Unit 2: Entropy and Thermodynamic Potentials

Concept of Entropy

Entropy is a fundamental thermodynamic quantity that serves as a quantitative measure of molecular disorder, randomness, or the unavailability of a system's thermal energy for conversion into mechanical work. Conceptually, it determines the direction of spontaneous thermal processes.

Thermodynamic Definition: For an infinitesimal reversible process, the change in entropy (dS) of a system is defined as the ratio of the heat exchanged (dQ_rev) to the absolute temperature (T) at which the exchange occurs:
dS = dQ_rev / T

Where:

• dS is the change in entropy (measured in Joules per Kelvin, J/K).
• dQ_rev is the reversible heat added or removed.
• T is the absolute temperature in Kelvin.

Physical Significance of Entropy

At the microscopic level, Boltzmann related entropy to the number of microstates (W) corresponding to a macrostate:

S = k * ln(W)

Where k is the Boltzmann constant. This shows that higher entropy directly translates to greater statistical molecular disorder.

Second Law of Thermodynamics in Terms of Entropy

While the Kelvin-Planck and Clausius statements of the Second Law describe limits on heat engines and heat flow, entropy provides a unified mathematical statement of this law.

Statement: The entropy of an isolated system always increases during any spontaneous (irreversible) process and remains constant during a reversible process.

Mathematically, for any process in an isolated system:

dS >= 0

If the system is not isolated and exchanges heat with its surroundings, the Second Law dictates that the total change in entropy of the universe (system + surroundings) must be non-negative:

dS_universe = dS_system + dS_surroundings >= 0

Entropy of a Perfect Gas

To calculate the entropy of a perfect (ideal) gas, we use the First Law of Thermodynamics and the ideal gas equation of state.

For 1 mole of an ideal gas, the First Law of Thermodynamics is:

dQ = dU + PdV

Since dU = C_v * dT (where C_v is the molar heat capacity at constant volume) and the process is reversible, we substitute dQ = TdS:

TdS = C_v * dT + PdV

Dividing by T:

dS = C_v * (dT / T) + (P / T) * dV

From the ideal gas equation, PV = RT, which gives P / T = R / V. Substituting this:

dS = C_v * (dT / T) + R * (dV / V)

Integrating both sides gives the entropy of a perfect gas in terms of temperature and volume:

S = C_v * ln(T) + R * ln(V) + S_0

Where S_0 is the constant of integration.

Alternative Expressions

We can express entropy in terms of other combinations of state variables using the relation PV = RT:

Note: C_p is the molar heat capacity at constant pressure, and C_p - C_v = R.

Entropy Changes in Reversible and Irreversible Processes

The behavior of entropy differs fundamentally depending on whether a process is reversible or irreversible.

1. Reversible Processes

During a reversible process, the system is always in thermodynamic equilibrium with its surroundings. The heat lost by one part equals the heat gained by another, keeping total entropy constant.

Example: Reversible Isothermal Expansion of an Ideal Gas

A gas expands slowly in contact with a heat reservoir at temperature T, absorbing heat Q_rev.

• Change in system entropy: dS_system = Q_rev / T
• Change in surroundings entropy: dS_surroundings = -Q_rev / T
• Total change: dS_universe = dS_system + dS_surroundings = Q_rev / T - Q_rev / T = 0

2. Irreversible Processes

All natural processes are spontaneous and irreversible. These processes always generate entropy, leading to a net increase in the entropy of the universe.

Example A: Heat Conduction (Thermal Equilibrium)

Consider heat Q flowing from a hot reservoir at temperature T1 to a cold reservoir at temperature T2 (where T1 > T2).

• Entropy change of the hot reservoir: dS_hot = -Q / T1
• Entropy change of the cold reservoir: dS_cold = Q / T2
• Total entropy change: dS_universe = Q / T2 - Q / T1 = Q * (T1 - T2) / (T1 * T2)

Since T1 > T2, the term (T1 - T2) is positive, which means dS_universe > 0.

Example B: Free Expansion of an Ideal Gas

A gas is confined to one half of an insulated container, with the other half evacuated. When the partition is removed, the gas undergoes free expansion to fill the entire container without doing work (W = 0) or exchanging heat (Q = 0).

• Because the container is insulated, dS_surroundings = 0.
• To find the change in the system, we choose a reversible isothermal path between the same initial and final volumes (V1 to V2). The entropy change is dS_system = R * ln(V2/V1).
• Total change: dS_universe = R * ln(V2/V1) > 0 (since V2 > V1).

Principle of Increase of Entropy and Entropy of the Universe

The Principle of Increase of Entropy states that the entropy of any isolated system, including the universe as a whole, must increase over time during any spontaneous process.

dS_universe = dS_system + dS_surroundings >= 0

If a process is completely reversible, the entropy of the universe remains constant. Since all actual processes occurring in nature are irreversible (due to friction, heat transfer across finite temperature differences, unrestrained expansion, etc.), the entropy of the universe is continually increasing.

Physical Consequence (The Heat Death of the Universe): As the entropy of the universe increases toward a maximum value, the energy of the universe becomes uniformly distributed as degraded thermal energy. Once maximum entropy is reached, no useful work can be extracted, and all physical processes will cease.

Temperature-Entropy (T-S) Diagrams for Carnot Cycle

A Temperature-Entropy (T-S) diagram plots absolute temperature (T) on the vertical axis and entropy (S) on the horizontal axis. This diagram is highly useful because the area under any curve represents the heat transferred during a reversible process:

Q_rev = Integral(T dS)

In a T-S diagram, the four stages of a Carnot Cycle form a perfect rectangle, simplifying its analysis compared to a P-V indicator diagram.

Analysis of the Carnot Cycle on a T-S Diagram

1. Isothermal Expansion (Step 1 to 2): The working substance absorbs heat Q1 reversibly from a hot reservoir at constant high temperature T1. The entropy increases from S1 to S2.
   Representation: A horizontal line from (S1, T1) to (S2, T1).
   Heat absorbed: Q1 = T1 * (S2 - S1).
2. Adiabatic Expansion (Step 2 to 3): The substance expands adiabatically and reversibly (isentropically). Heat exchange is zero (dQ = 0), so entropy remains constant at S2. The temperature drops from T1 to T2.
   Representation: A vertical line downwards from (S2, T1) to (S2, T2).
3. Isothermal Compression (Step 3 to 4): The substance rejects heat Q2 reversibly to a cold sink at constant low temperature T2. The entropy decreases from S2 back to S1.
   Representation: A horizontal line from (S2, T2) to (S1, T2).
   Heat rejected: Q2 = T2 * (S2 - S1).
4. Adiabatic Compression (Step 4 to 1): The substance is compressed adiabatically and reversibly. Entropy remains constant at S1. The temperature increases from T2 back to T1.
   Representation: A vertical line upwards from (S1, T2) to (S1, T1).

Work Done and Efficiency from T-S Diagram

The work done (W) is equal to the net heat absorbed in the cycle, which is represented by the area of the rectangle on the T-S diagram:

W = Q1 - Q2 = T1 * (S2 - S1) - T2 * (S2 - S1) = (T1 - T2) * (S2 - S1)

The efficiency (eta) is the ratio of work done to heat absorbed:

eta = W / Q1 = [(T1 - T2) * (S2 - S1)] / [T1 * (S2 - S1)] = (T1 - T2) / T1 = 1 - T2 / T1

This shows that the efficiency depends solely on the operating temperatures of the source and the sink.

Third Law of Thermodynamics and Unattainability of Absolute Zero

The Third Law of Thermodynamics deals with the behavior of systems as their temperature approaches absolute zero (0 Kelvin).

The Third Law (Nernst Heat Theorem Statement)

Statement: The entropy of a pure, perfectly crystalline substance at absolute zero temperature (0 K) is exactly equal to zero.

Mathematically:

lim (T -> 0) S = 0

For any physical or chemical transformation of a condensed system at absolute zero, the change in entropy approaches zero:

lim (T -> 0) delta S = 0

Unattainability of Absolute Zero

The Unattainability of Absolute Zero is an alternative statement of the Third Law. It states:

It is impossible by any procedure, no matter how idealized, to reduce the temperature of any system to absolute zero in a finite number of steps.

Explanation: Cooling a system involves alternating between isothermal steps (reducing entropy by applying an external field or parameter change) and adiabatic steps (reducing temperature by removing the field). Near absolute zero, the entropy curves for different parameters (such as magnetic fields B1 and B2) converge to the same value (zero) as T approaches 0 K. Because of this convergence, any adiabatic step starting from a non-zero temperature will always land on a temperature strictly greater than zero, requiring an infinite number of steps to reach absolute zero.

Thermodynamic Potentials

Thermodynamic potentials are state functions used to describe the thermodynamic state of a system and determine equilibrium conditions under various external constraints.

1. Internal Energy (U)

• Definition: The total microscopic kinetic and potential energy of all particles within a system.
• Fundamental Relation: From the First and Second Laws: dU = TdS - PdV
• Natural Variables: Entropy (S) and Volume (V).
• Properties:
  • State function and extensive property.
  • For an isochoric (dV=0) and isentropic (dS=0) process, internal energy is conserved or minimized at equilibrium.
• Applications: Used to define the first law of thermodynamics; acts as the base potential from which other potentials are derived.

2. Enthalpy (H)

• Definition: The total heat content of a system, equal to internal energy plus the product of pressure and volume.

  H = U + PV

• Fundamental Relation:
  dH = dU + d(PV) = (TdS - PdV) + (PdV + VdP)

  dH = TdS + VdP

• Natural Variables: Entropy (S) and Pressure (P).
• Properties:
  • State function and extensive property.
  • For an isobaric (dP=0) and isentropic (dS=0) process, enthalpy remains constant.
• Applications: Extremely useful in studying steady-flow processes (like turbines or compressors) and chemical reactions at constant pressure, where the change in enthalpy equals the heat exchanged (dQ = dH).

3. Helmholtz Free Energy (F)

• Definition: The portion of internal energy available to do work at a constant temperature.

  F = U - TS

• Fundamental Relation:
  dF = dU - d(TS) = (TdS - PdV) - (TdS + SdT)

  dF = -SdT - PdV

• Natural Variables: Temperature (T) and Volume (V).
• Properties:
  • State function and extensive property.
  • During an isothermal (dT=0) and isochoric (dV=0) process, F decreases and reaches a minimum at thermodynamic equilibrium.
  • The decrease in Helmholtz free energy represents the maximum work obtainable from a system at constant temperature: dW_max = -dF.
• Applications: Useful in closed systems kept at constant volume and temperature, such as gas reactions inside rigid containers.

4. Gibbs Free Energy (G)

• Definition: The maximum amount of non-expansion (useful) work that can be extracted from a closed system at constant temperature and pressure.

  G = H - TS

• Fundamental Relation:
  dG = dH - d(TS) = (TdS + VdP) - (TdS + SdT)

  dG = -SdT + VdP

• Natural Variables: Temperature (T) and Pressure (P).
• Properties:
  • State function and extensive property.
  • For processes occurring at constant temperature (dT=0) and pressure (dP=0), G decreases during spontaneous changes and is minimized at equilibrium (dG <= 0).
  • The decrease in Gibbs free energy represents maximum non-expansion work (such as electrical or chemical work): dW_useful = -dG.
• Applications: Widely used in chemistry, biology, and materials science because most natural processes and laboratory experiments occur at constant atmospheric pressure and ambient temperature. It is the core potential for predicting phase transitions and chemical reaction spontaneity.

Comparison and Summary of Thermodynamic Potentials