Animated Solution for Physics - Thermodynamics: A Carnot's engine working between $400\text{ K}$ and $800\text{ K}$ has a work output of $1200\text{ J}$ per cycle. The amount of heat energy supplied to the engine from the source in each cycle is

The Carnot Engine

Welcome to one of the most elegant and profound concepts in all of physics: the Carnot Engine. Imagine you are standing in front of a massive, roaring steam engine from the industrial revolution. It consumes burning coal, turns wheels, and exhausts hot steam into the cold air.

In the early 19th century, a brilliant French engineer named Sadi Carnot asked a fundamental question: What is the absolute maximum efficiency any heat engine can achieve? He discovered that no engine can be 100% efficient. Every engine must take heat from a hot "source," use some of it to do useful "work," and dump the rest into a cold "sink."

In our specific problem, we are given a theoretical Carnot engine operating between two distinct temperature reservoirs. The hot source is at $T_1=800\text{ K}$, and the cold sink is at $T_2=400\text{ K}$. We are also told that the engine performs a useful work output of $W=1200\text{ J}$ per cycle. Our mission is to find out exactly how much heat energy ($Q_1$) was supplied to the engine from the hot source to make this happen.

To solve this, we need to understand the concept of efficiency, denoted by the Greek letter $\eta$ (eta). In simple terms, efficiency is "what you get" divided by "what you pay for."

What do we get? We get useful work, $W$. What do we pay for? We pay for the heat supplied from the source, $Q_1$.

Therefore, the general formula for the efficiency of any heat engine is:

However, Sadi Carnot proved that for a perfectly reversible, ideal engine (the Carnot engine), the efficiency depends only on the absolute temperatures of the source and the sink. The Carnot efficiency is given by:

This is a beautiful and powerful equation. Notice that the temperatures must strictly be in Kelvin (the absolute temperature scale). If they were in Celsius, the ratio would be physically meaningless.

Since both expressions represent the efficiency of our Carnot engine, we can set them equal to each other. This creates our master equation for the problem:

This equation is the bridge between the thermal world (temperatures) and the mechanical world (heat and work). Let's take a breath and look at what we have. We know $T_1$, we know $T_2$, and we know $W$. The only unknown is $Q_1$, which is exactly what we are trying to find.

Let's carefully substitute our known values into the master equation.

Look at the fraction on the left side. The numbers are incredibly clean. $400$ divided by $800$ simplifies perfectly to $\frac{1}{2}$.

Subtracting $\frac{1}{2}$ from $1$ leaves us with $\frac{1}{2}$. This tells us a profound physical fact about our engine: its efficiency is exactly $50\%$. Half of the heat energy it absorbs is converted into useful work, and the other half is wasted.

Now, we are just one algebraic step away from the finish line. To isolate $Q_1$, we simply cross-multiply the equation.

And there we have it! The engine requires $2400\text{ Joules}$ of heat energy from the hot source in every single cycle to produce $1200\text{ Joules}$ of work.

Where Does the Rest Go?

Before we wrap up, let's push our understanding a little further. We supplied $2400\text{ J}$ of heat, but we only got $1200\text{ J}$ of work out. Where did the missing $1200\text{ J}$ go?

According to the First Law of Thermodynamics (the law of conservation of energy), energy cannot be destroyed. The heat supplied ($Q_1$) must equal the work done ($W$) plus the heat rejected to the cold sink ($Q_2$).

If we rearrange this to solve for the rejected heat, we get:

Exactly $1200\text{ Joules}$ of heat is dumped into the cold sink. This perfectly aligns with our earlier realization that the engine is $50\%$ efficient. Half the energy becomes work, and half becomes waste heat.

Understanding this balance of energy is what separates a good physics student from a great one. Keep visualizing the flow of energy, and these thermodynamics problems will become second nature to you!