Entropy Calculations for Ideal Gases

Table of Contents
1. Entropy Calculations for Ideal Gases
2. Fundamental relations and starting point
3. Entropy differential for any substance
4. Ideal-gas simplification
5. Integration between two equilibrium states
6. Special cases
7. Notes on variables and units
8. Worked symbolic derivation (stepwise)
9. Example (symbolic)
10. Applications and remarks
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Entropy Calculations for Ideal Gases

Entropy is a state property that quantifies the degree of molecular disorder and the availability of energy for useful work. For closed systems undergoing simple compressible-work interactions, entropy changes can be obtained directly from thermodynamic relations. This section derives expressions for the entropy change of an ideal gas between two equilibrium states (T0, P0) and (T, P) and discusses special cases such as reversible adiabatic (isentropic) processes and the constant heat-capacity approximation.

Fundamental relations and starting point

The first law for a closed system is

dU = dQ + dW

For a simple compressible substance undergoing a reversible process with only PdV work, the combined first and second laws yield the Gibbs relation

dU = T dS - P dV

Using the definition of enthalpy H = U + PV and differentiating, we obtain

d(H - PV) = T dS - P dV

Expanding the left-hand side gives

dH - P dV - V dP = T dS - P dV     (4.18)

Entropy differential for any substance

Cancelling -P dV on both sides of equation (4.18) leads to a convenient Gibbs-Duhem form for the entropy differential

T dS = dH - V dP

or equivalently

dS = dH/T - (V/T) dP

Ideal-gas simplification

For an ideal gas, the enthalpy depends only on temperature. In differential form,

dH = C_p^ig dT     (4.19)

Substituting (4.19) into the entropy differential gives

dS = (C_p^ig / T) dT - (V / T) dP

dS = (C_p^ig / T) dT - (R / P) dP     (4.20)

Here the ideal-gas relation PV = RT (per mole) or p v = R T (per unit mass) has been used to substitute V/T = R/P (R is the appropriate gas constant: molar R or specific R depending on the chosen unit basis).

Integration between two equilibrium states

Integrate equation (4.20) between an initial state (T0, P0) and a final state (T, P) to obtain the entropy change of the ideal gas:

ΔS = ∫_T0^T (C_p^ig/T) dT - ∫_P0^P (R/P) dP

ΔS = ∫_T0^T (C_p^ig/T) dT - R ln(P/P0)     (4.21)

This expression gives the exact entropy change for an ideal gas when the temperature dependence of C_p^ig is known. If molar quantities are used, ΔS is molar entropy change; if specific quantities are used, Δs is specific entropy change. Care must be taken to use a consistent R (molar or specific) and C_p basis.

Special cases

1. Reversible adiabatic (isentropic) process

For a reversible adiabatic process, there is no heat transfer (dQ = 0) and hence the entropy change is zero for the closed system:

dS = 0

Therefore, for an isentropic change between two states, ΔS = 0 and the relation from (4.20) gives

(C_p^ig/T) dT = (R/P) dP

2. Constant heat-capacity approximation

If the ideal gas heat capacity is approximately constant over the temperature range of interest (C_p^ig ≈ constant), the temperature integral in (4.21) simplifies to a logarithm yielding a compact formula:

ΔS = C_p^ig ln(T/T0) - R ln(P/P0)     (4.22)

When expressed in terms of specific heats and the ratio of specific heats k = C_p/C_v, the isentropic relations for ideal gases can be written as

• T2/T1 = (P2/P1)^(k-1)/k
• P2/P1 = (V1/V2)^k
• T2/T1 = (V1/V2)^k-1

These are standard isentropic relations obtained by setting ΔS = 0 and using C_p - C_v = R.

Notes on variables and units

• Use molar properties (molar enthalpy, molar entropy, R = 8.314 J mol^-1 K^-1) when working per mole.
• Use specific properties (per unit mass enthalpy, specific entropy, R specific) when working per unit mass.
• Ensure consistency between the heat-capacity basis and the gas constant R.
• If C_p^ig is temperature-dependent, evaluate the integral ∫(C_p(T)/T) dT using appropriate polynomial or tabulated expressions for C_p(T).

Worked symbolic derivation (stepwise)

The following lines present the derivation in clear sequential steps.

T dS = dH - V dP.

For an ideal gas, dH = C_p^ig dT.

Substitute into the previous relation.

dS = (C_p^ig/T) dT - (V/T) dP.

Use the ideal-gas relation PV = RT to write V/T = R/P.

dS = (C_p^ig/T) dT - (R/P) dP.

Integrate between states (T0, P0) and (T, P):

ΔS = ∫_T0^T (C_p^ig/T) dT - ∫_P0^P (R/P) dP.

Evaluate the pressure integral: ∫_P0^P (R/P) dP = R ln(P/P0).

Thus the general expression is ΔS = ∫_T0^T (C_p^ig/T) dT - R ln(P/P0).

If C_p^ig is constant, carry out the temperature integral to obtain ΔS = C_p^ig ln(T/T0) - R ln(P/P0).

Example (symbolic)

Find the entropy change of an ideal gas between (T0, P0) and (T, P) using constant C_p approximation.

Start with equation (4.22).

ΔS = C_p^ig ln(T/T0) - R ln(P/P0).

Substitute numerical values if given, ensuring consistent units and whether quantities are molar or specific.

Applications and remarks

• The expressions derived are used in thermodynamic analysis of gas turbines, compressors, nozzles, and heat-exchange processes where ideal-gas behaviour is a reasonable approximation.
• When precise entropy values are required over wide temperature ranges, use temperature-dependent C_p(T) functions (polynomial fits or tables) and perform the integral numerically or analytically as appropriate.
• For real gases or processes involving phase change or chemical reaction, the ideal-gas relations no longer apply and more general thermodynamic relations or property tables must be used.

Summary: The entropy change for an ideal gas between two states is given exactly by ΔS = ∫(C_p(T)/T) dT - R ln(P/P0). Under the constant heat-capacity approximation this reduces to ΔS = C_p ln(T/T0) - R ln(P/P0). Reversible adiabatic processes are isentropic (ΔS = 0), yielding the standard isentropic relations between T, P and V.