First slide

Radiation

Question

Certain quantity of water cools from 70°C to 60°C in the first 5 minutes and to 54°C in the next 5 minutes. The temperature of the surroundings is

Moderate

A

45°C
B

20°C
C

42°C
D

10°C

Solution

$Let T_{s} be the temperature of the surroundings.$
According to Newton's law of cooling
$\frac{T_{1} - T_{2}}{t} = K (\frac{T_{1} + T_{2}}{2} - T_{s})$
for first 5 minutes,
$\begin{matrix} T_{1} = 70^{\circ} C, T_{2} = 60^{\circ} C, t = 5 minutes \\ ∴ \frac{70 - 60}{5} = K (\frac{70 + 60}{2} - T_{s}) \\ \frac{10}{5} = K (65 - T_{s}) \end{matrix}$
$\begin{matrix} T_{1} = 60^{\circ} C, T_{2} = 54^{\circ} C, t = 5 minutes \\ ∴ \frac{60 - 54}{5} = K (\frac{60 + 54}{2} - T_{s}) \\ \frac{6}{5} = K (57 - T_{s}) \end{matrix}$
Divide eqn. (i) by eqn. (ii), we get
$\frac{5}{3} = \frac{65 - T_{s}}{57 - T_{s}}$
$\begin{array}{l} 285 - 5 T_{s} = 195 - 3 T_{s} \\ 2 T_{s} = 90 or T_{s} = 45^{\circ} C \end{array}$

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