Question

In a surrounding medium of temperature of ${10^ \circ }C$ , a body takes 7 min for a fall of temperature from ${60^ \circ }C$ to ${40^ \circ }C$ .In what time the temperature of the body will fall from ${40^ \circ }C$ to $28^\circ C$ .
A) 7 min
B) 11min
C) 14 min
D) 21 min

Answer 1

we have to use newton’s law of cooling i.e $\dfrac{{d\theta }}{t} = - k\left( {\dfrac{{{\theta _1} + {\theta _2}}}{2} - {\theta _\circ }} \right)$ & the with the given data we first determine the unknown value of $k$= constant . Then using this formula again in the given condition we determine the value of $t$.

Formula used:- $\dfrac{{d\theta }}{t} = - k\left( {\dfrac{{{\theta _1} + {\theta _2}}}{2} - {\theta _\circ }} \right)$
$d\theta$= temperature change of the body
$k$= constant
${\theta _1}$= initial body temperature
${\theta _2}$= final body temperature
${\theta _\circ }$= temperature of surrounding

Complete step by step solution:-
Given that , a body takes 7 min for a fall of temperature from ${60 \circ }C$ to ${40\circ }C$
$d\theta$= temperature change of the body = 40 – 60 = $20\circ C$
${\theta _1}$= initial body temperature = ${60\circ }C$
${\theta _2}$= final body temperature = ${40\circ }C$
${\theta _\circ }$= temperature of surrounding = ${10\circ }C$
Putting the values in the formula we have to find the value of $k$.
$\dfrac{{ - 20}}{7} = - k\left( {\dfrac{{60 + 40}}{2} - 10} \right)$
Now further simplifying the equation we get
$\dfrac{{ - 20}}{7} = - k(50 - 10)$
Sending $k$to left hand side we get
$k = \dfrac{{20}}{{7(40)}}$
Further solving we get the value of constant ($k$)obtained here is
$k = \dfrac{1}{{14}}$.
Also , we have to find in what time the temperature of the body will fall from ${40\circ }C$ to $28\circ C$.
We have given that
$d\theta$= temperature change of the body= $28 - 40 = - 12\circ C$
${\theta _1}$= initial body temperature = ${40 \circ }C$
${\theta _2}$= final body temperature = $28\circ C$
${\theta _\circ }$= temperature of surrounding = ${10\circ }C$
$k$= constant = ($k = \dfrac{1}{{14}}$)
We have to find the value of time ($t$)
Use the formula $\dfrac{{d\theta }}{t} = - k\left( {\dfrac{{{\theta _1} + {\theta _2}}}{2} - {\theta _\circ }} \right)$
Putting all the value in the formula , we get
$\dfrac{{ - 12}}{t} = - \dfrac{1}{{14}}\left( {\dfrac{{40 + 28}}{2} - 10} \right)$
$\dfrac{{ - 12}}{t} = - \dfrac{1}{{14}}(34 - 10)$
Sending ($t$) to the left hand side
$t = \dfrac{{12 \times 14}}{{(24)}}$
Further solving we get the value of ($t$)
$t = 7\min$

Hence option (A) is correct .

Note :- There are three modes of transferring the heat
Conduction – this transfer of heat takes place due to molecular collisions and the process is heat conduction.
Convection – In convection heat is transferred from one place to another by the actual motion of heated material.
Radiation – The radiation process does not need any material medium for heat transfer . Energy is emitted by a body and this energy travels in space just as the light.