A Carnot engine takes 3 * 10^6 cal of heat from a reservoir at 627°C and gives it to a sink at 27°C.

Q: Why must temperature be in Kelvin?

Thermodynamic formulas like Carnot's theorem are derived using the absolute temperature scale. Using Celsius will give completely wrong ratios.

Q: Can we use the efficiency formula directly?

Absolutely! You can find efficiency $\eta = 1 - T_2/T_1$ and then use $W = \eta Q_1$. It's often faster!

Q: Why is $W = Q_1 - Q_2$?

This is simply the First Law of Thermodynamics (Conservation of Energy). The energy entering the engine must equal the energy leaving it.

Q: What if the options were in calories?

Then $2 \times 10^6$ would be the correct answer. Always keep an eye on the units of the options before starting your calculation.

Q: Is the conversion factor $4.2$ exact?

It's an approximation for the mechanical equivalent of heat ($1\text{ cal} \approx 4.184\text{ J}$). In JEE, using $4.2$ is standard unless specified otherwise.

Question

A Carnot engine takes $3 	imes 10^6$ cal of heat from a reservoir at $627^\circ	ext{C}$ and gives it to a sink at $27^\circ	ext{C}$. The work done by the engine is

Options:

(A) $4.2 	imes 10^6$ J

(B) $8.4 	imes 10^6$ J

(C) $16.8 	imes 10^6$ J

(D) zero

Sigma Percentile · Accepted Answer

### The Anatomy of a Carnot Engine

Imagine a majestic, perfectly reversible machine—the Carnot Engine. It operates between two thermal reservoirs: a blazing hot source and a chilling cold sink. The engine's job is simple yet profound: it drinks in heat energy ($Q_1$) from the source, transforms a portion of it into useful mechanical work ($W$), and dumps the leftover, unusable heat ($Q_2$) into the sink.

This flow of energy is governed by the strict rules of thermodynamics, ensuring that energy is always conserved. But before we can crunch the numbers, we must prepare our data.

### The Absolute Temperature Trap

The most common pitfall in thermodynamics is forgetting to convert temperatures to the absolute scale (Kelvin). The laws of thermodynamics, especially Carnot's theorem, are built on the foundation of absolute zero. Using Celsius will lead to disastrously wrong ratios.

Let's secure our temperatures:
Source Temperature: $T_1 = 627^\circ	ext{C} + 273 = 900	ext{ K}$
Sink Temperature: $T_2 = 27^\circ	ext{C} + 273 = 300	ext{ K}$

### The Flow of Energy

Carnot's theorem gives us a beautiful, elegant relationship for a reversible engine: the ratio of heat exchanged is exactly equal to the ratio of their absolute temperatures.

$$\frac{Q_1}{T_1} = \frac{Q_2}{T_2}$$

We know the engine absorbs $Q_1 = 3 	imes 10^6	ext{ cal}$. Let's find out how much heat it rejects ($Q_2$):

$$Q_2 = Q_1 	imes \frac{T_2}{T_1}$$
$$Q_2 = (3 	imes 10^6) 	imes \frac{300}{900}$$
$$Q_2 = (3 	imes 10^6) 	imes \frac{1}{3} = 1 	imes 10^6	ext{ cal}$$

Now, by the First Law of Thermodynamics (Conservation of Energy), the work done is simply the difference between the heat taken in and the heat thrown away:

$$W = Q_1 - Q_2$$
$$W = 3 	imes 10^6 - 1 	imes 10^6 = 2 	imes 10^6	ext{ cal}$$

### The Final Catch: Units Matter

We have our work done: $2 	imes 10^6	ext{ cal}$. But wait! A quick glance at the options reveals they are all in Joules (J). This is a classic examiner's trap.

To convert calories to Joules, we multiply by the mechanical equivalent of heat, which is approximately $4.2	ext{ J/cal}$.

$$W = 2 	imes 10^6 	imes 4.2	ext{ J}$$
$$W = 8.4 	imes 10^6	ext{ J}$$

And there we have it! The engine performs $8.4 	imes 10^6	ext{ J}$ of work.

**Bonus Insight:** You could also solve this using the efficiency formula $\eta = 1 - \frac{T_2}{T_1}$. The efficiency is $1 - \frac{300}{900} = \frac{2}{3}$. Since $W = \eta Q_1$, the work done is $\frac{2}{3} 	imes 3 	imes 10^6 = 2 	imes 10^6	ext{ cal}$. Both paths lead to the same beautiful truth!

Animated Solution for Physics - Thermodynamics: A Carnot engine takes $3 \times 1 0^{6}$ cal of heat from a reservoir at $62 7^{\circ} C$ and gives it to a sink at $2 7^{\circ} C$ . The work done by the engine is

Select Answer:

Visualized Solution

Visualizing the Carnot Engine

Absolute Temperatures

Carnot's Theorem

Substituting Values

Calculating Rejected Heat

First Law of Thermodynamics

Calculating Work Done

Unit Conversion

The Way Forward

The Sigma Insight: Heat Engines and Refrigerators

The Anatomy of a Carnot Engine

The Absolute Temperature Trap

The Flow of Energy

The Final Catch

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