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Question
A hot body placed in a surrounding of temperature θ0 obeys Newton's law of cooling `(d theta)/(dt) = -K(theta - theta_0)` . Its temperature at t = 0 is θ1. The specific heat capacity of the body is sand its mass is m. Find (a) the maximum heat that the body can lose and (b) the time starting from t = 0 in which it will lose 90% of this maximum heat.
Solution
According to Newton's law of cooling,
`(d theta)/(dt) = -K (theta - theta_0)`
(a) Maximum heat that the body can lose, ΔQmax = ms (θ1 - θ0)
(b) If the body loses 90% of the maximum heat, then the fall in temperature will be θ.
`ΔQ_maxxx90/100 = ms (theta_1 - theta)`
⇒ `ms (theta_1 - theta_0)xx9/10 = ms (theta_1 - theta)`
⇒ θ = θ1 - (θ1-θ0) × `9/10`
⇒ θ = `(theta_1 - 9theta_0)/10`..............(i)
From Newton's law of cooling,
`(d theta)/(dt) = -K(theta_1 - theta)`
Integrating this equation within the proper limit, we get
At time t = 0,
θ = θ1
At time t,
θ = θ
`int_ {θ 1}^θ (dθ)/(θ _1 - θ ) = -K int_0^t dt`
`rArr In (theta_1 - theta)/(theta_1 - theta_0) = -kt`
`⇒ θ_1 - θ= θ_1 - θ_0e^-"kt"`...........(ii)
From (i) and (ii),
`(θ_1 - 9θ_0)/10 - θ_0 = (θ_1 - θ_0)e^-kt`
`⇒ t =(In (10))/k`
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