#### Question

A hot body placed in a surrounding of temperature θ_{0} obeys Newton's law of cooling `(d theta)/(dt) = -K(theta - theta_0)` . Its temperature at *t* = 0 is θ_{1}. The specific heat capacity of the body is *s*and 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, ΔQ_{max} = ms (θ_{1} - θ_{0})

(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 = θ_{1} - (θ_{1}-θ_{0}) × `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 = θ_{1}

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`