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

Hint: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.