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Thermal Properties Of Matter

Question

A body initially at 80°C cools to 64°C in 5 min and to 52°C in 10 min. The temperature of the surrounding is

26^oC

16^oC

36^oC

40^oC

Solution

16^oC

According to Newton's law of cooling

$\frac{θ_{1} - θ_{2}}{t} = K [\frac{θ_{1} + θ_{2}}{2} - θ_{0}]$

In the first case,

$\frac{80 - 64}{5}$ = K $[\frac{80 + 64}{2} - θ_{0}]$

⇒ 3.2 = K [ 72 $-$ θ₀ ] ....(i)

In the second case,

$\frac{64 - 52}{5} = K [\frac{64 + 52}{2} - θ_{0}]$

⇒ 2.4 = K [ 58 $-$ θ₀ ] ......(ii)

Dividing (i) by (ii),

$\frac{3.2}{2.4} = \frac{72 - θ_{0}}{58 - θ_{0}}$

⇒ 185.6 $-$ 3.2 θ₀ = 172.8 $-$ 2.4 θ₀

⇒ 185.6 $-$ 172.8 = ( 3.2 $-$ 2.4 ) θ₀

⇒ θ₀ = $\frac{185.6 - 172.8}{3.2 - 2.4}$

⇒ θ₀ = 16⁰C