Entropy

Entropy means the amount of disorder or randomness of a system. It is a measure of thermal energy per unit of the system which is unavailable for doing work. The concept of entropy can be applied in various contexts and stages, including cosmology, economics, and thermodynamics. Entropy is a concept that essentially discusses the spontaneous changes that take place in ordinary phenomena or the Universe’s inclination towards disorder.

In this article, we will learn what is meaning of Entropy, the entropy change formula, and how it is associated with the laws of thermodynamics.

What is Entropy?

Entropy is a scientific concept commonly associated with disorder, randomness, or uncertainty. It is a measure of the unavailable energy in a closed thermodynamic system that is also usually considered a property of the system’s state. Entropy is dynamic – the energy of the system is constantly being redistributed among the possible distributions as a result of molecular collisions. It is a state function, like temperature or pressure, instead of a path function, like heat or work. The letter “S” serves as the symbol for entropy.

Entropy Definition

Entropy is the degree of randomness or the amount of energy in the system which can’t be used to do any work

Meaning of Entropy

Entropy can be understood as a measure of how dispersed and random the energy and mass of a system are distributed.

Entropy in Thermodynamics

• Entropy, from a thermodynamic viewpoint, is a measure of the unavailable energy in a closed system and is considered a property of the system’s state.
• It varies directly with any reversible change in heat in the system and inversely with the degree of disorder or uncertainty in a system.

Entropy in Statistical Thermodynamics

• In statistical thermodynamics, entropy measures the number of possible microscopic states consistent with the system’s macroscopic properties.
• It measures uncertainty, disorder, or randomness in a system after its observable macroscopic properties, such as temperature, pressure, and volume, have been considered.
• This definition is based on the statistical distribution of the motions of the particles constituting a system, whether classical (e.g., atoms or molecules in a gas) or quantum-mechanical (e.g., photons, phonons, or spins).
• Entropy is a fundamental concept that connects the microscopic and macroscopic worldviews in thermodynamics.

Learn, Thermodynamics

Interpretations of Entropy

• Entropy in science, denoted as ΔS, gives us a valuable instinct on randomness and information. Entropy is used in information theory to indicate randomness in data. The origin of information entropy can be traced back to Claude Shannon’s 1948 paper titled “A Mathematical Theory of Communication.”
• The interpretation of entropy in statistical mechanics measures uncertainty, or disorder, which remains about a system after its observable macroscopic properties, such as temperature, pressure, and volume, have been considered.
• Entropy is the most misunderstood of thermodynamic properties. While temperature, pressure and volume can be easily measured, entropy cannot be observed directly, and there are no entropy meters.
• Qualitatively, entropy measures how much the energy of atoms and molecules becomes more spread out in a process and can be defined in statistical terms. Entropy is a state function often mistakenly called a system’s ‘state of disorder’.

Properties of Entropy

Some of the critical properties of entropy include:

• Thermodynamic function: Entropy is a thermodynamic function, which means it is a property of the system’s state and not a property of the process. It is represented by the letter “S” and is usually denoted in units of J/K or cal/K

• State function: Entropy is a state function, meaning it depends only on the system’s current state and not on the path taken to reach that state. This property ensures that the second law of thermodynamics is obeyed.

• Monotonicity: Entropy is monotonic for adiabatic availability, meaning that it increases in a spontaneous process and decreases in a non-spontaneous process. This property is crucial for understanding the direction of change in a system.

• Additivity on composite systems: Entropy is additive on composite systems, meaning that a system’s entropy of multiple parts is the sum of the entropies of its individual parts. This property helps analyze the behavior of complex systems.

• Extensivity: Entropy is extensive, meaning that it scales with the size or extent of a system. This property implies that the entropy of a large system is much larger than that of a small system.

Entropy Formula

Entropy is a scientific concept often associated with a state of disorder, randomness, or uncertainty. In the context of thermodynamics, entropy is related to the energy and temperature of a system, and it is represented by the equation

S = k_b ln Ω

where,

• S is the Entropy
• k_b is the Boltzmann Constant
• Ω is the number of microscopic configuration

Entropy Change Formula

The formula for calculating the change in entropy (ΔS) for a system depends on the type of process or reaction. Here are some standard formulas for entropy change:

Entropy Change General Formula: For a general case, where the process is not isothermal or isobaric, the entropy change can be calculated using the first law of thermodynamics:

ΔS= ΔQ/T

where,

• ΔQ is the heat absorbed or released by the system
• T is the absolute temperature

Isothermal Processes: For an isothermal process, such as the expansion of an ideal gas, the entropy change can be calculated using the following formula:

ΔS = nRln (V₂/V₁)

where,

• n is the number of moles
• R is the ideal gas constant
• V₁ and V₂ are the initial and final volumes, respectively.

Isobaric Process: For an isobaric process, such as a chemical reaction, the entropy change can be calculated using the following formula:

ΔS=∑ΔS_products −∑ΔS_reactant

where,

• ΔS_products and ΔS_reactants are the entropies of the products and reactants, respectively.

Unit of Entropy

Entropy has units of energy divided by temperature, which is represented as:

Joules per Kelvin (J/K)

This unit is derived from the Boltzmann constant, a fundamental physics constant that relates entropy to temperature and energy.

Dimension of Entropy

Dimensional Formula for Entropy is:

[ML²T⁻²K^-1 ]

Change in Entropy

Change in Entropy is a state function, meaning that it depends only on the initial and final states of the system and not on the path taken to get there.

Entropy Change during Reversible Adiabatic Expansion

Reversible Adiabatic Expansion is a process in which no heat flows in or out of the system, and the process is reversible. In this case, the change in entropy is given as

dS = dQ/T = 0

Now, let’s consider a system undergoing a reversible adiabatic expansion. Since no heat flows in or out of the system, the entropy change (ΔS) is zero.

• This is because the process is reversible, and the system returns to its original state after the expansion.
• In this case, there is no net change in the disorder or randomness of the system; thus, the entropy change is zero.

Entropy Changes During Phase Transition

During a phase transition, such as melting or vaporization, there is a change in the entropy of the system as follows:

Entropy of Fusion

The entropy of fusion is the increase in entropy when melting a solid substance. It is almost always positive since the degree of disorder increases in the transition from a solid to a liquid state. The entropy of fusion is denoted as ΔS_fus and is typically expressed in joules per mole-kelvin (J/(mol·K))

The entropy of fusion is related to the heat of fusion and the melting point. The entropy of fusion can be calculated using the equation:

ΔS_fus = Q_fus/T_m

Where ΔS_fus is the entropy of fusion, Q_fus is the heat of fusion, and T_m is the melting point.

Key Points on Entropy of Fusion:

• The entropy of fusion is always positive for most substances, as the liquid state has a higher disorder than the solid state
• Helium has a negative entropy of fusion at temperatures below 0.3 K, while helium-4 has a very slightly negative entropy of fusion below 0.8 K.
• At temperatures well above their melting points, the entropy of fusion becomes positive, and the entropy approaches zero as the temperature approaches the melting point.

Entropy of Vaporisation

Entropy of Vaporization is the increase in entropy when a liquid substance evaporates. It is always positive since the degree of disorder increases in the transition from a liquid in a relatively small volume to a gas in a larger volume. The entropy of vaporization is denoted as ΔS_vap and is typically expressed in joules per mole-kelvin (J/(mol·K))

ΔS_vap = Q_vap/T_b

where,

• ΔS_vap is Entropy of Vaporisation
• Q_vap is the heat of Vaporisation
• T_b is boiling point

Key Points on Entropy of Vaporization:

• The entropy of vaporization is related to the heat of vaporization and the boiling point. According to Trouton’s rule, the entropy of vaporization (at standard pressure) of most liquids has similar values, typically around 85 J/(mol·K), 88 J/(mol·K), or 90 J/(mol·K)

• This rule is based on the observation that the entropy of a gas is considerably larger than that of a liquid, and the entropy of the initial state (e.g., the liquid) is negligible in determining the overall entropy change during vaporization.

• Hydrogen-bonded liquids have somewhat higher values of entropy of vaporization, while ordered gases and some other substances deviate from Trouton’s rule.

• Entropy of vaporization provides insight into the degree of disorder in the vapor and liquid phases and helps explain why specific processes, such as vaporization, occur spontaneously.

Entropy in Thermodynamics

The relationship between entropy and different laws of thermodynamics is as follows:

First law of Thermodynamics and Entropy

• The first law of thermodynamics, which is a version of the law of conservation of energy, states that the total energy of an isolated system is constant; energy can be transformed from one form to another but can be neither created nor destroyed.

• Entropy, on the other hand, is a measure of the unavailable energy in a closed system and is not a conserved quantity.

• While the first law of thermodynamics deals with energy conservation, entropy is related to the quality of energy and the direction or outcome of spontaneous changes in a system.

• Therefore, the first law of thermodynamics and entropy are distinct concepts describing different thermodynamic systems.

Second Law of Thermodynamics and Entropy

• Second law of Thermodynamics and entropy are closely related. The second law of thermodynamics implies that the Universe’s entropy is continuously increasing, and any spontaneous process will increase the total entropy of the system and its surroundings.

• The relationship between the second law of thermodynamics and entropy is that the second law provides a quantitative measure of the direction of spontaneous processes. In contrast, entropy measures a system’s degree of disorder or randomness.

• For example, when a hot object is placed in contact with a cold object, heat flows spontaneously from the hot object to the cold object. This process increases the total entropy of the system and its surroundings as the energy becomes more dispersed and less available to do work.

• Another example is ice melting, a spontaneous process that increases the system’s entropy. The solid ice has a more ordered structure than the liquid water, and the melting process increases the degree of disorder or randomness in the system.

Third Law of Thermodynamics and Entropy

• Third law of thermodynamics states that the entropy of a system approaches a constant value as the temperature approaches absolute zero.

• The entropy of a system at absolute zero is typically zero, and in all cases, it is determined only by the number of different ground states it has.

• This law has two significant consequences: It defines the sign of the entropy of any substance at temperatures above absolute zero as positive, and it provides a fixed reference point that allows the absolute entropy of any substance to be measured.

Entropy and Enthalpy

Entropy and Enthalpy are two most important factors of Thermodynamics. Entropy is measure of randomness and disorder of a system while enthalpy is total amount of internal energy of the system. To learn more difference between Enthaloy and Entropy check, Enthalpy vs Entropy

Learn, More

• Thermodynamic Cycle
• Thermodynamic Processes
• Stefan Boltzmann Law

Entropy – Solved Examples

Example 1. Calculate the entropy change when 10 moles of an ideal gas expands reversibly and isothermally from an initial volume of 10L to 100L at 300K.

Solution:

First we will see the given information in the question:

Number of moles = 10

Initial volume of gas (V1) = 10L

Final volume of gas (V2) = 100L

Now, we will calculate the entropy of the gas using the formula:

ΔS = 2.303 nRlog(V2/V1)

On putting the values, we get,

ΔS=2.303×10×8.314 log(100/10)

ΔS=191.47JK⁻¹

Example 2. Calculate the entropy change for 1.00 mol of an ideal gas expanding isothermally from a volume of 24.4 L to 48.8 L.

Solution:

ΔS = nRln (V2/V1)

= (1.00mol) (8.314J/(molK)) ln(44.8L/22.4L)

= 5.76J/K

Example 3. The enthalpy of fusion for water is 6.01 kJ/mol. Calculate the entropy change for 1.0 mole of ice melting to form liquid at 273 K.

Solution:

This is a phase transition at constant pressure (assumed)

ΔS = (1 mol) (6010 J/mol)/273 K

= 22 J/K

Example 4. The contents of a large constant-temperature reservoir maintained at 500 K are continuously stirred by a paddle wheel driven by an electric motor. Estimate the entropy change of the reservoir if the paddle wheel is operated for two hours by a 250 W motor.

Solution:

Paddle wheel work converted into internal energy- an irreversible process. Imagine a reversible process with identical energy addition

S = ∫ (dQ/dt)_R = Q/T = 0.25 × 2(3600)/500 = 0.6 kJ

Example 5. Calculate entropy change if 1kg of water at 30⁰ C is heated to 80⁰C at 1 bar pressure. The specific heat of water is 4.2kJ/kg.K

Solution:

ΔS = C_p ln(T₂/T₁) = 4.2 × 103 × ln(273+80/273+30)

= 0.6415 kJ/kg.K

Entropy JEE Mains and Advanced Solved Questions

1. The process with negative entropy change is

1. dissolution of iodine in water
2. sublimation of dry ice
3. synthesis of ammonia from N₂ and H₂
4. dissociation of CaSO_4(s) to CaO_(s) and SO_3(g)

Solution:

N_2(g) + 3H₂ → 2NH_3(g)

∆s = 2 – 4 = – 2 < 0

So, entropy is negative.

Hence, option (3) is the answer.

2. Identify the correct statement regarding a spontaneous process

1. For a spontaneous process in an isolated system, the change in entropy is positive
2. Endothermic processes are never spontaneous
3. Exothermic processes are always spontaneous
4. Lowering of energy in the reaction process is the only criterion for spontaneity

Solution:

The change in entropy is positive for a spontaneous process in an isolated sys­tem.

Hence, option (1) is the answer.

3. For a particular reversible reaction at temperature T, ∆H and ∆S were found to be both +ve. If T_e is the temperature at equilibrium, the reaction would be spontaneous when

1. T_e > T
2. T > T_e
3. T_e is 5 times T
4. T_e = T

Solution:

At equilibrium, ∆G = 0.

∆G = ∆H – T∆S

For a reaction to be spontaneous ∆G should be negative.

So T > T_e

Hence, option (2) is the answer.

4. The entropy change involved in the isothermal reversible expansion of 2 moles of an ideal gas from a volume of 10 dm³ to a volume of 100 dm³ at 27°C is

1. 32.3 J mol^-1K^-1
2. 42.3 J mol^-1K^-1
3. 38.3 J mol^-1K^-1
4. 35.8 J mol^-1K^-1

Solution:

Given V₂ = 100

V₁ = 10

n = 2

∆S = 2.303nR log (V₁/V₂)

= 2.303 × 2 × 8.314 × log (100/10)

= 2.303 × 2 × 8.314 × log 10

= 38.29 J mol^-1 K^-1

Hence, option (3) is the answer.

5. An ideal gas undergoes isothermal expansion at constant pressure. During the process

1. enthalpy increases, but entropy decreases
2. enthalpy remains constant, but entropy increases
3. enthalpy decreases, but entropy increases
4. both enthalpy and entropy remain constant.

Solution:

During isothermal expansion at constant pressure, ∆H = nC_p ∆T = 0

∆S = nRln (V_f/V_i) > 0

Hence, option (2) is the answer.

6. The entropy of an isolated system can never __.

1. increase
2. decrease
3. be zero
4. none of the mentioned

Answer: 2

Explanation: The entropy of an isolated system always increases and remains constant only when the process is reversible.

7. Clausius summarised the first and second laws of thermodynamics as ___.

1. the energy of the world is constant
2. the entropy of the world tends towards a maximum
3. both of the above
4. none of the above

Solution:

Answer: 3

Explanation: Clausius gave these two statements.

Entropy Numericals

Solve the following Numericals based on Entropy

Q1. A body at 200^oC undergoes an reversible isothermal process. The heat energy removed in the process is 7875 J. Determine the change in the entropy of the body.

Q2. Air is compressed isothermally from 100 kPa to 800 kPa. Determine the change in specific entropy of the air.

Q3. A rigid insulated container holds 5 kg of an ideal gas. The gas is stirred so that its state changes from 5 kPa and 300 K to 15 kPa. Assuming C_p =1.0 kJ/kgK and g =1.4, determine the change of entropy of the system.

Entropy – Frequently Asked Questions(FAQs)

What is Entropy in Thermodynamics?

Entropy is a measure of disorder or randomness in a system. The higher the entropy, the more disordered the system.

What is Entropy and Enthalpy?

Entropy measures disorder, while enthalpy measures total heat content. They are related to thermodynamics and often describe energy changes in a system.

What is the Principle of Entropy?

The entropy principle states that in isolated systems, natural processes tend to increase entropy, moving towards greater disorder.

What is Unit of Entropy?

Entropy is measured in joules per kelvin (J/K).

What is Entropy of the Universe?

The entropy of the Universe tends to increase over time, reflecting the trend towards greater disorder in natural processes.

What is Spontaneity?

Spontaneity refers to processes that occur without external intervention. In thermodynamics, spontaneous processes often lead to increased entropy.

What is Order of Entropy of Matter?

Gases typically have higher entropy than liquids, and liquids have higher entropy than solids, reflecting the increasing disorder in these states.

Does Freezing Increase Entropy?

No, freezing generally decreases entropy as a substance transitions from a more disordered liquid state to a more ordered solid state.

Why is Entropy Constant at the Triple Point of Water?

At the triple point of water, all three phases (solid, liquid, gas) coexist in equilibrium, resulting in a constant entropy due to the unique conditions.

Can Entropy be Negative?

Entropy change is generally positive, indicating an increase in disorder. However, in specific processes, like crystallization, local decreases in entropy can occur while the overall system entropy increases.

What is Negentropy?

Negentropy is a measure of order or organization in a system. It’s the opposite of entropy, indicating a system’s tendency to move towards higher organization.

What is Entropy as Disorder?

Entropy as disorder refers to the tendency of a system to move from a state of order to disorder or randomness. It reflects the natural tendency of things to become more chaotic over time.