A solid sphere of density 'ρ' specific heat capacity ‘C’ and radius ‘R’ is at an initial temperature 300 K. On allowing it to cool under the process of radiation then the time taken for it to cool to 100 K is found to be 26RρC81Kσ, micro second. If 'σ' is Stefan’s constant, then find the value of ‘K’. Assume the surrounding temperature to be extremely small and emissivity of the outer surface of the sphere to be unity

From Stefan’s law−1AdQdt=σT4 −mcdTdt=σAT4 ∫300 K100 KT−4dT=−σAmc∫01dt 1(100)3−1(300)3=σ4πR243πR3ρCt ⇒t=23243RρCσ micro sec

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A solid sphere of density $' ρ'$ specific heat capacity ‘C’ and radius ‘R’ is at an initial temperature 300 K. On allowing it to cool under the process of radiation then the time taken for it to cool to 100 K is found to be $\frac{26 R ρ C}{81 K σ},$ micro second. If $' σ'$ is Stefan’s constant, then find the value of ‘K’. Assume the surrounding temperature to be extremely small and emissivity of the outer surface of the sphere to be unity

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Detailed Solution

From Stefan’s law
$- \frac{1}{A} \frac{d Q}{d t} = σ T^{4} - m c \frac{d T}{d t} = σ A T^{4} \int_{300 K}^{100 K} T^{- 4} d T = \frac{- σ A}{m c} \int_{0}^{1} d t \frac{1}{{(100)}^{3}} - \frac{1}{{(300)}^{3}} = \frac{σ 4 π R^{2}}{\frac{4}{3} π R^{3} ρ C} t \Rightarrow t = \frac{23}{243} \frac{R ρ C}{σ} m i c r o s e c$