The second law of thermodynamics is the most interesting and yet least understood of all. It helps us understand why does any change occur in the universe? The discussion on the second law of thermodynamics started in19^{th} century—steam engines were a hot subject in those days.

There are generally three main parts of a steam engine:

- A hot source—to extract heat.
- A piston or a turbine that converts heat into work.
- A cold sink—to dump the waste heat.

The cold sink, however, is not part of an actual physical design. It can be difficult to figure out—or maybe the immediate surrounding of the system. Let us dive back into history.

**Evolution of the Second Law of Thermodynamics**

It is the early 19^{th} century. The Britishers are using steam to drive their factories and pump out water from the wells. It is a hot topic—how to increase the efficiency of steam engines?

The great minds say—what if we change the working fluid—may be air or some other gas. Then comes a French engineer Saadi Carnot. This guy has some opposite views (though he was initially convinced of the prevalent thought).

He then comes up with the astonishing idea—”the efficiency of a perfect steam engine does not depend on the working fluid”. Carnot says, “the efficiency is totally dependent on the temperature difference of the source and the sink”.

To support his idea with mathematical interpretation he presents the following formula:

Efficiency = 1 – (T_{sink} / T_{source})

where:

T_{sink} = absolute temperature of the cold reservoir to dump waste heat.

T_{source} = absolute temperature of the hot reservoir to extract heat.

By this, he lays the foundation of his famous **“Carnot’s Theorem”**. The genius says, Ok guys! you do whatever you can but this formula gives you the maximum theoretical efficiency of a perfect heat engine. You make any changes to your final design of the engine but you would never be able to go beyond the efficiency—this formula gives you.

**Interpretation of Carnot’s Theorem**

Let us take an example to understand—what was Carnot actually meant to say?

Suppose we provide a steam turbine with superheated steam at 350 ˚C (623.15 K) to make electricity. It spreads the waste heat to its surrounding at 30 ˚C (303.15 K) after extracting the useful work.

If we substitute these values into Carnot’s formula, we get the value of maximum theoretical efficiency of our system under the same parameters. Let us do so:

Efficiency = 1- (303.15/623.15) = 0.51

So the efficiency is 51%. What does this tell us? It says, no matter how perfect your system’s design is you can’t go beyond the 51% efficiency. To increase the efficiency you would either have to increase your source temperature or decrease the temperature of the surrounding.

Another important consequence of Carnot’s theorem is that you can never achieve 100% efficiency out of a heat engine. To achieve 100% efficiency, the term T_{sink}/T_{source} must be equal to 0. To get (T_{sink}/T_{source})= 0.

Either T_{sink} should be at absolute zero (T_{sink} = 0), or T_{source} should be infinitely hot (T_{source} = ∞). Both of the situations are not possible practically. Yes, Carnot was a genius.

**Kelvin’s and Clausius’s statements on the second law**

Later in the 19^{th }century, Carnot’s theorem served a great purpose in reviving the discussion on heat. Then came two more giants of physics.

**Lord Kelvin**

One of them was Lord Kelvin. His query was, what should be an essential or truly important structure of a heat engine? He considered a cold sink—though difficult to acknowledge—was still inevitable. His statement of the second law of thermodynamics was:

“In a cyclic process, it is impossible to extract heat from a source and completely convert it into useful work”.

There has to be a certain amount of heat energy—that we must dump into a cold sink. Without doing so we can’t drive an engine.

**Rudolph Clausius**

The second giant was Rudolph Clausius. He observed the obvious fact that heat only travels from a hot body to a cold body on its own. But it doesn’t do the reverse on its own.

His statement of the second law of thermodynamics was:

“To transfer heat from a cold body to a hot body, there has to be some change occurring somewhere else”. In other words, we have to do work.

The daily example of his statement is a refrigerator. Cooling in refrigerators accompanies the transfer of heat from a cold object to a warmer surrounding. But to drive a refrigerator we need electricity. We generate this electricity in power stations (located somewhere else) by doing work.

Both Kelvin’s and Clausius’s statements are logically equivalent. If we falsify one—it results in falsifying the other.

**Entropy**

To capture both the statements discussed above, we needed a property. We call it entropy. The entropy gives us an alternative definition of the second law of thermodynamics and is logically convincing. Mathematically, entropy is:

S= heat supplied reversibly (in a small amount of temperature difference between system and surrounding)/ Temperature

We can say that entropy is randomness or disorder. If the molecules are disordered as in gases their entropy would be high. Perfectly crystalline solids have comparatively low entropy. If we add heat to a substance, the randomness in the molecules increases and so does the entropy. Conversely, if we remove heat from an object its entropy decreases.

The second law’s interpretation of entropy is, “The entropy of the universe is always increasing”. The universe is an important consideration here. The universe is the sum of both the system and its surroundings.

The first law tells us that a process is feasible only if the total energy of the universe doesn’t change.

However, the second law identifies that a feasible process is spontaneous only if the total entropy of the universe increases.

**Explanation with respect to Kelvin and Clausius’s statements**

According to the Kelvin statement, when we extract heat from a hot source its entropy decreases as given by the equation:

△S = △Q/ T

This decrease is small since the source is at a very high temperature. After converting that heat into some useful work—we dump the waste heat into the sink. Its entropy increases and this increase in entropy is greater than the entropy decrease of the source. Because the sink is at a very low temperature.

Hence the overall result is an increase in the entropy of the universe. Therefore, the process is spontaneous.

Now if we consider the Clausius statement, it says: When we extract heat from a cold object, its entropy decreases. And transferring that heat to a hotter object increases its entropy. The overall result is a decrease in the entropy of the universe. Because the entropy decrease in the cold object is greater due to its low temperature.

Therefore, by the second law’s definition of entropy, the process is not spontaneous. That is why to run a refrigerator, the power stations have to increase the entropy of the universe.

**How does entropy affect the efficiency of a heat engine?**

The efficiency of a heat engine is given by Carnot’s expression. It says efficiency is greater when the source is at a higher temperature and the sink is at a lower temperature. We had understood it with an example already.

But what does it actually mean in terms of entropy?

Suppose the object is at a very high temperature. When we extract heat from the source, it causes a very low reduction in its entropy.

To compensate for the low reduction of entropy in the source, a low entropy generates in the cold sink. Therefore, more useful energy is available to do the work.

**Microscopical view of entropy**

We have discussed above that according to the second law, “Entropy is increasing”. Let us observe this in another way to understand it deeply. The most common definition of entropy is:

“It is the measure of the disorder of a system”.

This definition does not give us a clear perspective of entropy. So why not dive a bit deeper? Consider a container with gas, isolated on one side with a wall.

What happens if we remove the wall? The gas will expand into the full volume of the container. The entropy of the gas gets increased since it gets more disordered. The reason for this disorder is that as the volume increases, there are more places for the individual molecules to go.

The number of arrangements for the gas molecules increases as the volume increases. This greater probability into which the gas molecules can arrange themselves is entropy.

Ludwig Boltzmann gave this concept. His definition of entropy was: “Entropy is the way of counting the number of arrangements of atoms/molecules inside a system”. His famous formula **S= k log(W)** is even craved on the stone of his grave. This guy was also a genius.

**What if we push the gas to its initial position?**

Pushing the gas with the help of a piston to its initial position won’t reduce its entropy. Because compression adds heat and heat increases entropy. Now the question is why does adding heat/temperature increase entropy?

The answer is that the temperature increases due to the increase in K.E of the molecules. As a result, the molecules start moving faster. This abrupt motion increases entropy because the number of ways to describe the gas also increases.

Another question is, can the gas restore its initial position spontaneously?

This can’t happen because it would result in a decrease in entropy. Here is an important consideration:

If the gas restores to its initial position, the laws of physics don’t prevent it from doing so. But it is very unlikely to happen. The probability that all these molecules arrange themselves back to the initial state is 1 in 10^{150,000,000,000,000,000,000,000}. This number is so large that it is statistically impossible to happen.

Plancks volume is the smallest possible volume to exist. The scientists say that only 10^{183} Plancks volumes could fit into the known universe. If we compare this number to the one discussed above, we would be able to understand why the gas doesn’t restore to its initial position.

The law of entropy is not an absolute law. It is rather a statistical law. This means it can be violated, but the probability of happening so is extremely low.

**Maxwell’s Demon**

A Scottish Physicist James Maxwell 1867 presented a thought experiment. In this experiment, he reasoned that the second law of thermodynamics can be violated. This seemed so plausible that scientists were unable to reject it for 100 years. The experiment says:

Consider a container which has two chambers divided by a wall. We fill gas into one of the chambers. There is a door which lets the gas expand into both chambers. The door is frictionless, so adds no heat to the system.

After the gas has expanded. Suppose there is a demon which controls the door in a way that only allows molecules to go into one chamber. If a molecule comes from the left he opens the door to let it go towards the right. But he doesn’t allow the molecules on the right to travel towards the left.

At one point in time, all the molecules would be in the right chamber (same as they were initially). We didn’t add any work and the temperature also remained the same. This means that the entropy has decreased and the second law of thermodynamics is violated.

This seems to be a thought experiment. But now, with the concept of nanorobots and microprocessors, this could be conceived in the near future. Scientists couldn’t reject this idea until 1982.

**Entropy as a measure of information**

In 1982, Rolf Landauer and Charles Bennett, from IBM research, presented the solution to Maxwell’s thought experiment. According to them, for the demon to reduce the entropy of the system, he must keep the information about various molecules.

He increases his record of information as, when to close the door and when to open it. Thus he increases the amount of information for that system. Because he is part of the system. This increase in information leads to an increase in entropy.

One could say that for a robot we can erase this information from the hard drive. But again the two greats showed that erasing the information from the hard drive would generate heat. Though this heat is very little but big enough to increase the overall entropy of the universe.

This explanation gives another perspective to the concept of entropy. Which says, entropy is the measure of that information which helps in expressing the state of a system.

**How time is related to entropy and the second law of thermodynamics?**

The laws of physics seem to be symmetric with time, i.e. if time goes backwards these laws don’t get violated. The only law that would be violated if time reverses its direction is the second law of thermodynamics, the entropy.

Entropy always goes forward, i.e. it gets increased. The British scientist Arthur Eddington assumed that due to the increase in entropy, time flows.

But the question is, does the time flow forward due to the increase in entropy? This would be a wrong assumption. Because it suggests that the time would flow backwards if entropy decreases. Let’s take an example.

In our refrigerators where things get cooled, the entropy inside the refrigerator decreases. But does the time flow backwards inside our refrigerators? No, it doesn’t. Similarly, at night, the entropy of our earth decreases, but time doesn’t flow backwards. Then the question is, what causes the time to flow forward?

To answer this, we must first answer why does entropy increase?

The scientific laws don’t force entropy to increase. Since entropy is a statistical law, therefore, there are more possible arrangements for the microstates of a system in future than in the past.

For example, in a checkerboard, the checker pieces can arrange themselves in more ways in future (when they get disturbed) as compared to the start of the game. As entropy requires much information to describe its microstates, therefore, it is a statistical phenomenon.

Thus we can assume that time is also a statistical phenomenon. It can go backwards, but it is statistically impossible. For more information, you can read, “Entropy as an arrow of time“.