A hot body placed in air is cooled down according to newton's law of cooling the rate of decrease of temperature being k times the temperature difference from the surrounnding Starting from `t =0` The time in which the body will lose half of the maximum heat is `(xLn2)/(2k)` Find the value of x .

Answer

2

Explanation

We have `- (dT)/(dt) =k (T -T_(0))`
where `T_(0)` is the temperature of the surrounding If `T_(1)` is the initial temperature and `T` is the temperature at any time t then `underset(T_(1))overset(T)int(dT)/((T-T_(0)))=-kunderset(0)overset(t0)intdt`
`or |"in"(T-T_(0))|_(T)^(T)=-ktorIn[(T-T_(0))/(T_(1)-T_(0))]=-kt`
or `T =T_(0) + (T_(1) -T_(0))e^(-kt)`
The body continues to lose heat till its temperature becomes equal to that of the surrounding The loss of heat `Q = mc(T_(1) -T_(0))`
If the body lose half of the maximum loss that it can then decrease in temp `(Q)/(2) =mc ((T_(1)-T_(0))/(2))`
If body loses this heat in time t then its temperature at time t' will be `((T-T_(0))/(2))=(T_(1)+T_(0))/(2)`
Putting these valuse in Eq (i) we have
`(T_(1)-T_(0))/(2)=T_(0)+(T_(1)-T_(0))^(-kt)`
or `(T_(1)-T_(0))/(2)=(T_(1)-T_(0))e^(-kt)ore^(-kt)=(1)/(2)ort'=("In2")/(k)` .