CIE iGCSE Co-Ordinated Science P5.2.3 Radioactive decay Exam Style Questions Paper 4
Question
Most-appropriate topic codes (Cambridge IGCSE Co-ordinated Sciences 0654):
• Topic P5.1 — The nucleus (Part (a)(i), (a)(ii), (a)(iii))
• Topic P5.2.3 — Radioactive decay (Part (b)(i))
• Topic P5.2.4 — Half-life (Part (b)(ii), (b)(iii))
▶️ Answer/Explanation
(a)(i) Number of protons = 79.
In the nuclide notation \( ^{A}_{Z}\text{X} \), the bottom number (Z) is the proton number (atomic number), which represents the number of protons in the nucleus. For \( ^{197}_{79}\text{Au} \), Z = 79, so there are 79 protons.
(a)(ii) Number of neutrons = 197 − 79 = 118.
The top number (A) is the nucleon number (mass number), which is the total number of protons and neutrons in the nucleus. The number of neutrons is calculated by subtracting the proton number from the nucleon number: \( A – Z = 197 – 79 = 118 \) neutrons.
(a)(iii) The word that describes these two gold atoms is isotopes.
Atoms of the same element (same number of protons) but with different numbers of neutrons are called isotopes. \( ^{197}_{79}\text{Au} \) and \( ^{198}_{79}\text{Au} \) both have 79 protons (so they are both gold), but they have different numbers of neutrons (118 and 119 respectively). Therefore, they are isotopes. They are not ions (which are charged particles formed by loss or gain of electrons) and they are not charged overall as they are neutral atoms.
(b)(i) \( ^{198}_{79}\text{Au} \rightarrow ^{198}_{80}\text{Hg} + ^{0}_{-1}\beta \)
In beta decay, a neutron in the nucleus changes into a proton and an electron (beta particle). The beta particle is emitted from the nucleus. The atomic number increases by 1 (from 79 to 80), so the atom becomes mercury (Hg). The mass number remains unchanged at 198 because the total number of nucleons is conserved (a neutron is replaced by a proton, which has approximately the same mass). The beta particle is represented as \( ^{0}_{-1}\beta \) (or \( ^{0}_{-1}\text{e} \)).
(b)(ii) Half-life is the (average) time taken for half the nuclei of the isotope in a sample to decay.
Half-life is a measure of the rate of radioactive decay. It is defined as the average time taken for the activity of a radioactive sample to decrease to half its original value, or equivalently, the time taken for half of the nuclei in a sample to decay. It is a constant for a given isotope and does not depend on the size of the sample or external conditions.
(b)(iii) Mass remaining = 35 g.
Number of half-lives = 8.1 ÷ 2.7 = 3 half-lives.
Mass remaining = 280 ÷ 2³ = 280 ÷ 8 = 35 g.
To calculate the mass remaining after a certain time:
- Determine the number of half-lives that have passed: \( \text{Number of half-lives} = \frac{\text{Time elapsed}}{\text{Half-life}} = \frac{8.1}{2.7} = 3 \) half-lives.
- After each half-life, the mass halves. After n half-lives, the remaining mass is: \( \text{Remaining mass} = \text{Initial mass} \times \left(\frac{1}{2}\right)^n \).
\( \text{Remaining mass} = 280 \times \left(\frac{1}{2}\right)^3 = 280 \times \frac{1}{8} = 35 \text{ g} \).
Question
Most-appropriate topic codes (Cambridge IGCSE Co-ordinated Sciences 0654):
• Topic P6.2.3 — Galaxies and the Universe / Big Bang Theory (Part (a)(i) & (a)(ii))
• Topic P6.2.1 — The Sun as a star / Speed of light calculations (Part (b))
• Topic P5.2.3 — Radioactive decay / Alpha decay equations (Part (c)(i))
• Topic P5.2.4 — Half-life calculations (Part (c)(ii))
▶️ Answer/Explanation
(a)(i) 13.8 billion years
According to the Big Bang Theory, the Universe is approximately 13.8 billion years old, determined from cosmic microwave background radiation and the expansion rate of the Universe.
(a)(ii) The Universe began from a single point of extremely high temperature and density, which then expanded rapidly.
The Big Bang Theory states that all matter and energy in the Universe were initially concentrated in a singularity. This singularity then expanded and continues to expand today.
(b) Shortest distance = \( 5.6 \times 10^{10} \) m
\( \text{time} = \frac{\text{distance}}{\text{speed}} = \frac{5.6 \times 10^{10}}{3.0 \times 10^8} = 1.87 \times 10^2 = 187 \text{ s} \)
The shortest possible time uses the minimum distance between Earth and Mars. Light travels at \( 3.0 \times 10^8 \) m/s in a vacuum.
(c)(i) \( ^{210}_{84}\text{Po} \rightarrow ^{206}_{82}\text{Pb} + ^{4}_{2}\alpha \)
Alpha decay reduces both the mass number and atomic number. The mass number decreases by 4 (210 → 206) and the proton number decreases by 2 (84 → 82), forming lead (Pb).
(c)(ii) Number of half-lives = \( \frac{12.4}{3.1} = 4 \)
\( \text{Mass remaining} = \frac{560}{2^4} = \frac{560}{16} = 35 \text{ g} \)
After 4 half-lives, the mass has halved four times: 560 → 280 → 140 → 70 → 35 g. The remaining mass is 35 g.