U^238 decays to 90 Th^234 with half-life 4.5 ×10^9 year. The resulting 90 Th^234 is in excited

step 1 In this problem I can write the reaction as U238, U2382 in slow process it changes into TH 234 a

U^238 decays to 90 Th^234 with half-life 4.5 ×10^9 year....

$\mathrm{U}^{238}$ decays to ${ }_{90} \mathrm{Th}^{234}$ with half-life $4.5 \times 10^{9}$ year. The resulting ${ }_{90} \mathrm{Th}^{234}$ is in excited state and hence, emits further a gamma ray to come to the ground state, with halflife of $10^{-8} \mathrm{~s}$. A sample of ${ }_{92} \mathrm{U}^{238}$ emits 20 gamma rays per second. In what time, the emission rate will drop to 5 gamma rays per second? (a) $2 \times 10^{-8} \mathrm{~s}$ (b) $0.25 \times 10^{-8} \mathrm{~s}$ (c) $9 \times 10^{9}$ year (d) $1.125 \times 10^{9}$ year

$\mathrm{U}^{238}$ decays to ${ }_{90} \mathrm{Th}^{234}$ with half-life $4.5 \times 10^{9}$ year. The resulting ${ }_{90} \mathrm{Th}^{234}$ is in excited state and hence, emits further a gamma ray to come to the ground state, with halflife of $10^{-8} \mathrm{~s}$. A sample of ${ }_{92} \mathrm{U}^{238}$ emits 20 gamma rays per second. In what time, the emission rate will drop to 5 gamma rays per second?
(a) $2 \times 10^{-8} \mathrm{~s}$
(b) $0.25 \times 10^{-8} \mathrm{~s}$
(c) $9 \times 10^{9}$ year
(d) $1.125 \times 10^{9}$ year

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Key Concepts

Half-life and Decay Constant

The half-life is the time required for half of the radioactive nuclei in a sample to decay, and it is inversely related to the decay constant (?) through the relation T1/2 = ln2/?. Understanding half-life is crucial for calculating the time elapsed for a radioactive substance to reduce to a fraction of its initial activity, as demonstrated in the decay rate drop from 20 to 5 gamma rays per second.

Exponential Decay Law

This law describes how the quantity of a radioactive substance decreases over time. It states that the number of undecayed nuclei, and consequently the decay rate, declines exponentially at a rate defined by the decay constant. The law is foundational for determining how long it takes for a sample to diminish by a specific factor, which is mathematically expressed by N(t) = N0 * exp(-?t).

Decay Chain and Prompt Gamma Emission

In a decay chain where the daughter nucleus is produced in an excited state, it often quickly relaxes to the ground state by emitting a gamma ray. When the gamma emission has a very short half-life compared to the parent, the gamma rate directly reflects the decay rate of the parent nucleus. Thus, calculations of the overall decay process can be focused on the parent’s decay kinetics.

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