Testing a decay model with background counts

Part (a) marking points

Final answer: 8.00 s⁻¹. Numerical checks assess the final value only; they do not check working or the number of digits typed.

1. Background rate is 120/600 = 0.200 s⁻¹, equivalent to 12 counts in 60 s. (1 method mark)
2. Corrected source rate is (492 − 12)/60 = 8.00 s⁻¹. (1 accuracy mark)

No earlier numerical result is required. Credit a valid alternative method. Accuracy credit requires a physically valid method.

Part (b) marking points

1. Subtract 12 from each count and divide by 60 to obtain 16.0, 11.3167, 8.00, 5.65 and 4.0167 s⁻¹. (1 independent mark)
2. Plot ln(R/Rref) vertically against time / min horizontally, using Rref = 1.00 s⁻¹, and label both axes. (1 independent mark)
3. Plot about 2.773, 2.426, 2.079, 1.732 and 1.390; a straight fitted line tests a constant decay constant. (1 independent mark)

Credit equivalent correct explanations. Award each distinct point once; a contradiction invalidates the affected point.

Acceptable alternatives: Plot ln(corrected counts / 1 count) against time, or use a base-10 logarithm with the corresponding half-life relation.

Part (c) marking points

Final answer: 4.00 min. Numerical checks assess the final value only; they do not check working or the number of digits typed.

1. Determine gradient about −0.173 min⁻¹ and λ = −gradient for natural logarithms. (1 method mark)
2. Use T½ = ln 2/λ. (1 method mark)
3. Obtain a value in the accepted range 3.8–4.2 min. (1 accuracy mark)

Accept valid graph fits within the stated range. Credit a correctly applied logarithm/half-life relationship after a numerical gradient error. A negative half-life is not creditable.

Acceptable alternatives: For base-10 logs use T½ = log10(2)/magnitude of gradient.

Part (d) marking points

1. As the source decays, background forms a larger fraction of the observed rate. (1 independent mark)
2. The uncorrected log-rate plot flattens at late times and is not a straight line with the source decay gradient. (1 independent mark)
3. A fitted late-time slope would have too small a magnitude, giving too long a half-life. (1 independent mark)

Credit equivalent correct explanations. Award each distinct point once; a contradiction invalidates the affected point.

Part (e) marking points

1. Identify a valid improvement to source counting or sampling. (1 independent mark)
2. Explain how it improves precision or detects departures from a single exponential. (1 independent mark)
3. Identify a distinct valid background/geometry/sampling improvement. (1 independent mark)
4. Explain its role, including an appropriate limitation. (1 independent mark)

Credit equivalent correct explanations. Award each distinct point once; a contradiction invalidates the affected point.

Acceptable alternatives: Any two distinct justified changes earn up to two marks each. Do not award two versions of the same repetition suggestion.