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

The Mole – Cambridge AS & A‑Level Physics (9702)

1. Placement in the Syllabus

This note covers Topic 15.1 – The Mole, an A‑level extension. The mole is the quantitative link used throughout the rest of the syllabus:

• Ideal‑gas equations (Topic 15.2)
• Thermodynamics – internal energy, enthalpy, specific heat (Topic 16)
• Nuclear physics – binding energy, decay calculations (Topic 23)
• Particle & medical physics – activity, dose calculations (Topic 24)

Later topics also require the concepts of molar concentration, the molar gas constant R, and mass‑energy equivalence. These connections are highlighted in the Further Reading box (see below).

Assessment Objective mapping

• AO1 (knowledge): definitions, constants, fundamental equations.
• AO2 (application): conversions between mass, moles and number of particles; stoichiometric and gas‑law calculations.
• AO3 (experimental): design of a gravimetric or gas‑evolution experiment to determine \(N_{\!A}\).

2. Objective

Use molar quantities confidently: recognise that one mole of any substance contains exactly \(N_{\!A}=6.02214076\times10^{23}\) elementary entities, and convert between mass, amount of substance and number of particles.

3. Key Definitions & Constants

4. Fundamental Relations

The three core quantities are linked by two equivalent equations:

Combining them gives a direct mass–particle conversion:

For gases the Ideal‑gas equation introduces the mole:

5. Converting Between Units – Step‑by‑Step Checklist

1. Identify the known quantity – mass \(m\), amount \(n\) or particles \(N\).
2. Choose the appropriate relation:

   • If \(m\) is known → \(n=m/M\).
   • If \(n\) is known → \(N=nN_{\!A}\) (or \(m=nM\)).
   • If \(N\) is known → \(n=N/N{\!A}\) (or \(m=(N/N{\!A})M\)).

3. Insert the correct units – remember \(1\;\text{kg mol}^{-1}=1000\;\text{g mol}^{-1}\). Convert volumes to litres or dm³ as required.
4. Solve for the unknown and retain the appropriate number of significant figures (usually 3 sf for exam work).

6. Extended Molar‑Mass Table (selected common substances)

7. Worked Examples

Example 1 – Number of molecules in a sample

Problem: How many molecules are present in \(5.00\ \text{g}\) of \(\mathrm{CO_2}\)?

1. Amount of substance:
   
   \(n = \dfrac{m}{M}= \dfrac{5.00\ \text{g}}{44.009\ \text{g mol}^{-1}} = 0.1136\ \text{mol}\)
2. Convert to molecules:
   
   \(N = nN_{\!A}=0.1136\ \text{mol}\times6.02214076\times10^{23}\ \text{mol}^{-1}=6.84\times10^{22}\ \text{molecules}\)

Example 2 – Mass from a given number of atoms

Problem: What mass of copper corresponds to \(2.00\times10^{22}\) atoms?

1. Amount of substance:
   
   \(n = \dfrac{N}{N_{\!A}} = \dfrac{2.00\times10^{22}}{6.02214076\times10^{23}} = 3.32\times10^{-2}\ \text{mol}\)
2. Mass:
   
   \(m = nM = 3.32\times10^{-2}\ \text{mol}\times63.546\ \text{g mol}^{-1}=2.11\ \text{g}\)

Example 3 – Gas‑volume to moles (link to Ideal‑gas topic)

Problem: A container holds \(11.2\ \text{L}\) of nitrogen gas at STP. How many moles are present?

At STP, \(1\ \text{mol}\) of an ideal gas occupies \(22.4\ \text{L}\).

Example 4 – Energy released per mole of fuel (nuclear/thermodynamics link)

Problem: The fission of \(1\ \text{mol}\) of \(\mathrm{^{235}U}\) releases \(200\ \text{MeV}\) per nucleus. Find the total energy in joules.

8. Practical Skills (AO3) – Determining \(N_{\!A}\) by Gas‑Evolution

Goal: Estimate Avogadro’s number using a simple acid‑metal reaction.

1. Apparatus: 250 mL volumetric flask, analytical balance (±0.01 g), gas‑evolution set‑up, thermometer, barometer.
2. Procedure outline (adapted from the Cambridge Practical Handbook):

   • Weigh the empty flask (\(m_{\text{empty}}\)).
   • Weigh \(m_{\text{Zn}}\) g of zinc metal and place it in the flask.
   • Add \(V{\text{acid}}\) mL of dilute \(\mathrm{H2SO_4}\) and immediately seal the flask with a delivery tube leading to an inverted graduated cylinder over water.
   • Collect the evolved \(\mathrm{H2}\) gas, record its volume \(V{\mathrm{H_2}}\) at the measured temperature \(T\) and pressure \(P\).
   • Dry the gas (remove water vapour) and calculate the number of moles using \(n{\mathrm{H2}} = PV/RT\).

3. Calculations:

   • Number of moles of zinc used: \(n{\text{Zn}} = m{\text{Zn}}/M_{\text{Zn}}\).
   • Stoichiometry of the reaction \(\mathrm{Zn + H2SO4 \rightarrow ZnSO4 + H2}\) gives \(n{\mathrm{H2}} = n{\text{Zn}}\). Compare this with the experimental \(n{\mathrm{H2}}\) from the gas‑law to obtain \(N{\!A}=n{\text{Zn}}/n{\mathrm{H2}}\times N{\!A}^{\text{(assumed)}}\) – effectively solving for the constant.

4. Uncertainty analysis – propagate errors from mass (balance), volume (graduated cylinder), temperature, and pressure. Discuss systematic effects such as gas leakage or incomplete reaction.

9. Common Mistakes & How to Avoid Them

• Units of molar mass: \(M\) is normally given in g mol⁻¹. If you work in kilograms, convert: \(1\ \text{kg mol}^{-1}=1000\ \text{g mol}^{-1}\).
• Molecular mass vs. molar mass: 1 u = 1 g mol⁻¹. Do not mix the two scales.
• Rounded Avogadro constant: Use the exact defined value \(6.02214076\times10^{23}\) unless the question explicitly states otherwise.
• Significant figures: Keep at least three sf for typical exam work; propagate correctly through multiplication/division.
• STP volume: \(22.4\ \text{L mol}^{-1}\) applies only at \(0^{\circ}\text{C}\) and \(1\ \text{atm}\). For other conditions use \(PV=nRT\).
• Molar concentration: Remember \(c=n/V\). Confusing \(c\) with mass concentration leads to errors in solution‑based questions.
• Mass‑energy equivalence: When converting energy per mole to energy per particle, always divide by \(N{\!A}\) (or multiply by \(1/N{\!A}\)).

10. Quick Reference Sheet

11. Self‑Check Questions

1. What mass corresponds to \(2.5\ \text{mol}\) of \(\mathrm{NaCl}\)?
2. How many atoms are present in \(0.250\ \text{g}\) of copper (\(M=63.546\ \text{g mol}^{-1}\))?
3. A gas occupies \(22.4\ \text{L}\) at STP. How many moles does it contain?
4. Calculate the number of molecules in \(0.750\ \text{g}\) of glucose (\(\mathrm{C6H{12}O_6}\), \(M=180.156\ \text{g mol}^{-1}\)).
5. Design a brief experimental plan (max 150 words) to determine \(N_{\!A}\) using a gas‑evolution method.
6. State the molar concentration of a solution prepared by dissolving \(5.00\ \text{g}\) of \(\mathrm{NaCl}\) in enough water to make \(250\ \text{mL}\) of solution.
7. Using \(E=mc^2\), find the energy released when \(1\ \text{mol}\) of \(\mathrm{^{235}U}\) (mass defect = 0.1 % of its mass) undergoes fission. (Take \(c=3.00\times10^8\ \text{m s}^{-1}\).)

12. Answers & Marking Scheme

1. \(m = nM = 2.5\ \text{mol}\times58.44\ \text{g mol}^{-1}=146.1\ \text{g}\) (3 sf).
2. \(n = 0.250/63.546 = 3.93\times10^{-3}\ \text{mol}\);
   

   \(N = nN_{\!A}=3.93\times10^{-3}\times6.02214076\times10^{23}=2.37\times10^{21}\ \text{atoms}\) (3 sf).

3. At STP, \(22.4\ \text{L}\) = \(1.00\ \text{mol}\) (definition).
4. \(n = 0.750/180.156 = 4.16\times10^{-3}\ \text{mol}\);
   

   \(N = nN_{\!A}=4.16\times10^{-3}\times6.02214076\times10^{23}=2.51\times10^{21}\ \text{molecules}\) (3 sf).

5. Sample answer (≈130 words): Weigh an empty 250 mL flask, then add a measured mass of zinc metal. Add excess dilute \(\mathrm{H2SO4}\) and immediately seal the flask with a delivery tube leading to an inverted graduated cylinder over water. Record the displaced water volume \(V{\mathrm{H2}}\), temperature \(T\) and pressure \(P\). Convert the wet volume to dry gas using the vapour pressure of water, then calculate \(n{\mathrm{H2}} = PV/RT\). Since the reaction stoichiometry is 1 mol Zn → 1 mol H₂, the experimental \(n{\mathrm{H2}}\) should equal the moles of Zn used (\(m{\text{Zn}}/M{\text{Zn}}\)). Any discrepancy allows estimation of \(N{\!A}\) via \(N{\!A}=n{\text{Zn}}/n{\mathrm{H2}}\times N{\!A}^{\text{(assumed)}}\). Include uncertainties from volume, temperature, pressure and balance.
6. Mass of NaCl = 5.00 g → \(n = 5.00/58.44 = 0.0856\ \text{mol}\).
   

   Molar concentration \(c = n/V = 0.0856\ \text{mol}/0.250\ \text{L}=0.342\ \text{mol L}^{-1}\) (3 sf).

7. Mass of 1 mol \(\mathrm{^{235}U}\) ≈ 235 g = \(0.235\ \text{kg}\). Mass defect = 0.001 × 0.235 kg = \(2.35\times10^{-4}\ \text{kg}\).
   

   Energy = \(E = \Delta m\,c^{2}=2.35\times10^{-4}\times(3.00\times10^{8})^{2}=2.12\times10^{13}\ \text{J}\) (2 sf).

13. Further Reading (Link‑in to Later Topics)

14. Summary

The mole is the essential bridge between the macroscopic quantities measured in the laboratory and the microscopic world of atoms, molecules and ions. Mastery of the relations \(n=m/M\), \(N=nN_{\!A}\) and \(PV=nRT\), together with an awareness of units, significant figures and the broader context (concentration, energy, gas laws), equips you to solve the full range of A‑level physics problems and to design experiments that satisfy the AO3 requirements of the Cambridge assessment.