The radiochemist, Will I. Glow, studied thorium-232 and found that 2.82 *10¯⁷ moles emitted 8.42 *10⁶ $\alpha$ particles in one year. What is the decay constant for thorium-232?

Question

Quizplus · Accepted Answer

The decay constant ($\lambda$) can be calculated using the formula: 
$$N = N_0 e^{-\lambda t}$$ where $N_0$ is the initial number of atoms, $N$ is the number of atoms at time $t$, and $\lambda$ is the decay constant.

The number of $\alpha$ particles emitted in one year ($8.42	imes10^{10}$) is proportional to the number of thorium-232 atoms present. Thus, we can write: 
$$8.42	imes10^{10} = \lambda N$$

The number of moles of thorium-232 ($2.82	imes10^{-10}$) can be converted to the number of atoms using Avogadro's number: 
$$N_0 = 2.82	imes10^{-10}	imes6.022	imes10^{23} = 1.6972	imes10^{14}$$

Substituting these values, we get: 
$$\lambda = \frac{8.42	imes10^{10}}{1.6972	imes10^{14}} = 4.96	imes10^{-11}$$

Converting this to years$^{-1}$, we get: 
$$\lambda = 4.96	imes10^{-11}	ext{ s}^{-1}	imes3.1536	imes10^7	ext{ s yr}^{-1} = 1.56	imes10^{-3}	ext{ yr}^{-1}$$

The closest answer choice is B) 4.96 × 10$^{10}$ yr$^{-1}$, which is the correct answer.

The Radiochemist, Will I α\alphaα Particles in One Year

The Radiochemist, Will I $\alpha$ Particles in One Year