Carnot cycle & Gibbs Energy Change Spontaneous & Non Spontaneous Process

Carnot Cycle

The Carnot cycle is an idealised reversible heat engine cycle that establishes the maximum possible efficiency any engine operating between two heat reservoirs can achieve. It consists of four reversible processes performed on an ideal gas between two temperatures: a hot reservoir at absolute temperature T₁ and a cold reservoir at absolute temperature T₂ (T₁ > T₂). The cycle is usually represented on a P-V diagram as a closed loop made of two isothermal and two adiabatic (isentropic) processes.

The four reversible processes (A → B → C → D → A)

• Reversible isothermal expansion (A → B) at T₁

  The system absorbs heat q₁ from the hot reservoir while expanding isothermally at temperature T₁. For an ideal gas the internal energy change is zero for an isothermal process, so

  ΔE_AB = 0

  

  Heat absorbed (isothermal):

  q₁ = nR T₁ ln(V_B/V_A)

• Reversible adiabatic expansion (B → C)

  No heat is exchanged with surroundings, so q = 0 and the process is adiabatic and reversible (isentropic). The internal energy change equals the work done by the gas:

  ΔE_BC = nC_V(T₂ - T₁)

  W_BC = ΔE_BC

  
• Reversible isothermal compression (C → D) at T₂

  The system rejects heat q₂ to the cold reservoir while being compressed isothermally at T₂. Again, internal energy change is zero:

  ΔE_CD = 0

  

  Heat rejected (isothermal):

  q₂ = nR T₂ ln(V_D/V_C)

• Reversible adiabatic compression (D → A)

  No heat exchange; compression raises the temperature from T₂ back to T₁. The internal energy change equals work done on the gas:

  ΔE_DA = nC_V(T₁ - T₂)

  W_DA = ΔE_DA

Because the process is a complete cycle, the net internal energy change over one cycle is zero:

ΔE_cycle = 0

Work and heat over the whole cycle

The net work done by the engine in one cycle equals the difference between heat absorbed from the hot reservoir and heat rejected to the cold reservoir:

W_cycle = q₁ - q₂

Since q₁ and q₂ are given by isothermal expressions for an ideal gas:

q₁ = nR T₁ ln(V_B/V_A)

q₂ = nR T₂ ln(V_D/V_C)

For the Carnot cycle the adiabatic relations connect volumes so that the logarithmic ratios are equal in magnitude:

ln(V_B/V_A) = ln(V_C/V_D)

Using those relations we obtain the useful relationship between heats and temperatures:

q₁ / q₂ = T₁ / T₂

Therefore the net work can also be written as:

W_cycle = q₁ (1 - T₂/T₁)

Efficiency

The efficiency (fraction of absorbed heat converted to useful work) of any heat engine is defined as:

η = (work output) / (heat input) = W_cycle / q₁

For the Carnot engine, using the expression above:

η_Carnot = 1 - T₂ / T₁

This is the maximum possible efficiency of any heat engine operating between these two temperatures. No real engine working between the same two reservoirs can be more efficient than the Carnot engine.

Entropy in the Carnot cycle

For the two isothermal processes, entropy changes for the system are given by:

ΔS_{isothermal at T} = q_rev / T

Over a complete reversible Carnot cycle the net entropy change of the system is zero because entropy is a state function and the cycle returns to its initial state. The entropy exchanged with the reservoirs cancels:

ΔS_cycle = 0

This demonstrates that entropy is a state function and that reversibility and the Carnot construction give the fundamental temperature dependence of entropy transfer in reversible isothermal processes.

Gibbs Free Energy

Gibbs free energy (commonly called simply G) is a thermodynamic potential useful for predicting the spontaneity of processes at constant temperature and pressure. It combines enthalpy and entropy into a single quantity.

Definition and basic relation

The Gibbs free energy is defined as:

G = H - T S

For an initial state 1 and a final state 2:

G₁ = H₁ - T S₁

G₂ = H₂ - T S₂

Therefore the change in Gibbs free energy is:

ΔG = G₂ - G₁ = (H₂ - H₁) - T (S₂ - S₁)

ΔG = ΔH - T ΔS

This relation is known as the Gibbs-Helmholtz equation when used in different forms relating temperature derivatives; here it is the fundamental expression for ΔG at constant temperature.

Meaning of ΔG

• Maximum non-expansion work: The decrease in Gibbs free energy of a system during a process at constant temperature and pressure equals the maximum useful work other than expansion work that can be obtained from the process. In other words, ΔG is the available free energy for non-expansion work.
• First-law context at constant pressure:

  The first law for a process including expansion and non-expansion work is

  q = ΔU + w_expansion + w_{non-expansion}

  At constant pressure, w_expansion = P ΔV, so

  q = ΔH + w_{non-expansion}

• Reversible change at constant T and P:

  For a reversible process at constant temperature:

  ΔS = q_rev / T

  q_rev = T ΔS

  Combining with q = ΔH + w_{non-expansion} gives

  T ΔS = ΔH + w_{non-expansion}

  Rearranging:

  ΔG = ΔH - T ΔS = - w_{non-expansion}

  Thus ΔG equals minus the maximum non-expansion (useful) work obtainable reversibly from the process at constant T and P.

• Free energy and work:

  -ΔG = w_max (useful, non-expansion work)

• Electrical work (cells):

  If the useful work is electrical, w = n F E (electrical work = charge × potential), then

  -ΔG = n F E

  where n is the number of electrons transferred, F is Faraday's constant, and E is the cell EMF.

  For standard states (298 K, 1 bar):

  -ΔG° = n F E°

The heat evolved in a fuel cell is not completely converted into useful electrical work; some energy is necessarily lost as heat. The ratio ΔG / ΔH is often used to describe the thermodynamic efficiency of conversion of chemical energy into electrical work (the fraction of enthalpy change available as useful work).

Spontaneity in terms of Gibbs free energy

(1) Derivation from entropy considerations

For a system exchanging heat with its surroundings at constant temperature and pressure, the total entropy change is:

ΔS_total = ΔS_system + ΔS_surroundings

If q_P is the heat absorbed by the system at constant pressure, then the surrounding loses heat q_P, so:

ΔS_surroundings = - q_P / T

At constant pressure q_P = ΔH, therefore:

ΔS_surroundings = - ΔH / T

Thus:

ΔS_total = ΔS - ΔH / T

Multiplying both sides by T:

T ΔS_total = T ΔS - ΔH

Using ΔG = ΔH - T ΔS gives:

T ΔS_total = - ΔG

or

ΔG = - T ΔS_total

From this relation we obtain the spontaneity criteria at constant T and P:

• If ΔS_total > 0 then ΔG < 0 and the process is spontaneous.
• If ΔS_total = 0 then ΔG = 0 and the system is at equilibrium.
• If ΔS_total < 0 then ΔG > 0 and the forward process is non-spontaneous (the reverse may be spontaneous).

(2) Direct criteria from ΔG = ΔH - T ΔS

The Gibbs free energy equation combines two competing factors: the enthalpy term ΔH (energy factor) and the entropy term T ΔS (randomness factor). The sign of ΔG depends on their magnitudes and signs. The possibilities are:

• Both ΔH and TΔS negative (energy favours, entropy opposes): If |ΔH| > |TΔS| then ΔG < 0 and process is spontaneous; if |ΔH| < |TΔS| then ΔG > 0 and process is non-spontaneous; if equal, ΔG = 0 (equilibrium).
• Both ΔH and TΔS positive (energy opposes, entropy favours): If |ΔH| > |TΔS| then ΔG > 0 (non-spontaneous); if |ΔH| < |TΔS| then ΔG < 0 (spontaneous); if equal, ΔG = 0 (equilibrium).
• ΔH negative and TΔS positive (both favour): ΔG will be negative and the process is spontaneous at all temperatures.
• ΔH positive and TΔS negative (both oppose): ΔG will be positive and the process is non-spontaneous at all temperatures; the reverse process may be spontaneous.

Summarised criteria:

• If ΔG < 0 the process is spontaneous at the given T and P.
• If ΔG = 0 the system is at equilibrium; no net spontaneous change occurs.
• If ΔG > 0 the process is non-spontaneous in the forward direction at the given T and P.

Effect of temperature on spontaneity

The temperature appears explicitly in ΔG = ΔH - T ΔS. Changing temperature can change the relative importance of the entropy term and thus change spontaneity.

Endothermic reactions (ΔH > 0)

• If T ΔS is negative, both terms oppose spontaneity and the reaction is non-spontaneous.
• If T ΔS is positive, entropy favours the reaction. The reaction may be spontaneous at sufficiently high temperature if T ΔS > ΔH. This is called an entropy-driven process.
• Temperature effect summary:

  At low T: T ΔS < ΔH ⇒ ΔG > 0 ⇒ non-spontaneous.

  At intermediate T: T ΔS ≈ ΔH ⇒ ΔG ≈ 0 ⇒ near equilibrium or slow spontaneous behaviour.

  At high T: T ΔS > ΔH ⇒ ΔG < 0 ⇒ spontaneous and often faster.

This is why many endothermic reactions or processes are carried out at high temperature to make them spontaneous (or practically feasible).

Exothermic reactions (ΔH < 0)

• If T ΔS is positive, both terms favour spontaneity and the reaction will be spontaneous at all temperatures.
• If T ΔS is negative, entropy opposes spontaneity. The reaction may still be spontaneous if |ΔH| > |T ΔS|; such reactions are called enthalpy-driven.
• Temperature effect summary:

  At very high T: T ΔS may dominate and if T ΔS > |ΔH| then ΔG > 0 ⇒ non-spontaneous.

  At low T: T ΔS is smaller so |ΔH| > T ΔS ⇒ ΔG < 0 ⇒ spontaneous.

Standard free energy change and standard free energy of formation

Standard free energy change of a reaction (Δ_rG°) is the change in Gibbs free energy when reactants in their standard states are converted to products in their standard states (standard state usually 1 bar and specified temperature, commonly 298.15 K).

Δ_rG° can be calculated from standard free energies of formation of products and reactants:

Δ_rG° = Σ Δ_fG°(products) - Σ Δ_fG°(reactants)

Standard free energy of formation (Δ_fG°) of a compound is the free energy change when one mole of the compound is formed from its elements in their standard states under standard conditions. The Δ_fG° of elements in their standard state is taken as zero.

Summary

The Carnot cycle sets the upper limit on the efficiency of heat engines and illustrates the reversible exchange of heat and entropy between reservoirs. Gibbs free energy is the thermodynamic potential that determines spontaneity at constant temperature and pressure: ΔG < 0 indicates spontaneity, ΔG = 0 indicates equilibrium, and ΔG > 0 indicates a non-spontaneous forward process. ΔG also quantifies the maximum non-expansion work obtainable from a process and connects directly to electrical work through the relation -ΔG = n F E for electrochemical cells.