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

Cambridge International AS & A Level Physics – Topic 15.2: Equation of State for an Ideal Gas

Learning Objectives (Assessment Objectives)

• AO1 – Knowledge & Understanding: Recall the two forms of the ideal‑gas equation, the meaning of each symbol and the numerical values of the universal constants.
• AO2 – Application: Rearrange and use the equation to calculate any one variable when the other three are known, performing all required unit conversions (Pa ↔ atm ↔ bar, L ↔ m³, °C ↔ K).
• AO3 – Analysis & Evaluation: Discuss the assumptions behind the ideal‑gas model, identify conditions where it breaks down and compare it with the Van der Waals equation.
• AO4 – Practical Skills (Paper 5): Design a simple experiment to verify the ideal‑gas law and outline the data‑analysis steps required.

Syllabus Context

This topic belongs to the Thermodynamics section of the A‑Level extension (syllabus items 15.1–15.5). It links directly to:

• 15.1 – Temperature and the kinetic theory of gases.
• 15.3 – Internal energy, specific heat capacities and the first law of thermodynamics.
• 15.4 – Real gases and the Van der Waals equation.
• Practical work (Paper 5) – Measuring pressure, volume and temperature to test the gas law.

Constants at a Glance

Quick‑Reference Table of Symbols

Two Equivalent Forms of the Ideal‑Gas Equation

Both forms describe the same physical relationship; they differ only in whether the amount of gas is expressed in moles or in individual molecules.

\[

\boxed{pV = nRT}\qquad\text{(mole‑based form)}

\]

\[

\boxed{pV = NkT}\qquad\text{(molecule‑based form)}

\]

Link between the two forms:

\[

N = nNA\qquad\text{and}\qquad k = \frac{R}{NA}

\]

Derivation (Mole‑Based → Molecule‑Based)

1. Start with the experimentally established mole‑based law: \(pV = nRT\).
2. Express the amount of substance in molecules: \(n = \dfrac{N}{N_A}\).
3. Substitute into the law:

   \[

   pV = \left(\frac{N}{NA}\right)RT = N\left(\frac{R}{NA}\right)T.

   \]

4. Recognise that \(\displaystyle\frac{R}{N_A}=k\); thus

   \[

   pV = NkT.

   \]

Link to the Kinetic Theory of Gases (Topic 15.1)

• From kinetic theory: \(p = \frac{1}{3}\rho\overline{c^{2}}\), where \(\rho\) is the density and \(\overline{c^{2}}\) the mean‑square speed.
• Combining this with \(pV = NkT\) gives the familiar result \(\frac{1}{2}m\overline{c^{2}} = \frac{3}{2}kT\).
• This connection satisfies AO3 – students must be able to explain why the ideal‑gas law works and where its assumptions (point particles, no intermolecular forces, perfectly elastic collisions) may fail.

Limits of Applicability (Real Gases – Topic 15.4)

The ideal‑gas equation is an approximation. It fails when:

• High pressure – molecular volumes become non‑negligible.
• Low temperature – attractive forces dominate and condensation may occur.

In these regimes the Van der Waals equation provides a better description:

\[

\left(p + \frac{aN^{2}}{V^{2}}\right)(V - Nb) = NkT,

\]

where \(a\) and \(b\) are gas‑specific constants.

Rearrangements for Quick Reference

• Pressure: \(\displaystyle p = \frac{nRT}{V} = \frac{NkT}{V}\)
• Volume: \(\displaystyle V = \frac{nRT}{p} = \frac{NkT}{p}\)
• Temperature: \(\displaystyle T = \frac{pV}{nR} = \frac{pV}{Nk}\)
• Moles: \(\displaystyle n = \frac{pV}{RT}\)
• Molecules: \(\displaystyle N = \frac{pV}{kT}\)

Unit‑Conversion Cheat‑Sheet

Worked Examples

Example 1 – Pressure (Mole Form)

Problem: 0.500 mol of an ideal gas occupies 12.0 L at 298 K. Find the pressure in kPa.

Solution:

1. Convert volume: \(12.0\ \text{L}=1.20\times10^{-2}\ \text{m}^{3}\).
2. Use \(p = \dfrac{nRT}{V}\) with \(R = 8.314\ \text{J mol}^{-1}\text{K}^{-1}\):

   \[

   p = \frac{(0.500)(8.314)(298)}{1.20\times10^{-2}} = 1.03\times10^{5}\ \text{Pa}.

   \]

3. Convert to kPa: \(p = 103\ \text{kPa}\).

Example 2 – Volume (Molecule Form)

Problem: A sample contains \(3.0\times10^{22}\) molecules at 1.00 atm and 350 K. Find its volume in cm³.

Solution:

1. Convert pressure: \(1.00\ \text{atm}=1.013\times10^{5}\ \text{Pa}\).
2. Apply \(V = \dfrac{NkT}{p}\) with \(k = 1.381\times10^{-23}\ \text{J K}^{-1}\):

   \[

   V = \frac{(3.0\times10^{22})(1.381\times10^{-23})(350)}{1.013\times10^{5}}

   = 1.42\times10^{-3}\ \text{m}^{3}.

   \]

3. Convert to cm³: \(1.42\times10^{-3}\ \text{m}^{3}=1.42\times10^{3}\ \text{cm}^{3}\).

Example 3 – Combined Gas Law (Change of State)

Problem: 2.00 mol of an ideal gas is initially at \(p{1}=1.0\ \text{atm}\), \(V{1}=5.0\ \text{L}\) and \(T{1}=300\ \text{K}\). The gas is heated to \(T{2}=450\ \text{K}\) while the pressure is kept constant. Find the final volume \(V_{2}\).

Solution:

1. With constant pressure, use Charles’s law: \(\displaystyle \frac{V{1}}{T{1}} = \frac{V{2}}{T{2}}\).
2. Rearrange: \(V{2}=V{1}\frac{T{2}}{T{1}} = 5.0\ \text{L}\times\frac{450}{300}=7.5\ \text{L}\).

Example 4 – Mixed‑Unit Problem (Mole Form)

Problem: A 0.250‑mol sample of an ideal gas is measured at a pressure of 0.750 atm and a temperature of 25 °C. Determine the volume in litres.

Solution:

1. Convert temperature: \(25\ ^\circ\text{C}=298.15\ \text{K}\).
2. Convert pressure to pascals (or use the L·atm form of \(R\)).

   Using \(R = 0.082057\ \text{L atm mol}^{-1}\text{K}^{-1}\) avoids a pressure conversion.

3. Apply \(V = \dfrac{nRT}{p}\):

   \[

   V = \frac{(0.250)(0.082057)(298.15)}{0.750}

   = 8.15\ \text{L}.

   \]

Common Pitfalls & How to Avoid Them

• Confusing \(n\) and \(N\): Remember \(N = nN_A\). If the number is of order 10²³ you are dealing with molecules, not moles.
• Temperature units: Always use kelvin. Convert with \(T(\text{K}) = T(^{\circ}\text{C}) + 273.15\).
• Pressure units: The SI constant \(R=8.314\) requires pressure in pascals. If you prefer atm, use \(R=0.082057\ \text{L atm mol}^{-1}\text{K}^{-1}\) consistently.
• Volume units: Use m³ with the SI form of \(R\); use litres when the L·atm form of \(R\) is used.
• Assuming ideal behaviour at extremes: Check reduced temperature \(Tr = T/Tc\) and reduced pressure \(pr = p/pc\). If \(Tr < 2\) and \(pr > 0.5\), the ideal‑gas law may give >5 % error.

Experimental Design (AO4 – Paper 5)

Objective: Verify \(pV = nRT\) for a fixed amount of gas.

1. Apparatus: sealed syringe (variable volume), digital pressure sensor, thermometer, gas‑tight holder, analytical balance (to determine mass → moles).
2. Method:

   • Weigh a known mass of a dry gas (e.g., N₂) and calculate \(n = \dfrac{m}{M}\) where \(M\) is the molar mass.
   • Set the syringe to a series of volumes (e.g., 50 mL, 75 mL, 100 mL).
   • Record pressure and temperature for each volume.
   • Plot \(p\) against \(1/V\); the slope should equal \(nRT\).

3. Data‑analysis checklist:

   • Convert all readings to SI units before calculation.
   • Determine the experimental value of \(R\) from the slope and compare with the accepted 8.314 J mol⁻¹ K⁻¹.
   • Identify systematic errors (e.g., thermal expansion of the syringe, pressure‑sensor calibration drift, gas leakage) and suggest mitigations.

Summary

The ideal‑gas equation of state links the macroscopic variables pressure, volume and temperature to the amount of gas present. It can be expressed in two interchangeable forms:

\[

pV = nRT \qquad\text{or}\qquad pV = NkT

\]

Mastery of the equation enables rapid calculation of any missing variable, conversion between moles and molecules, and provides a foundation for more advanced topics such as kinetic theory, real‑gas corrections, and thermodynamic cycles.

Practice Questions

1. (AO2 – 2 marks) Calculate the pressure exerted by 0.500 mol of an ideal gas occupying 12.0 L at 298 K. Give your answer in kPa.
2. (AO2 – 3 marks) A sample contains \(3.0\times10^{22}\) molecules of an ideal gas at 1.00 atm and 350 K. Find its volume in cm³. Show all unit conversions.
3. (AO2 – 2 marks) How many moles of gas are required to fill a 5.00 L container at 2.00 atm and 273 K?
4. (AO3 – 4 marks) Explain why the ideal‑gas equation may fail for a gas at 0.1 K and 100 atm. Include a brief reference to the Van der Waals equation.
5. (AO4 – 5 marks) Outline an experiment to test the ideal‑gas law using a syringe and a digital pressure sensor. Include at least three sources of systematic error and how they could be minimised.

Suggested Diagram