Animated Solution for Physics - Thermodynamics: Even Carnot engine cannot give $100\%$ efficiency, because we cannot

Imagine you are an engineer during the height of the Industrial Revolution. Steam engines are powering trains, factories, and ships, but they are incredibly inefficient. They consume mountains of coal while producing relatively little useful work. You might ask yourself: Is there a limit to how efficient an engine can be? Can we build a perfect engine that converts $100\%$ of its heat into work?

This exact question was pondered by a brilliant French physicist named Sadi Carnot in 1824. To answer it, he didn't look at the messy reality of friction, leaking steam, or imperfect materials. Instead, he imagined an idealized, theoretical engine—what we now call the Carnot Engine.

To understand Carnot's genius, we must first understand the basic anatomy of any heat engine. Every heat engine operates between two thermal reservoirs:

1. The Source: A hot reservoir at a high temperature $T_1$. The engine extracts a certain amount of heat, $Q_1$, from this source. 2. The Sink: A cold reservoir at a lower temperature $T_2$. The engine rejects the leftover heat, $Q_2$, into this sink.

The difference between the heat taken in and the heat rejected is the useful work done by the engine:

Efficiency, denoted by the Greek letter $\eta$ (eta), is simply a measure of what you get out compared to what you put in. Mathematically, it is the ratio of the useful work done to the total heat absorbed:

Carnot proved a profound theorem: for a perfectly reversible, ideal engine, the ratio of the heat exchanged is exactly equal to the ratio of the absolute temperatures of the reservoirs. That is, $\frac{Q_2}{Q_1}=\frac{T_2}{T_1}$.

Substituting this into our efficiency equation gives us the famous Carnot efficiency formula:

This elegant equation tells us that the maximum possible efficiency of an engine depends only on the temperatures of the hot source and the cold sink. It doesn't matter if you use steam, air, or a magical futuristic gas; you cannot beat this limit.

Now, let's return to our original dream: a perfect engine with $100\%$ efficiency. In decimal form, $100\%$ efficiency means $\eta =1$. Let's plug this into our formula and see what the universe demands of us:

If we subtract $1$ from both sides, we get:

For this fraction to be zero, there are only two mathematical possibilities. Either the denominator $T_1$ must be infinitely large ($T_1\to \infty$), or the numerator $T_2$ must be exactly zero ($T_2=0$).

Creating a source with an infinite temperature is physically impossible; it would require an infinite amount of energy. Therefore, the only remaining option is that the sink temperature, $T_2$, must be exactly $0\text{ K}$.

Here is where the dream of the perfect engine shatters against the cold, hard reality of physics. $0\text{ K}$ is Absolute Zero, the theoretical temperature where all atomic motion ceases.

According to the Third Law of Thermodynamics, it is impossible to cool any system to absolute zero in a finite number of steps. Absolute zero is an asymptotic limit—you can get incredibly close to it (scientists have reached fractions of a billionth of a Kelvin), but you can never actually reach it.

Because we can never possess a thermal sink at exactly $0\text{ K}$, the term $\frac{T_2}{T_1}$ will always be a positive number greater than zero. Consequently, the efficiency $\eta =1-\frac{T_2}{T_1}$ will always be strictly less than $1$.

This mathematical conclusion is the physical embodiment of the Second Law of Thermodynamics, specifically the Kelvin-Planck statement: It is impossible to construct an engine which will work in a complete cycle and produce no other effect except the raising of a weight and the cooling of a heat reservoir.

In simpler terms: You cannot convert all absorbed heat into work. Some heat must inevitably be rejected to a colder body. The universe demands a thermal exhaust.

So, the next time you hear a car engine rumbling or see the cooling towers of a power plant, remember Sadi Carnot. That wasted heat isn't just a design flaw; it is a fundamental tax levied by the universe itself, a reminder that perfection is mathematically and physically forbidden.