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Question: Temperature of a piece of metal is increased from \(27^\circ \,C\) to \(327^\circ \,C\). the rate of...

Temperature of a piece of metal is increased from 27∘ C27^\circ \,C to 327∘ C327^\circ \,C. the rate of heat of radiation by the metal will become
A. Double
B. Four times
C. Eight times
D. Sixteen times

Explanation

Solution

It is given in the question that a metal is radiating heat when the temperature will increase from 27∘ C27^\circ \,C to 327∘ C327^\circ \,C . Therefore, we will use Stefan Boltzmann law to find the rate of heat radiated by a metal. Stefan-Boltzmann law states that the total energy emitted by a black body per unit surface area is directly proportional to the fourth power of the temperature of a black body.

Complete step by step answer:
It is given in the question that the temperature of the metal is increased from 27∘ C27^\circ \,C to 327∘ C327^\circ \,C. Now, changing the temperature from Celsius into kelvin.
Therefore, the initial temperature = 27∘C =27+273 =300 K = \,27^\circ C\, = 27 + 273\, = 300\,K
Also, final temperature = 327∘C = 327+273= 600K = \,327^\circ C\, = \,327 + 273 = \,600K

Now, the heat radiated by the metal can be calculated by using Stefan Boltzmann law. Stefan-Boltzmann law states that the total energy emitted by a black body per unit surface area is directly proportional to the fourth power of the temperature of a black body is given by,
ε=σT4\varepsilon = \sigma {T^4}
Where, σ\sigma is the constant of proportionality.
Now, ε=PA\varepsilon = \dfrac{P}{A}
Putting this value in the above equation, we get
P=AσT4P = A\sigma {T^4}
From here, we can say that, P∝ T4P \propto \,{T^4}

Now, taking the ratio of the heat radiated by the metal during the initial and final temperatures, we get
P1P2=(T1T2)4\dfrac{{{P_1}}}{{{P_2}}} = {\left( {\dfrac{{{T_1}}}{{{T_2}}}} \right)^4}
Now, putting the values of temperatures in the above equation, we get
P1P2=(300600)4\dfrac{{{P_1}}}{{{P_2}}} = {\left( {\dfrac{{300}}{{600}}} \right)^4}
⇒ P1P2=(12)4\Rightarrow \,\dfrac{{{P_1}}}{{{P_2}}} = {\left( {\dfrac{1}{2}} \right)^4}
⇒ P1P2=116\Rightarrow \,\dfrac{{{P_1}}}{{{P_2}}} = \dfrac{1}{{16}}
∴ P2=16P1\therefore \,{P_2} = 16{P_1}

Therefore, the rate of heat radiation by the metal will become sixteen times.Hence, option D is the correct option.

Note: Here, remember to change the temperature from Celsius to kelvin. We change Celsius to kelvin because when we will talk about 0K0K it represents zero kinetic energy or temperature. But the Celsius or Fahrenheit scale does not begin with zero. We use kelvin as a unit of temperature because it is directly related to the kinetic energy and the temperature.