What is Entropy and Enthalpy - It's Introduction, Mathematical Expression, Examples,

 

Entropy:

The first concept of entropy was given by Lazare Carnot. Lazare was very interested in the relation between work put into a system compared to work that comes out of it. He named this output useful work, while calling the lost work transformation energy. This would later be known as entropy. Sadi Carnot continued the work of his father studying engines by creating Carnot Cycle, a cycle which acts as the upper bound of efficiency for classical thermodynamic engines. Sadi realized that some energy must always be lost in its conversion from heat to work. This is another way of stating the second law of thermodynamics, which essentially says that entropy can never decrease.

A little over half a century later Lord Kelvin built further on the theories of Sadi Carnot. Imagine standing in front of a nineteenth-century steam engine. What will be its most essential component? The piston or the hot reservoir. The key component is the cold sink. It is the surroundings in which the waste heat is discarded. Take away the cold sink and the engine stops running even though plenty of fuel is available and all the gears and pistons are perfectly lubricated and greased. Kelvins statement asserts that no steam engine will work without a cold sink into which some heat must be wasted. Around the same time the German physicist Rudolph Clausius saw that heat never spontaneously flows from a cooler to a hotter body. By spontaneous, Clausius meant natural, without the help of a heat pump or refrigerator.

Entropy is the measure of how spread out energy is in a system at a given temperature. Energy will disperse if it is allowed and as energy disperses, the entropy will go up. Entropy is the tax you pay to nature for converting one form of energy to another. It is the measure of interference of nature in the human effort to make perfect use of its resources. No system is perfect, says the second law of thermodynamics, there is always some loss of energy when we try to make it useful. Entropy increase is the story of energy's journey from being concentrated and useful to being dispersed and useless. Entropy is the reason heat flows from hot to cold because it's more probable for energy to spread out than to concentrate. It is reason engines are inefficient because they must exhaust waste heat, increasing entropy elsewhere.

Low entropy :-

If anything in universe have concentrated energy then it has low entropy. Low entropy system have energy which is concentrated, useful, and can do work. This is a valuable, improbable state.
e.g. tightly coiled spring, a tank of compressed gas, or a hot cup of coffee in a cool room, a charged battery, a hot star, a raised weight.

High Entropy :-

If we use concentrated energy i.e. low entropy system to do work some energy. The energy will disipited in universe. The energy that wasn't converted into work is dissipated into the surroundings as waste heat, spreading out and becoming useless. If energy is spread out, dissipated, and useless for doing work, it is high entropy.
e.g. released spring, gas expanded to fill the room, or that same coffee now at room temperature, the warm air around a car engine, the heat from a lightbulb, the low-grade warmth radiating from your computer.

Examples:

Imagine you light a campfire. The wood and oxygen represent a low-entropy state i.e. concentrated, high-quality chemical energy. When you burn it, that energy doesn't disappear because according to 1st law, energy is conserved. Instead, it disperses. It becomes heat that warms the air around you i.e. energy spread among countless air molecules. It becomes light that travels off into space. It becomes the sound of crackling. The energy is now spread thinly throughout the environment. It's still there, but it's so dispersed that it's virtually impossible to gather it back up and use it to form a new wooden piece. This is a high-entropy state.

Take an ice cube in a glass of warm water. All the cold is concentrated in the cube, all the heat is in the water. The ice melts. The cold from the cube disperses into the water, and the heat from the water disperses into the melting ice. Everything reaches a uniform temperature i.e. a high entropy state. The reverse process of turning water into ice cube can't happen itself. This will violate the Second Law. It would require energy to undisperse itself, which would mean a decrease in entropy.

Why total entropy of the universe always increases?

The waste heat is so dispersed among countless atoms in the environment that it is statistically impossible to gather it all back up and reconcentrate it into the original, useful form. You utilize your fuel to create energy and heat, but you can't convert that heat and energy into fuel. You can't collect the heat from the ocean and use it to power a ship unless you have an even colder reservoir.

This is what it means for the total entropy of the universe to always increase. Every time we use concentrated energy to do anything, we are degrading the quality of the universe's energy, spreading it out, and making it less useful for future work.

Imagine if our universe is a closed system then eventually all concentrated energy sources will be used up. Our fuel will be exhausted. All energy will have been converted to work and then dissipated as waste heat. The entire universe will reach a uniform temperature a state of maximum entropy. In this state, there will be no more temperature differences, no more energy gradients, and therefore, no possibility for any further work to be done. We are spending the universe's finite supply of low-entropy energy.

Entropy as energy spread at atomic level?

At the atomic level, entropy is not about energy spreading in space like a gas expanding. It's about energy spreading into the different ways atoms and molecules can store energy their degrees of freedom. A microstate is a single, specific arrangement of which atom has how much energy and in what form. Entropy is a measure of the number of possible microstates for a given total energy.

Let's take a system with a fixed total energy and what happens as we increase its entropy. A small amount of energy is added to a group of atoms.

Imagine three atoms in a box, with a total of 3 units of energy. In a low entropy state, the energy is highly concentrated. Initially it is at low entropy state.
Microstate 1: Atom A has 3 units, Atoms B and C have 0. Energy is concentrated in one atom. This is an improbable, low-entropy state.

Through countless collisions will cause the energy to rapidly spread out into the most probable configuration. The same 3 units of energy can now be distributed in many more ways:
Microstate 2: A=3, B=0, C=0
Microstate 3: A=2, B=1, C=0
Microstate 4: A=2, B=0, C=1
Microstate 5: A=1, B=2, C=0
Microstate 6: A=1, B=1, C=1
Microstate 7: A=1, B=0, C=2 and so on.

The system will naturally evolve into the configuration with the highest number of microstates like Microstate 6, where energy is spread in equal amount. This is the state of maximum entropy. It happens because temperature is the average kinetic energy per atom. It's a measure of how full the primary energy slots i.e. translational motion already are.

e.g. Imagine two objects in contact, A hot object i.e. fast atoms and a cold object i.e. slow atoms. Through collisions at the boundary, fast atoms transfer energy to slow atoms. The energy of the hot object spreads into the atoms of the cold object. Both objects reach the same temperature. The total energy is now distributed in the absolute maximum number of ways across all the atoms of the combined system. This is the state of highest entropy. The energy is now useless for doing work because there is no longer a concentration of fast atoms to harness.

How energy is stored at atomic level?

Entropy is related to the number of ways energy can be distributed among
the atoms or molecules of a system. At the atomic level, atoms can have energy
in different modes like translation, rotation, vibration etc. The more ways to distribute a given amount of energy among the atoms, the higher the entropy.

Imagine a single molecule. It can store energy in several ways. When we add energy to that molecule, there are three ways the molecule or atom can store that energy. If added energy causes its movement through space in x, y, z directions, so this stored energy is known as translational kinetic energy. If stored energy causes its rotation around its axes, it is known as rotational kinetic energy. If added or stored energy from atoms within the molecule causes stretching and bending within the molecule, it is known as vibrational potential & kinetic energy.

Imagine these energy levels of a molecule like a set of buckets. Each of these three are buckets where energy can be placed. The more buckets available, the more ways the total energy can be distributed. Some buckets are low and easy to fill. Others are high and require more energy to reach.

When we add energy it will first goes to fill translational bucket. They are easiest to fill. These bucket have the lowest energy thresholds. Even a tiny amount of heat can make molecules move faster. At very low temperatures, atoms moves in x, y, z direction. It causes high level of increment in entropy because, more translational energy means molecules zip around faster and explore more volume, leading to a large increase in the number of possible microstates. But entropy still lower than rotational modes.

Low temperature is enough to fill translational bucket. When we add additional energy i.e. medium temperature, it fills rotational bucket. Energy causes molecule spinning around its axes. It takes a bit more energy to get a molecule spinning than to just speed it up. As the temperature rises and translational modes are somewhat saturated, energy starts spilling over into rotation. Rotation adds another way for molecules to move and interact, increasing the number of possible arrangements.

Now both translational and rotational buckets are filled. Now added energy at high temperature causes the atoms in the molecule stretch and bend like springs. Chemical bonds are like stiff springs. It takes a lot of energy to make them vibrate significantly. These modes only become active at relatively high temperatures. While it does increase disorder, vibration is more contained within the molecule itself compared to translation or rotation, which affect the molecule's position and orientation in space. Each type of motion has a different energy threshold to activate it.

This sequence is most clear for molecules like N₂, O₂, H₂O.

For single atom like Helium, is a point-like particle. It only has Translational Motion. It can move in three directions: x, y, and z. It cannot have rotational or vibrational motion in the same way a molecule can. There's no structure to rotate or vibrate. So, if you have a gas of single atoms like a monatomic gas, increasing the temperature only makes the atoms move faster. It increases their translational kinetic energy. The energy can't spread into other modes because there are no other modes available. Single atoms can't rotate or vibrate meaningfully.

For a molecule like Nitrogen N₂, Water is made of two or more atoms bonded together. This structure creates new ways to store energy. At low temperature, energy goes into translational motion. At medium temperature, energy also goes into rotational motion i.e. molecule spins like a dumbbell. There is enough energy in collisions to start making molecules spin noticeably. At high temperature, energy also goes into vibrational motion i.e. atoms in the molecule stretch and bend like springs. The bonds between atoms are strong, so it takes high temperatures to make them vibrate significantly.

e.g. Heating a Nitrogen Gas (N₂)
At very low temperatures i.e. 10 K. Energy is concentrated in a few translational modes. Molecules are slow. Entropy is low.

At room temperature i.e. 300 K. Energy has spread into translational modes i.e. molecules zipping around and rotational modes i.e. molecule spinning. The vibrational modes are still not actived as the temperature isn't high enough to excite them significantly. Entropy is much higher.

At very high temperatures i.e 1000 K. Energy now spreads fully into translational, rotational, and vibrational modes. The bonds in the N₂ molecules are stretching and bending vigorously. Entropy is higher still.

The journey of energy through translational, then rotational, then vibrational modes is a beautiful and accurate depiction of entropy increasing at the atomic level. It's the process of energy finding more and more places to hide.

How heat transfer actually works at the molecular level?

It happens through collisions and energy transfer, and there's a specific order to how the cold molecule's buckets fill up. When the hot and cold molecules collide, the most direct energy transfer is through translational motion from hot molecule to the cold molecule. Now the cold molecule has more translational energy. Through internal collisions of atoms within the molecule, some of this new translational energy gets converted into rotational energy. The molecule starts spinning faster as it moves. The cold molecule's rotational bucket begins to fill from the bottom up. Vibrational energy usually requires multiple collisions. Once the translational and rotational buckets have enough energy, some of that energy can finally couple into the vibrational modes. The cold molecule's vibrational bucket is the last to fill.

This is what formula for entropy change, dS = dQ/T tells us

Imagine adding energy dQ to a cold system at low T. The atoms are slow. The translational energy buckets are relatively empty. Adding energy can easily fill these buckets, opening up a huge number of new microstates i.e. a large increase in entropy, dS.

At atomic level, imagine adding heat to ice at -10°C. The molecules are locked in a crystal. The added energy allows them to break free and vibrate, rotate, and eventually translate a massive increase in possible arrangements.

Imagine adding the same energy dQ to a hot system at high T. The atoms are already moving very fast. The easy to fill translational buckets are already full. The added energy must go into harder to excite modes like vibrations, which opens up fewer new microstates i.e. a small increase in entropy, dS.

At atomic level, imagine adding heat to steam at 200°C. The molecules are already zipping around chaotically. The extra energy makes them zip slightly faster, but it doesn't change much.

Why engines would be useless without entropy ?

The Second Law of Thermodynamics, which governs entropy, states that heat will not spontaneously flow from a cold object to a hot one. It must flow from hot to cold. An engine is a machine that forces this heat flow to happen in a controlled way, extracting some useful work in the process. Second Law forces waste heat as a necessary byproduct of work extraction.

Imagine you have low entropy source of energy i.e. fuel. The energy is concentrated and useful. You burn fuel, creating a high temperature energy. The engine uses this heat to do useful work, like moving the car. To complete the cycle and let the engine run continuously, you must reject some waste heat to the cold reservoir i.e. the atmosphere. This waste heat is energy that has become dispersed and useless. Its entropy is very high. By ejecting this high entropy energy, you are exporting entropy out of the engine system. Entropy isn't an optional inefficiency; it's the fundamental reason engines can operate at all. If you didn't reject this waste heat, entropy would build up inside the engine.

What happens when outside temperature is higher than working temperature of engine?

If the temperature of the environment is hotter than or equal to the temperature inside the engine's cylinders, your engine will not produce any useful work. This is a direct and unavoidable consequence of the Second Law of Thermodynamics and the concept of entropy.

The heat would flow from the hot environment into the relatively cooler engine parts. The engine would heat up until it was at the same temperature as its surroundings. Once everything is at the same temperature, heat flow stops entirely. This state is called thermal equilibrium. This is the state of maximum entropy for that system. The energy is completely dispersed and useless for doing work. Without a temperature gradient that allows heat to flow out, the engine cycle cannot be completed. The piston won't be pushed; the turbine won't spin. No useful work is produced. The engine is now just a hot, inert piece of metal.

The maximum possible efficiency (η_max) :

ηmax = 1 - (Tcold / Thot)

Thot - temperature of the hot source
Tcold - temperature of the cold sink.

It tells us,
• Efficiency is always less than 100%, because Tcold is never zero.
• The colder the heat sink, the more efficient the engine can be. This is why power plants are built next to cold rivers or use giant cooling towers.
• The hotter the heat source, the more efficient the engine can be. This is why we try to burn fuel at the highest possible temperatures.

So what is entropy, it is wasteful energy or energy spread?

You might be combining the consequence of entropy i.e. waste heat with its fundamental nature i.e. spread. Entropy itself is not energy but a property of energy distribution. Entropy is the measure of the energy spread. The result of that spread is what we perceive as wasteful or useless energy.

Entropy feels like waste to us. We humans value concentrated, usable energy like gasoline, a battery, a hot furnace. The energy is concentrated and can be directed to do useful work. When energy spread so thin among so many molecules that it's impossible to harness for any practical work. We call this waste heat.

So Spreading where? Spreading in space? What kind of space?

You may be thinking that the spread of energy is some kind of density of energy in some volume. Why then in dS=Q/T the Q factor gets divided by T and not by some volume measure? Why is energy measured in units J/K and not in J/m³ ?

Entropy is fundamentally about energy distribution per degree of freedom, not just volume. Temperature reflects how energy is partitioned among translational, rotational, and vibrational modes, which is more fundamental than spatial spread.

The formula ds=dq/T emerges naturally from Carnot's analysis of heat engines, showing how work potential depends on temperature gradients, not just volume. The car engine example can tie it together. Waste heat becomes useless not because it's in a large volume, but because it's low temperature energy that can't be harnessed without a colder reservoir.


Entropy change:

Entropy change as resulting from changes in specific properties of our system. This properties are Temperature, Volume, Phase, Mixing, Composition. Changes in temperature will lead to changes in entropy. The higher the temperature the more thermal energy the system has; the more thermal energy the system has, the more ways there are to distribute that energy; the more ways there are to distribute that energy, the higher the entropy. Increasing the temperature will increase the entropy. Changes in volume will lead to changes in entropy. The larger the volume the more ways there are to distribute the molecules in that volume; the more ways there are to distribute the molecules (energy), the higher the entropy.

Changes in phase will lead to changes in entropy. Some phases have larger numbers of microstates and thus higher energy. Solids have the fewest microstates and thus the lowest entropy. Liquids have more microstates since the molecules can translate and thus have a higher entropy. When a substance is a gas it has many more microstates and thus have the highest entropy. Mixing of substances will increase the entropy. This is because there are many, many more microstates for the mixed system than for the unmixed system. More microstates means greater entropy.

Lastly, the entropy can change as the result of chemistry. Different molecules have different entropies.

Enthalpy:

Enthalpy is a concept in thermodynamics that combines the internal energy of a system with the product of its pressure and volume. Enthalpy can be thought of as
the total energy of a system, including both its internal energy and the energy
required to make room for it in the environment i.e. PV term.

It is denoted by H and defined as,
H = U + PV,
where U is internal energy,
P is pressure and
V is volume.

PΔV is the energy spent or gained from expanding or contracting against the constant pressure of the surroundings.

Intuition of enthalpy:-

Imagine in college you are moving from one room to another room. Your belongings i.e. internal energy, is all your stuff like bags and books. This is the energy actually contained within the system like the molecules kinetic and potential energy. To get your belongings into another room, you have to push aside the air that was already in the apartment. Move the doors and windows. This takes energy, This is the work done to create space for your system i.e. PV term. The total effort required to get yourself established in the new room isn't just the value of your belongings. It's your belongings plus the work you did to move them in.

e.g.

Imagine you have a gas in a cylinder with a movable piston. The gas has internal energy (U) due to the motion and interactions of its molecules. Now, if you want to push the piston to make room for the gas, you have to do work against the atmospheric pressure. The work done to make room is the PV term. So, enthalpy (H) is like the total energy you would need to create the system and place it in an environment with pressure P.

When you boil water in an open pot at constant atmospheric pressure, the heat you add goes into increasing the internal energy of the water i.e. making the molecules move faster and also does work by pushing back the atmosphere as the water turns into steam which has a much larger volume.

You heat water in an open pot at constant atmospheric pressure. The heat you add does two things. Increases the water's internal energy i.e. making molecules move faster. Does work to push back the atmosphere as the liquid water expands into steam, which takes up 1700x more volume. The total heat you must add to boil the water is the change in enthalpy (ΔH). ΔH is positive.

Imagine burning candle, combustion reaction releases heat. This heat comes from a decrease in the chemical energy stored in the wax i.e. decrease in internal energy. The reaction also produces gases like CO₂, H₂O vapor that expand and push against the atmosphere. The total heat released to the surroundings is the change in enthalpy (ΔH). Since heat is leaving the system, ΔH is negative.

Imagine we add energy needed to melt ice at 0°C. Part of that energy goes into breaking the rigid ice lattice i.e. increasing internal energy and a smaller part goes into the slight expansion of water as it melts i.e. PΔV work.

Why we add internal energy and PV term together and call it enthalpy. Because this combination creates a new property that is perfectly suited for analyzing the vast majority of real-world processes. Think about where most chemistry and engineering happens in open containers, beakers, car engines, industrial reactors under the constant pressure of the Earth's atmosphere.

In these conditions, when a process happens like a chemical reaction, two things always occur simultaneously. One is the internal energy (U) changes i.e. chemical bonds break and form. And second is the system often expands or contracts i.e. gases are produced or consumed, doing PV work against the atmospheric pressure.

If we only tracked internal energy (ΔU), we would be missing a crucial part of the story. The heat we measure in an experiment (q) is the energy that accounts for both the change in internal energy and the expansion work.

By defining Enthalpy as H = U + PV, we create a new property whose change (ΔH) automatically captures this total heat flow at constant pressure. It means that for any reaction in an open container, the change in this new property, enthalpy, is exactly equal to the heat we can easily measure.

This makes enthalpy a very useful concept in chemistry and engineering, especially for studying heat transfer in reactions and processes that occur at constant pressure.

Another scientific reason is Heat (q) and Work (W) are path functions. Their value depends on how you perform a process. You can't say a system has a certain amount of heat or work. While internal Energy (U) is a state function. Its value depends only on the current state of the system (its T, P, V, etc.), not on its history.

Heat Flow (q) is a path function. It's messy to calculate and depends on the exact process. ΔH is a state function. It's easy to calculate and use in calculations. It only depends on the initial and final states. Separate Tracking You'd have to constantly calculate ΔU and PΔV separately for every process. H bundles them together, simplifying analysis for the most common lab conditions.