Question 2

SCQMEDIUM

A nuclear reactor starts producing a radioactive nuclide $X$ from $t = 0$ , at a constant rate of $\alpha$ per second. Each decay of $X$ produces energy $E_0$ , which is utilized to heat a liquid of mass $m$ and specific heat $s$ . Assuming no heat loss from the liquid and taking $\lambda$ as the decay constant of $X$ , the rate of increase in the temperature of the liquid is:

(A)

$\frac{\alpha E_0}{m s}(1 - e^{-\lambda t})$

(B)

$\frac{\alpha E_0}{m s}(e^{\lambda t} - 1)$

(C)

$\frac{\lambda E_0}{m s}(1 - e^{-\lambda t})$

(D)

$\frac{E_0}{m s}(\alpha - \lambda e^{-\lambda t})$

Detailed Solution

Find the number of nuclei ( $N$ ) at time $t$ : The rate of change of $N$ is $\frac{dN}{dt} = \alpha - \lambda N$ . Rearranging and integrating: $\int_0^N \frac{dN}{\alpha - \lambda N} = \int_0^t dt$ . $-\frac{1}{\lambda} \ln\left(\frac{\alpha - \lambda N}{\alpha}\right) = t \implies \alpha - \lambda N = \alpha e^{-\lambda t} \implies \lambda N = \alpha(1 - e^{-\lambda t})$
Find the rate of decay ( $R$ ): $R = \lambda N = \alpha(1 - e^{-\lambda t})$ .
Find the rate of heat production ( $P$ ): Each decay releases $E_0$ , so $P = R E_0 = \alpha E_0(1 - e^{-\lambda t})$ .
Find the rate of temperature increase: Power $P = ms \frac{dT}{dt} \implies \frac{dT}{dt} = \frac{P}{ms}$ $\frac{dT}{dt} = \frac{\alpha E_0}{ms}(1 - e^{-\lambda t})$ .

Question 2

Detailed Solution

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