First slide

Radiation

Question

A body initially at 60°C cools to 50°C in 10 min. What will be its temperature at the end of next 10 min. if the room temperature is 25°C. Assume Newton's law of cooling

Moderate

A

42.85°C
B

45°C
C

40.46°C
D

44.23°C

Solution

According to Newton s law of cooling $\frac{dT}{dt} = - k (T - T_{0})$
where T = temperature of body and T_o = temperature of surroundings
Now    $\int_{T_{1}}^{T_{2}} \frac{dT}{(T - T_{0})} = - k \int_{0}^{t} dt$
   ${[\log_{e} (T - T_{0})]}_{T_{1}}^{T_{2}} = - k [t]_{0}^{t}$
or    $\log_{e} (\frac{T_{2} - T_{0}}{T_{1} - T_{0}}) = - kt$
or $\log_{e} (\frac{60 - 25}{50 - 25}) = - k \times (10 \times 60) . . . (1)$
Let T' be the temperature at the end of 10 min. In this case,
$\log_{e} (\frac{50 - 25}{T^{'} - 25}) = - k \times (10 \times 60) . . . (2)$
Solving eqs. (1) and (2), we get
   $T^{'} = 42 \cdot 85^{\circ} C$

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