Heat and Thermodynamics 3 Question 33
33. A solid copper sphere (density $\rho$ and specific heat $c$ ) of radius $r$ at an initial temperature $200 K$ is suspended inside a chamber whose walls are at almost $0 K$. The time required for the temperature of the sphere to drop to $100 K$ is ……
(1991, 2M)
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Solution:
- Area of sphere, $A=4 \pi r^{2}$
Mass of sphere, $\quad m=\left(\frac{4}{3} \pi \rho r^{3}\right)$
Now, energy radiated per second $=\left(\sigma A T^{4}\right)$
$\therefore \quad m c\left(-\frac{d T}{d t}\right)=\sigma A T^{4}$
or $\quad \int _0^{t} d t=\left(\frac{-m c}{\sigma A}\right) \int _{200}^{100} T^{-4} d T$
or $\quad t=\frac{m c}{3 \sigma A}\left[\frac{1}{(100)^{3}}-\frac{1}{(200)^{3}}\right]$
$$ =\frac{7 m c}{3 \times 8 \sigma A} \times 10^{-6}=\frac{7 m c \times 10^{-6}}{24 \sigma A} $$
Substituting the values,
$$ t=\frac{\left(7 \times \frac{4}{3} \pi \rho r^{3}\right)(c) \times 10^{-6}}{3 \times 8 \times 5.67 \times 10^{-8} \times 4 \pi r^{2}}=1.71 \rho r c $$
