Carnot Cycle

Step 1: Recall the formula for the efficiency of a Carnot engine.
The efficiency ( ) of a Carnot engine operating between a hot reservoir at temperature and a cold reservoir at temperature is given by:

where and are absolute temperatures (in Kelvin).

Step 2: Understand the given conditions.
In this problem, the cold reservoir temperature is kept constant, while the hot reservoir temperature is varied. We need to determine the graphical representation of the efficiency ( ) as a function of the hot reservoir temperature ( ).

Step 3: Analyze the behavior of efficiency as changes.
- When :

So, when the hot reservoir temperature is equal to the cold reservoir temperature, the efficiency is zero.
- As increases ( ):
As increases, the ratio decreases because is constant and positive.
Since , as decreases, the efficiency increases.
- As :
As approaches infinity, the term approaches 0.
Thus,
The efficiency approaches 1 (or 100%) as the hot reservoir temperature becomes infinitely large, but it can never reach 1 for finite .

Step 4: Determine the shape of the graph.
To see how the efficiency increases with , let's consider the rate of change of efficiency with respect to .

Since and , .
This confirms that efficiency increases as increases.
The second derivative is:

Since and , .
This means the rate of increase of efficiency decreases as increases.
The curve is concave down.

Step 5: Match the analysis with the given graphs.
- Option (A) and (D) are incorrect because they show efficiency decreasing as increases beyond .
- Option (C) is incorrect because it shows a linear increase in efficiency, whereas the relationship is not linear and should be concave down.
- Option (B) starts from zero efficiency at , increases as increases, and the rate of increase decreases as increases (concave down shape), approaching a maximum value. This matches our analysis.

Final Answer:
The correct graphical representation of the relationship between efficiency and is Option (B)

Given:

• The engine uses 1 mole of an ideal gas.
• Heat supplied to the gas: J

Step 1: Understanding Work Done

The area enclosed by the P-V diagram represents the work done by the engine:

For a Carnot engine, efficiency is given by:

From the problem, we use the given heat supplied and known thermodynamic properties to determine the work done.

Step 2: Calculation

Using the given data:

Answer: The correct option is B (3000 J).