Question

Efficiency of Carnot engine is 100% if
A. ${{T}_{2}}=273K$
B. ${{T}_{2}}=0K$
C. ${{T}_{1}}=273K$
D. ${{T}_{1}}=0K$

Answer 1

Hint: We know that efficiencies of a carnot engine is ratio of total worddone per cycle to the amount heat absorbed per cycle from the source. The carnot engine is the only engine than can give us a maximum efficiency upto 1 or 100%. In this problem we will use the mathematical formula to calculate the temperature at which the efficiency of Carnot engine will be 100%.

Formula used:
Efficiency of heat engines,
$\eta =1-\frac{{{T}_{2}}}{{{T}_{1}}}$
Where temperature used is in kelvin scale.

Complete answer:
Efficiency of a heat engine is the ratio of work done to the heat absorbed to do that work. The second law of thermodynamics says that the efficiency of a heat engine cannot be 100%. It is because absolute zero temperature is unattainable. The first expression of the second law of thermodynamics says that heat transfer occurs spontaneously from higher to lower temperature region and heat transfer cannot be done from lower temperature region to higher temperature region.

Now let us calculate the temperature value for 100% efficiency. Given that,
The efficiency of the heat engine=$\eta= 100 \%=1$

Using the formula of efficiency of carnot engine, we have:
$\eta =1-\dfrac{{{T}_{2}}}{{{T}_{1}}}$
$1=1-\dfrac{{{T}_{2}}}{{{T}_{1}}}$
$\dfrac{{{T}_{2}}}{{{T}_{1}}}=0$
Therefore, the efficiency of the carnot engine will be 100% if ${T_2}=0 k$.

Therefore, the answer is option (B).

Additional Information:
Heat engines have two reservoirs- hot reservoir and a cold reservoir. Heat engines take heat from hot reservoirs and do some work and remaining heat is transferred to cold reservoirs. This hot reservoir is called source and cold reservoir is called sink.

Note: We can note that the efficiency of the carnot engine will be 100% if ${T_1} =\infty$ also. We have to be careful while choosing the options. In some questions even ${T_1} =\infty$ option will be there, in such case for two different values we can conclude that the efficiency of carnot engine is 100%. It should be remembered that the efficiency of a heat engine cannot be 100%. Carnot engine is the engine having maximum efficiency which is 100%. The Carnot engine is an ideal engine.