only the process conditions needed to reach that state.
After opening a valve at common fixed temperature, conserve total gas amount and use pf(V₁ + V₂) = p₁V₁ + p₂V₂.
When asked for a pressure change after joining vessels, subtract the original pressure from the common final pressure.
For connected vessels held at different temperatures, use their common pressure and n ∝ V/T to compare amounts or masses of the same gas.
For gas added at fixed temperature, find the final amount from pV and subtract the initial amount to answer the amount added.
In an open system at fixed pressure and volume, use NT = constant: heating reduces the gas amount and cooling can increase it.
Derive the gradient of a constant-pressure V–T graph as nR/p = mR/(Mp), using the sample mass and molar mass distinctly.
At fixed volume, use p–T gradient nR/V; doubling the particle number doubles the gradient while the kelvin graph still passes through the origin.
For a p-versus-Celsius-temperature graph, infer nR/V from the gradient or p/T in kelvin, rather than p divided by Celsius temperature.
In a tube with V = AH, relate the p-versus-1/H gradient b to nRT/A and calculate n = bA/(RT).
Use a gas-state graph to calculate n = pV/(RT), then multiply by molar mass if the question asks for the sample’s mass.
Use the given normal-pressure contributions and column geometry together: p = p_atm + p_column and V = AH for gas below a mercury plug.
For a fixed gas species, derive p = (R/M)ρT and identify pressure as proportional to density and absolute temperature.
Define R using PV = nRT and identify pressure, volume, amount in mol and absolute temperature when the question asks for the constant’s meaning.
For a stated approximate result, show full substitution or an extra significant figure rather than repeating the target value.

[Maximum number: 2]

(a)

An ideal monatomic gas is kept in a container of volume $2.1 \times 10^{-4} \mathrm{~m}^{3}$ , temperature 310 K and pressure $5.3 \times 10^{5} \mathrm{~Pa}$ .

[ 1 ]

(i)

Calculate the number of atoms in the gas.

[ 1 ]

$N=\frac{pV}{kT}=\frac{5.3\times10^{5}\times2.1\times10^{-4}}{1.38\times10^{-23}\times310}=2.6\times10^{22}$

(b)

The volume of the gas in (a) is increased to $6.8 \times 10^{-4} \mathrm{~m}^{3}$ at constant temperature.

[ 1 ]

(i)

Calculate, in Pa , the new pressure of the gas.

[ 1 ]

$p_2=5.3\times10^{5}\times\frac{2.1\times10^{-4}}{6.8\times10^{-4}}=1.6\times10^{5}\ \mathrm{Pa}$

Exam points

Explain wall pressure through molecular momentum changes in collisions, giving a force and hence force per unit area.
Use Newton’s third law: the wall changes a particle’s momentum, and the particle exerts an equal opposite force on the wall.
For isothermal expansion, keep mean molecular speed unchanged and link less frequent wall collisions to a lower momentum-transfer rate.
At fixed volume, link heating to greater mean molecular speed, increased wall momentum transfer or collision rate, and higher pressure.
Relate temperature to mean molecular kinetic energy, while explaining pressure through particle collisions with the container walls.

[Maximum number: 2]

The volume of the gas in (a) is increased to $6.8 \times 10^{-4} \mathrm{~m}^{3}$ at constant temperature.

Explain, in terms of molecular motion, this change in pressure.

Exam points

Calculate total monatomic ideal-gas internal energy with U = 3NkBT/2 or 3nRT/2, distinguishing particle number from amount in mol.
For an ideal monatomic gas, identify internal energy with total random translational kinetic energy; multiply mean particle energy by N.
After heating, use the final kelvin temperature to find total kinetic energy, and convert joules to kJ when requested.

[Maximum number: 2]

An ideal monatomic gas is kept in a container of volume $2.1 \times 10^{-4} \mathrm{~m}^{3}$ , temperature 310 K and pressure $5.3 \times 10^{5} \mathrm{~Pa}$ .

Calculate, in J , the internal energy of the gas.

For one atom $U=\frac{3}{2}kT$ $U=2.6\times10^{22}\times\frac{3}{2}\times1.38\times10^{-23}\times310=170\ \mathrm{J}$

Exam points

Select low pressure and high temperature as conditions under which a real gas approaches ideal-gas behaviour.
Identify low density together with low pressure when the question compares conditions for an ideal-gas approximation.
For a constant-area gas column, interpret large 1/H as small volume and high pressure, where an ideal-gas extrapolation can fail.

[Maximum number: 2]

The pump forces oil to move up the tube decreasing the volume of the trapped air.

The equation of the line of best fit is:

p=a+\frac{b}{H}

The equation above may be used to predict the pressure of the air at extremely large values of $\frac{1}{H}$ . Suggest why this will be an unreliable estimate of the pressure.

very large $\frac{1}{H}$ means very small volumes / very high pressures at very small volumes the ideal gas does not apply

OR at very small volumes some of the assumptions of the kinetic theory of gases do not hold

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