A-Level Physics May/June 2024 Q9(c): The mass of radioactive isotope X in the sample in (b) is 7.3 × 10⁻⁴ kg at time t = 0.…

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  The number of nuclei, $N$, is given as $4.0 \times 10^{22}$. The total mass of the sample, $M$, is $7.3 \times 10^{-4}$ kg.

  First, calculate the mass of a single nucleus, $m_{nucleus}$: $m_{nucleus} = \frac{\text{Total mass}}{\text{Number of nuclei}} = \frac{M}{N}$ $m_{nucleus} = \frac{7.3 \times 10^{-4}}{4.0 \times 10^{22}} = 1.825 \times 10^{-26}$ kg

  The mass of a nucleus is its nucleon number, $A$, multiplied by the unified atomic mass unit, $u$ ($1u = 1.66 \times 10^{-27}$ kg). $m_{nucleus} \approx A \times u$

  Rearranging to find the nucleon number, $A$: $A = \frac{m_{nucleus}}{u}$ $A = \frac{1.825 \times 10^{-26}}{1.66 \times 10^{-27}}$ $A = 10.9939...$

  Since the nucleon number must be an integer: $A = 11$

  Nucleon number = 11

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  How the marks are awarded

  • C1 — Correctly stating or using the formula to find the mass of one nucleus by dividing the total sample mass by the number of nuclei: (7.3 × 10⁻⁴) / (4.0 × 10²²).
  • C1 — Using the correct method to find the nucleon number by dividing the mass of a single nucleus by the unified atomic mass unit (1.66 × 10⁻²⁷ kg).
  • A1 — Obtaining the correct final answer of 11 and ensuring it is stated as an integer.

  Common mistakes

  • Leaving the final answer as a decimal (e.g., 10.99 or 11.0); the nucleon number is a count of particles and must be a whole number.
  • Calculation error when entering powers of ten into the calculator, for example, misplacing a bracket and effectively multiplying by 10⁻²⁷ instead of dividing.
  • Incorrectly rearranging the formula, for example, multiplying the mass of the nucleus by the unified atomic mass unit instead of dividing.
  • Confusing the number of nuclei in the sample with Avogadro's constant, or trying to use moles without knowing the molar mass.

  Examiner tip: Break down multi-step calculations by clearly defining the relationship between macroscopic quantities (like total mass) and microscopic properties (like nucleon number) using fundamental constants.

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