Entropy in Physics: Meaning, Definition of Entropy and Formula

Entropy

Edited By Vishal kumar | Updated on Jul 02, 2025 06:29 PM IST

Entropy is a concept in thermodynamics that measures the disorder or randomness within a system. It's crucial to understand why certain processes happen spontaneously and others do not. For students preparing for board exams and competitive exams like JEE and NEET, mastering entropy is essential because it helps explain phenomena in chemistry, physics, and biology. This article simplifies the concept of entropy and includes a solved example to illustrate how it affects real-world systems, providing a practical understanding that can be applied in exams and beyond.

This Story also Contains

What is Entropy?
2. Entropy for an ideal gas
Solved Examples Based on Entropy
Summary

What is Entropy?

Entropy is a measure of the disorder of the molecular motion of a system. i.e Greater is the disorder, greater is the entropy.

The change in entropy is given as

$d S=\frac{\text { Heat absorbed by system }}{\text { Absolute temperature }}$ or $d S=\frac{d Q}{T}$

The relation $d S=\frac{d Q}{T}$ is called the mathematical form of the Second Law of Thermodynamics

1. Entropy for solid and liquid

i. When heat is given to a substance to change its state at a constant temperature.

Then change in entropy is given as

$d S=\frac{d Q}{T}= \pm \frac{m L}{T}$

where the positive sign refers to heat absorption and the negative sign to heat evolution.

And $L=$ Latent Heat and T is in Kelvin.

ii. When heat is given to a substance to raise its temperature from $T_1$ to $T_2$

Then change in entropy is given as

$d S=\int \frac{d Q}{T}=\int_{T_1}^{T_2} m c \frac{d T}{T}=m c \log _e\left(\frac{T_2}{T_1}\right)=2.303 * \operatorname{mc} \log _{10}\left(\frac{T_2}{T_1}\right)$

where c = specific heat capacity

2. Entropy for an ideal gas

For n mole of an ideal gas, the equation is given as PV = nRT

I.Entropy change for ideal gas in terms of T & V

From the first law of thermodynamics, we know that $d Q=d W+d U$
and
$
\Delta S=\int \frac{d Q}{T}=\int \frac{n C_V d T+P d V}{T}
$
using $P V=n R T$
$
\begin{aligned}
& \Delta S=\int \frac{n C_V d T+\frac{n R T}{V} d V}{T}=n C_V \int_{T_1}^{T_2} \frac{d T}{T}+n R \int_{V_1}^{V_2} \frac{d V}{V} \\
& \Delta S=n C_V \ln \left(\frac{T_2}{T_1}\right)+n R \ln \left(\frac{V_2}{V_1}\right)
\end{aligned}
$
II. Entropy change for an ideal gas in terms of $T$ \& $P$
$
\Delta S=n C_P \ln \left(\frac{T_2}{T_1}\right)-n R \ln \left(\frac{P_2}{P_1}\right)
$

III. Entropy change for an ideal gas in terms of P \& V
$
\Delta S=n C_V \ln \left(\frac{P_2}{P_1}\right)+n C_P \ln \left(\frac{V_2}{V_1}\right)
$

Solved Examples Based on Entropy

Example 1: Which of the following is incorrect regarding the first law of thermodynamics?

1) It introduces the concept of the internal energy

2) It introduces the concept of entropy

3) It is applicable to any cyclic process

4) It is a restatement of the principle of conservation of energy

Solution:

The first law of Thermodynamics

Heat imported to a body is in general used to increase internal energy and work done against external pressure.

wherein

$
d Q=d U+d W
$

Entropy
It is a measure of the disorder of molecular motion of a system.
wherein
Greater is disorder greater is entropy
$
d S=\frac{d Q}{T}
$

The concept of entropy is introduced in the second law of thermodynamics.

The first law dealt with internal energy, work, and heat energy.

It is a statement of the first law of thermodynamics.

Hence, the answer is the option (2).

Example 2: The temperature- entropy diagram of a reversible engine cycle is given in the figure. Its efficiency is

1) $1 / 3$
2) $2 / 3$
3) $1 / 2$
4) $1 / 4$

Solution:

Entropy

It is a measure of disorder of molecular motion of a system.

wherein

Greater is disorder greater is entropy

$dS= \frac{dQ}{T}$

and

Efficiency of a cyclic process
$
\eta=\frac{\text { work done per cyclic }}{\text { gross heat supplied per cyclic }}
$
wherein
Gross heat supplied implies only part of heat absorbed.
The efficiency of cycle is:
$
\begin{aligned}
& \eta=\frac{\text { work done }}{\text { heat absorb }} \\
& =\frac{Q_1-Q_2}{Q_1} \\
& =\frac{\text { Area of cycle }}{\text { Area under AB curve }} \\
& \qquad \eta=\frac{\frac{1}{2} \times T_0 \times S_0}{T_0 S_0+\frac{1}{2} T_0 S_0}=\frac{1}{3} \\
& \text { }
\end{aligned}
$

Example 3: If $\Delta Q$ heat is given to a substance and changes its state at constant temperature T then a change in entropy ds will be
1) $\pm \frac{m L}{T}$
2) $\pm \frac{m C}{T}$
3) $\pm m L$
4) $\pm m C$

Solution:

Entropy for solid and liquid

$
\begin{aligned}
& \Delta S=\frac{m L}{T} \\
& L=\text { Latent Heat } \\
& \text { T in kelvin } \\
& \frac{d \theta}{d T}= \pm \frac{m L}{T}
\end{aligned}
$

When a substance changes its state at a constant temperature $d \theta= \pm m L$ where the (+ve) sign refers to heat absorption and the (-ve) sign to heat evolution

Hence, the answer is the option (1).

Example 4: When heat is given to a substance of mass $m$ and specific heat c its temperature raise from $27^{\circ} \mathrm{C}$ to $327^{\circ} \mathrm{C}$, then
1) $2.303 \log _{10}(2)$
2) $2.303 \log _{10}(1 / 2)$
3) $\log _{10}(3 / 2)$
4) $2.303 \log _{10}(3 / 2)$

Solution:

Entropy for solid and liquid

When heat is given to a substance to raise its temperature from $T_1$ to $T_2$
wherein
$
\begin{aligned}
& \Delta S=m s \ln \left(\frac{T_2}{T_1}\right) \\
& \mathrm{s}=\text { specific heat capacity } \\
& \mathrm{ds}=2.303 \\
& \log \left[\frac{273+327}{273+27}\right]=2.303 \log _{10} 2=0.693
\end{aligned}
$

Hence, the answer is the option 1,

Example 5: The change in the entropy of 1 mole of an ideal gas that went through an isothermal process from the initial position to the final state is equal to

${ }_{1)} R \ln \frac{V_2}{V_1}$
2) $R \ln \frac{V_1}{V_2}$
3) zero
4) $R \ln T$

Solution:

Entropy change for an ideal gas in terms of T & V

$
\Delta S=n c_p \ln \left(\frac{T_2}{T_1}\right)+n R \ln \left(\frac{V_2}{V_1}\right)
$

For an ideal gas, we have
$
\Delta S=n c_p \ln \left(\frac{T_2}{T_1}\right)+n R \ln \left(\frac{V_2}{V_1}\right)
$
for isothermal process
$
\begin{aligned}
& \log _e\left(\frac{T_2}{T_1}\right)=0 \\
& \Delta s=n R \log _e\left(\frac{V_2}{V_1}\right) \text { for one mole of gas } \\
& \text { So } \\
& \Delta s=R \log _e\left(\frac{V_2}{V_1}\right)
\end{aligned}
$

Hence, the answer is the option 1.

Summary

Entropy, or the shift in the energy state accompanied by heat, is connected to a number of daily actions. It aids in a shift in heat transport and is reflected by the first, second, and third laws of thermodynamics. The more unpredictability in the system, the higher the rate of entropy; conversely, the lower the randomness of the system's molecules, the lower the entropy.

Frequently Asked Questions (FAQs)

1. What is entropy in thermodynamics?

Entropy is a measure of the disorder or randomness in a system. In thermodynamics, it quantifies the unavailability of a system's thermal energy for conversion into mechanical work. As entropy increases, the energy becomes more spread out and less useful.

2. How is entropy related to heat transfer?

Entropy is closely related to heat transfer. When heat flows from a hot object to a cold one, the entropy of the universe increases. This process spreads energy more evenly, increasing the overall disorder and thus entropy.

3. What's the connection between entropy and the arrow of time?

Entropy provides a direction for time, known as the "arrow of time." As entropy increases, it defines the flow of time from past to future. This is why we remember the past but not the future, and why many processes (like breaking an egg) are irreversible.

4. Why does entropy always increase in closed systems?

Entropy always increases in closed systems due to the Second Law of Thermodynamics. This law states that natural processes tend to move towards a state of greater disorder. As particles spread out and energy disperses, the system becomes more disordered, increasing entropy.

5. Can entropy ever decrease?

Entropy can decrease in an open system that exchanges energy or matter with its surroundings. However, the total entropy of the universe always increases. Local decreases in entropy are always offset by larger increases elsewhere.

6. How does entropy relate to the efficiency of heat engines?

Entropy limits the efficiency of heat engines. No heat engine can convert all thermal energy into mechanical work due to entropy increase. Some energy is always lost to the environment, reducing efficiency. This is why perpetual motion machines are impossible.

7. How does temperature affect entropy?

Temperature generally increases entropy. As temperature rises, particles move faster and have more kinetic energy, allowing them to occupy more positions. This increases the number of possible microstates and thus entropy. However, there are exceptions, like the anomalous behavior of water between 0°C and 4°C.

8. How does the concept of entropy apply to black holes?

Black hole entropy, proposed by Jacob Bekenstein and Stephen Hawking, suggests that black holes have entropy proportional to their surface area. This concept connects thermodynamics, gravity, and quantum mechanics, and helps resolve the black hole information paradox.

9. What is the Boltzmann constant and how does it relate to entropy?

The Boltzmann constant (k) is a physical constant that relates the average kinetic energy of particles in a gas with the temperature of the gas. It appears in the Boltzmann entropy formula, S = k ln W, connecting the microscopic properties of matter (number of microstates, W) to the macroscopic quantity of entropy (S).

10. What is the entropy change in an ideal gas expansion?

When an ideal gas expands, its entropy increases. This is because the gas molecules have more volume to occupy, increasing the number of possible microstates. For an isothermal expansion, the entropy change is given by ΔS = nR ln(V2/V1), where n is the number of moles, R is the gas constant, and V2 and V1 are the final and initial volumes.

11. How does the Third Law of Thermodynamics affect our ability to reach absolute zero?

The Third Law implies that it's impossible to reach absolute zero temperature through any finite number of steps. As we approach absolute zero, it becomes increasingly difficult to remove energy from a system, and the entropy approaches a minimum value. This is why achieving absolute zero remains a theoretical limit.

12. What is the Gibbs paradox and how does it relate to entropy?

The Gibbs paradox arises when considering the entropy of mixing of identical gases. Classical thermodynamics predicts an increase in entropy when mixing two volumes of the same gas, which seems paradoxical. The resolution comes from quantum mechanics, which recognizes that identical particles are indistinguishable, leading to no entropy increase in this case.

13. What is the significance of the Sackur-Tetrode equation in relation to entropy?

The Sackur-Tetrode equation provides an expression for the entropy of an ideal gas. It relates the entropy to the number of particles, volume, and temperature of the gas. This equation is significant because it connects the macroscopic property of entropy to the microscopic properties of the gas particles, bridging classical thermodynamics and statistical mechanics.

14. What is the relationship between entropy and the efficiency of solar cells?

Entropy limits the efficiency of solar cells. The conversion of solar energy to electrical energy involves an increase in entropy, as not all the incoming photon energy can be converted to useful work. The Shockley-Queisser limit, which sets the maximum theoretical efficiency for single-junction solar cells, is fundamentally based on entropy considerations.

15. How does entropy relate to the concept of entanglement in quantum mechanics?

In quantum mechanics, entanglement entropy measures the amount of quantum information shared between two subsystems. This concept extends classical entropy to quantum systems. Highly entangled systems have higher entropy, reflecting the increased complexity and information content of the quantum state.

16. What is the role of entropy in the formation and evolution of stars?

Entropy plays a crucial role in stellar evolution. As stars form from collapsing gas clouds, the increase in gravitational entropy outweighs the decrease in thermal entropy, making the process spontaneous. Throughout a star's life, nuclear fusion increases entropy by converting matter to energy. In the final stages, stars evolve towards higher entropy states, eventually reaching maximum entropy in black holes or dispersed matter.

17. How does entropy relate to the concept of chemical potential in thermodynamics?

Chemical potential and entropy are closely related in thermodynamics. The chemical potential of a substance is defined as the partial derivative of the Gibbs free energy with respect to the number of particles, at constant temperature and pressure. Since Gibbs free energy incorporates entropy (G = H - TS), changes in entropy directly affect the chemical potential, influencing processes like diffusion and phase equilibria.

18. What's the difference between entropy and enthalpy?

Entropy (S) measures the disorder of a system, while enthalpy (H) measures the total heat content. Enthalpy includes both the internal energy of the system and the product of pressure and volume. Entropy focuses on the distribution of energy, while enthalpy quantifies the amount.

19. How does the concept of microstates relate to entropy?

Microstates are the different possible arrangements of particles in a system. Entropy is directly related to the number of microstates: the more microstates a system has, the higher its entropy. This connection is expressed in Boltzmann's entropy formula: S = k ln W, where W is the number of microstates.

20. What is the Third Law of Thermodynamics and how does it relate to entropy?

The Third Law of Thermodynamics states that the entropy of a perfect crystal at absolute zero temperature is zero. This provides a reference point for entropy calculations and implies that it's impossible to reach absolute zero temperature through a finite number of steps.

21. How does entropy affect the spontaneity of reactions?

Entropy is a key factor in determining the spontaneity of reactions. Generally, reactions that increase entropy are more likely to occur spontaneously. The Gibbs free energy equation (ΔG = ΔH - TΔS) combines entropy (S) with enthalpy (H) to predict reaction spontaneity.

22. What is the entropy of mixing?

The entropy of mixing is the increase in entropy that occurs when two or more different substances are mixed together. This increase happens because mixing increases the number of possible arrangements of particles, thus increasing disorder and entropy.

23. What is the relationship between entropy and probability?

Entropy and probability are closely linked. Systems tend to evolve towards states with higher probability, which coincides with states of higher entropy. The most probable state of a system is the one with the highest entropy, explaining why entropy tends to increase over time.

24. How does entropy relate to information theory?

In information theory, entropy measures the amount of information content in a message. This concept, developed by Claude Shannon, is analogous to thermodynamic entropy. Both measure the degree of randomness or unpredictability in a system, whether it's made of particles or bits of information.

25. What is the Clausius inequality and how does it relate to entropy?

The Clausius inequality states that for any cyclic process, the integral of dQ/T (heat transfer divided by temperature) is less than or equal to zero. This inequality leads to the definition of entropy and demonstrates that entropy always increases for irreversible processes.

26. How does entropy explain the impossibility of a perfect heat engine?

A perfect heat engine would convert all heat into work with 100% efficiency. However, this violates the Second Law of Thermodynamics. Some heat must always be expelled to a cold reservoir, increasing the entropy of the universe. This entropy increase makes perfect efficiency impossible.

27. What is the entropy change during a phase transition?

During a phase transition at constant temperature (like melting or boiling), there is a significant change in entropy. This is because the disorder of the system increases as it transitions from a more ordered state (like solid) to a less ordered state (like liquid or gas).

28. What is the relationship between entropy and the quality of energy?

Entropy is inversely related to the quality of energy. High-quality energy (like electrical energy) has low entropy and can be easily converted to other forms. Low-quality energy (like heat at low temperatures) has high entropy and is less useful for doing work.

29. How does entropy relate to the concept of free energy?

Free energy, particularly Gibbs free energy, incorporates both enthalpy and entropy. The equation ΔG = ΔH - TΔS shows that increasing entropy (ΔS) decreases free energy (ΔG). Systems naturally move towards lower free energy states, which often means higher entropy states.

30. How does the concept of entropy apply to living organisms?

Living organisms maintain low internal entropy by constantly exchanging matter and energy with their environment. They create local order (decreasing entropy) at the expense of increasing the entropy of their surroundings. This is consistent with the Second Law of Thermodynamics when considering the organism and its environment as a whole system.

31. What is the difference between reversible and irreversible processes in terms of entropy?

In a reversible process, the entropy of the universe remains constant. The system can be returned to its initial state without any net change in the surroundings. In an irreversible process, which includes all real-world processes, the entropy of the universe always increases. The system cannot be returned to its initial state without changing the surroundings.

32. How does entropy relate to the efficiency of refrigerators and heat pumps?

Entropy limits the efficiency of refrigerators and heat pumps. These devices move heat from a cold reservoir to a hot one, which decreases entropy locally but increases the total entropy of the universe. The work required to run these devices is always more than the theoretical minimum due to irreversible processes and entropy generation.

33. How does entropy relate to the concept of heat death of the universe?

The heat death of the universe is a hypothetical end state where the universe has reached maximum entropy. In this state, all energy is evenly distributed, and no more work can be extracted. This concept arises from extrapolating the Second Law of Thermodynamics to the entire universe.

34. What is the relationship between entropy and the spontaneous formation of crystals?

The formation of crystals might seem to decrease entropy locally, as it creates a more ordered structure. However, this process is typically exothermic, releasing heat to the surroundings and increasing their entropy. The total entropy of the universe still increases, making crystal formation spontaneous under the right conditions.

35. How does the concept of entropy apply to chemical reactions?

In chemical reactions, entropy changes can drive or inhibit the reaction. Reactions that increase entropy (like decomposition reactions) are often spontaneous. The entropy change in a reaction is calculated by comparing the entropy of products and reactants. This entropy change, along with enthalpy change, determines the reaction's spontaneity through the Gibbs free energy equation.

36. What is the relationship between entropy and the direction of time?

Entropy provides a thermodynamic arrow of time. The Second Law of Thermodynamics states that the total entropy of an isolated system can never decrease over time, only increase or remain constant. This unidirectional increase in entropy aligns with our perception of time flowing from past to future, explaining why we don't observe spontaneous decreases in entropy, like shattered glass reassembling itself.

37. How does entropy relate to the concept of equilibrium in thermodynamics?

In thermodynamics, a system reaches equilibrium when it attains its maximum entropy state under given constraints. At equilibrium, the system has explored all accessible microstates and has no tendency to change macroscopically. Any spontaneous process in an isolated system will proceed in the direction that increases the total entropy, eventually reaching equilibrium.

38. How does the concept of entropy apply to the expansion of the universe?

As the universe expands, its total entropy increases. This is partly due to the increasing volume, which allows for more possible arrangements of matter and energy (more microstates). Additionally, processes like star formation and black hole evolution contribute to the overall increase in cosmic entropy, aligning with the Second Law of Thermodynamics on a universal scale.

39. How does the concept of entropy apply to ecosystems?

In ecosystems, entropy is related to the flow of energy and matter. Ecosystems maintain low internal entropy by constantly importing low-entropy energy (like sunlight) and exporting high-entropy waste. The overall process increases the entropy of the larger system (Earth), aligning with the Second Law of Thermodynamics. This concept helps in understanding ecosystem stability and energy flow in food webs.

40. What is the connection between entropy and the arrow of time in cosmology?

In cosmology, the arrow of time is closely linked to the increase of entropy in the universe. The expansion of the universe creates more available volume, increasing the number of possible microstates and thus entropy. This cosmological arrow of time aligns with the thermodynamic arrow, providing a consistent direction for time's flow from the Big Bang onwards.

41. How does entropy relate to the concept of information erasure in computing?

Entropy is fundamental to information theory and computing. Landauer's principle states that erasing information increases entropy, requiring a minimum amount of energy dissipation. This principle connects the abstract concept of information with physical entropy, demonstrating that information processing has thermodynamic consequences.

42. What is the relationship between entropy and the spontaneity of dissolution processes?

The spontaneity of dissolution processes is often driven by an increase in entropy. When a solute dissolves in a solvent, there's usually an increase in disorder as the solute particles become dispersed. This entropy increase can make dissolution spontaneous even if it's endothermic, explaining why some salts dissolve spontaneously despite absorbing heat from the surroundings.

43. How does the concept of entropy apply to the formation of weather patterns?

Entropy plays a role in weather pattern formation. The uneven heating of Earth's surface by the Sun creates temperature gradients, which represent a state of low entropy. The atmosphere works to eliminate these gradients, increasing entropy through processes like wind, precipitation, and storm formation. This drive towards higher entropy helps explain the complexity and dynamics of weather systems.

44. What is the significance of entropy in understanding phase diagrams?

Entropy is crucial in understanding phase diagrams. Phase transitions often involve significant changes in entropy. For example, the slope of the phase boundary lines in a pressure-temperature diagram is related to the entropy change of the transition through the Clapeyron equation. This relationship helps predict how changes in pressure or temperature will affect phase transitions.

45. How does entropy relate to the concept of maximum work in thermodynamics?

The concept of maximum work is closely tied to entropy. The maximum work that can be extracted from a system is achieved in a reversible process, where the entropy of the universe remains constant. In real (irreversible) processes, some work potential is always lost due to entropy generation, reducing the available work below this theoretical maximum.

46. What is the role of entropy in the spontaneous folding of proteins?

The spontaneous folding of proteins involves a complex interplay of entropy and enthalpy. While the folding process decreases the conformational entropy of the protein chain, it often increases the entropy of the surrounding water molecules. This entropy increase, combined with favorable enthalpy changes from forming internal bonds, drives the overall folding process.

47. How does the concept of entropy apply to the efficiency of thermal insulation?

Entropy is key to understanding thermal insulation efficiency. Effective insulation slows down the natural flow of heat from hot to cold, which would increase entropy. By reducing heat transfer, insulation maintains a lower entropy state for longer. However, no insulation is perfect; some heat always leaks through, gradually increasing entropy over time.

48. What is the relationship between entropy and the spontaneity of nuclear decay?

Nuclear decay processes are typically spontaneous because they lead to an increase in entropy. When a large, unstable nucleus decays into smaller, more stable products, the total entropy of the system increases. This entropy increase, combined with the release of energy, makes nuclear decay processes thermodynamically favorable.