A Carnot engine operates between temperatures of $ 600 \, \text{K} $ and $ 300 \, \text{K} $. If it absorbs $ 900 \, \text{J} $ of heat from the source, how much work does it perform?

Question

A Carnot engine operating between two reservoirs has efficiency $\frac{1}{3}$ When the temperature of cold reservoir raised by $x$, its efficiency decreases to $\frac{1}{6}$ The value of $x$, if the temperature of hot reservoir is $99^{\circ} C$, will be :

Answer 1

A Carnot engine functions between a hot reservoir at $ T_H = 600 \, \text{K} $ and a cold reservoir at $ T_C = 300 \, \text{K} $. It takes in $ Q_H = 900 \, \text{J} $ of heat from the hot reservoir. The objective is to calculate the work $ W $ output by the engine.

The efficiency $ \eta $ for a Carnot engine is defined by the equation:

$\eta = 1 - \frac{T_C}{T_H}$

Plugging in the provided temperatures yields:

$\eta = 1 - \frac{300}{600} = 1 - 0.5 = 0.5$

The efficiency can also be represented as the ratio of work done to heat absorbed:

$\eta = \frac{W}{Q_H}$

Solving for work $ W $:

$W = \eta \times Q_H$

Substituting the known values:

$W = 0.5 \times 900 \, \text{J} = 450 \, \text{J}$

Consequently, the work performed by the Carnot engine is 450 J, aligning with the second option previously indicated.

A Carnot engine operates between temperatures of $ 600 \, \text{K} $ and $ 300 \, \text{K} $. If it absorbs $ 900 \, \text{J} $ of heat from the source, how much work does it perform?

Show Hint

The Correct Option is A

Solution and Explanation

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