Entropy change is a central thermodynamics tool for analyzing heat transfer, irreversibilities, and the direction of real processes. This cheat sheet summarizes how engineers compute entropy changes for pure substances, ideal gases, and incompressible materials. It is useful because entropy often connects property data, energy balances, and second-law performance in one calculation.
The Tds relations provide the bridge between measurable properties and entropy changes.
Key Facts
- The first Tds relation is T ds = du + P dv, where T is absolute temperature, s is specific entropy, u is specific internal energy, P is pressure, and v is specific volume.
- The second Tds relation is T ds = dh - v dP, where h is specific enthalpy and P is pressure.
- For an ideal gas with constant specific heats, the entropy change is s2 - s1 = cp ln(T2/T1) - R ln(P2/P1).
- For an ideal gas with constant specific heats, the entropy change can also be written as s2 - s1 = cv ln(T2/T1) + R ln(v2/v1).
- For an incompressible substance with constant specific heat, the entropy change is s2 - s1 = c ln(T2/T1).
- For a reversible adiabatic process, the entropy change is zero, so s2 = s1 and the process is isentropic.
- For an internally reversible process, entropy transfer with heat is ds = delta qrev/T or, for a finite process, delta s = integral(delta qrev/T).
- Entropy is a property, so the entropy change between two equilibrium states depends only on the end states, not on the process path.
Vocabulary
- Entropy
- A thermodynamic property that measures energy dispersal and helps determine whether a process can occur naturally.
- Tds relation
- A thermodynamic property equation that relates entropy change to changes in internal energy, enthalpy, volume, pressure, and temperature.
- Isentropic process
- A process with constant entropy, usually modeled as reversible and adiabatic.
- Ideal gas
- A gas model in which Pv = RT and internal energy and enthalpy depend only on temperature.
- Incompressible substance
- A substance whose specific volume is approximately constant, such as many liquids and solids.
- Specific heat
- The energy required to raise the temperature of a unit mass of a substance by one degree.
Common Mistakes to Avoid
- Using Celsius in logarithmic entropy formulas is wrong because ratios such as T2/T1 must use absolute temperature in kelvin or rankine.
- Calling every adiabatic process isentropic is wrong because adiabatic means no heat transfer, while isentropic also requires no internal irreversibilities.
- Mixing cp and cv in ideal gas formulas is wrong because cp belongs with pressure ratios and cv belongs with volume ratios in the constant-specific-heat forms.
- Treating entropy as path dependent is wrong because entropy is a property, although heat transfer and entropy generation do depend on the process path.
- Ignoring units for R, cp, and cv is wrong because inconsistent units can make the logarithmic entropy change numerically incorrect.
Practice Questions
- 1 Air is modeled as an ideal gas with cp = 1.005 kJ/kg·K and R = 0.287 kJ/kg·K. Find s2 - s1 when T1 = 300 K, P1 = 100 kPa, T2 = 600 K, and P2 = 500 kPa.
- 2 Liquid water is approximated as incompressible with c = 4.18 kJ/kg·K. Find the specific entropy change when it is heated from 20°C to 80°C.
- 3 An ideal gas undergoes a constant-volume process from T1 = 400 K to T2 = 800 K with cv = 0.718 kJ/kg·K. Calculate s2 - s1.
- 4 Explain why a turbine that is adiabatic but has friction cannot be considered isentropic, even though no heat crosses its boundary.