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An unstable radioactive nucleus X decays into two stable nuclei Y and Z simultaneously (branching decay). At t = 0, only X is present. The decay of X follows first-order kinetics with overall decay constant lambda. Three graphs are plotted: ln(N_X) vs t is a straight line with slope equal to -lambda (negative slope, y-intercept = a), N_Y vs t, and N_Z vs t are both growth curves. The slope of the ln(N_X) vs t graph is -tan(theta). The partial decay constants for X to Y and X to Z are lambda_Y and lambda_Z respectively, with lambda_Y / lambda_Z = b/c (from the ratio of N_Y to N_Z at large t). Which of the following statements is/are correct?

1. Ratio of N_Y: N_Z at any time is constant.
2. Decay constant for decay of X into Y is b * tan(theta) / (b + c).
3. Decay constant for decay of X into Z is c * tan(theta) / e^a.
4. Ratio of N_Y: N_Z is time dependent.

Correct answer: Ratio of N_Y: N_Z at any time is constant.

Solution

For branching radioactive decay: dN_X/dt = -(lambda_Y + lambda_Z) * N_X, so overall lambda = lambda_Y + lambda_Z = tan(theta) (magnitude of slope of ln N_X vs t graph). Since dN_Y/dt = lambda_Y * N_X and dN_Z/dt = lambda_Z * N_X, dividing gives dN_Y/dN_Z = lambda_Y/lambda_Z = constant. Integrating, N_Y/N_Z = lambda_Y/lambda_Z = b/c = constant at all times (option 1 correct, option 4 wrong). The partial decay constant lambda_Y = lambda * b/(b+c) = b*tan(theta)/(b+c) (option 2 correct). Option 3 has e^a in the denominator which is incorrect; the correct formula should not involve e^a.

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