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.
Understanding Entropy Change and the Tds Relations
The Tds relations are most useful when a problem gives properties that are easy to measure, such as temperature, pressure, or volume, but does not list entropy directly. They come from combining the energy statement for a simple compressible substance with the second law. One form is usually convenient when internal energy and volume are the natural variables.
The other is convenient when enthalpy and pressure are known. This is why the enthalpy form appears often in flow devices such as turbines, compressors, nozzles, and heat exchangers.
Temperature in these relations must be an absolute temperature. Celsius values cannot be used directly in temperature ratios or entropy calculations.
For an ideal gas, entropy responds to both temperature change and pressure or volume change. Heating a gas tends to raise its entropy because molecular energy spreads over more possible microscopic arrangements. Compressing a gas tends to lower its entropy because the gas occupies less space.
The logarithms in the ideal gas formulas matter because entropy depends on relative changes, not simple temperature or pressure differences. A temperature ratio must be dimensionless and positive. Students should choose one ideal gas form based on the data available.
Use the pressure form when temperatures and pressures are given. Use the volume form when temperatures and specific volumes are given. The gas constant, specific heat at constant pressure, and specific heat at constant volume must all use compatible units.
Liquids and solids are often treated as incompressible because their volumes change very little under ordinary pressure changes. In that model, temperature is the main cause of entropy change. This works well for water flowing through a pump, liquid coolant in a pipe, or a metal part being heated.
It becomes less reliable near a phase change, near a critical point, or across a very large temperature range where specific heat changes noticeably. Steam tables and property software are needed when water changes from liquid to vapor or when a substance is not close to ideal behavior. Property tables provide entropy values directly, which can be safer than forcing a simple approximation onto a complicated state.
An isentropic process is an ideal benchmark, not a guarantee for real equipment. A reversible adiabatic process has no heat transfer and no internal sources of entropy. Real turbines, compressors, valves, and pipes have friction, turbulence, pressure drops, or mixing.
These effects create entropy even when the device is insulated. Engineers compare an actual exit state with an ideal isentropic exit state to judge how much performance is lost.
A useful learning habit is to identify the system first, then list the known state properties, then decide whether the process is ideal gas, incompressible, phase changing, reversible, or irreversible. This prevents using an isentropic relation merely because a device is insulated.