Questions: Equations of State and Thermodynamic Properties

Question 1

An engineer needs to calculate the enthalpy change for steam undergoing compression in a turbine near its saturation curve. Which approach is most appropriate?

• A. Apply dh = cₚ dT with a constant specific heat, since steam is a vapor and all vapors behave as ideal gases
• B. Use the van der Waals equation, which is accurate enough for any engineering fluid
• C. Use steam tables (which encode a high-accuracy equation of state for water, pre-evaluated on a grid) because steam near saturation has a compressibility factor Z well below 1 and behaves strongly non-ideally
• D. Assume Z = 1 (ideal gas) and apply the ideal gas relation — steam is above its boiling point so it must be ideal

Answer: (C)

Steam near saturation is one of the clearest cases of non-ideal gas behavior: intermolecular attraction and molecular volume effects are significant, Z can be far from 1, and the simple ideal gas relations du = cᵥ dT and dh = cₚ dT do not hold. Steam tables encode highly accurate equations of state (IAPWS-IF97) pre-evaluated over a grid of temperature and pressure — they are the standard engineering tool precisely for this reason. The van der Waals equation (option B) is qualitatively correct but not accurate enough for engineering calculations. 'Above the boiling point' (option D) does not imply ideal behavior — it only means the fluid is a gas, not a liquid.

Question 2

Nitrogen at 300 K and 1 atm has a compressibility factor Z very close to 1. The same gas at 300 K and 300 atm has Z significantly different from 1. What is the practical implication for applying the ideal gas law at high pressure?

• A. The ideal gas law works better at high pressure because molecules are forced closer together, making behavior more uniform
• B. Z deviating from 1 at high pressure means the ideal gas assumptions are breaking down, and the simple relations du = cᵥ dT and dh = cₚ dT no longer hold accurately
• C. Z is always greater than 1 at high pressure for all gases, so the ideal gas law systematically underestimates volume
• D. Z deviating from 1 is a small rounding error that does not affect engineering calculations significantly

Answer: (B)

Z = Pv/(RT) = 1 exactly for an ideal gas. Deviation from 1 indicates that real intermolecular forces and molecular volume are influencing the gas behavior. When Z ≠ 1, the ideal gas assumption breaks down — which means that du = cᵥ dT (which requires (∂u/∂v)_T = 0) and dh = cₚ dT (which requires (∂h/∂P)_T = 0) are no longer valid. Engineers check the reduced temperature Tr = T/T_c and reduced pressure Pr = P/P_c using generalized correlations to determine whether the ideal gas assumption introduces acceptable error.

Question 3

Maxwell relations, derived from the fundamental thermodynamic relations (e.g., dh = T ds + v dP), allow engineers to calculate entropy changes from measurable P-v-T data without requiring direct calorimetric measurement.

• T. True
• F. False

Answer: True

The Maxwell relations are exact derivative equalities derived by applying the equality of mixed partial derivatives to the fundamental relations. For example, from dh = T ds + v dP, we get (∂T/∂P)_s = (∂v/∂s)_P, and from dg = −s dT + v dP we get (∂s/∂P)_T = −(∂v/∂T)_P. The last relation is particularly useful: it connects entropy (unmeasurable directly) to the derivative of specific volume with temperature (measurable from P-v-T data). This lets engineers compute entropy changes purely from equation-of-state data.

Question 4

Steam tables and refrigerant property charts are independent empirical lookup tables with no theoretical connection to equations of state — each entry should be separately measured by calorimetry.

• T. True
• F. False

Answer: False

Steam tables and refrigerant charts are equations of state pre-evaluated on a grid. For water, the IAPWS-IF97 standard equation uses a mathematical function of T and P (or T and v in the two-phase region) whose coefficients were fit to experimental data, and then the function is evaluated at thousands of T-P grid points to generate the tables. All properties in the table (h, s, u, v) are computed from this single underlying equation of state using the fundamental thermodynamic relations — they are not each independently measured. This is why all properties in a given table are thermodynamically consistent with each other.

Question 5

Why can't an engineer use du = cᵥ dT to calculate the internal energy change of a real gas at high pressure, and what additional information is needed to get the correct answer?

Model answer: The relation du = cᵥ dT is valid only for ideal gases, because it assumes (∂u/∂v)_T = 0 — that internal energy does not depend on volume at constant temperature. For an ideal gas this is true because there are no intermolecular interactions. For a real gas, molecules attract (or repel) each other, so expanding the volume at constant temperature requires work against those forces, changing u. The correct relation is du = cᵥ dT + [(∂u/∂v)_T] dv, where the second term is non-zero for real gases. Using a Maxwell relation, (∂u/∂v)_T = T(∂P/∂T)_v − P, which can be computed from the equation of state (P-v-T data). So the additional information needed is the equation of state for the specific fluid — whether van der Waals, Peng-Robinson, or a reference equation — to evaluate the departure from ideal behavior.

This is why engineers distinguish between 'ideal gas specific heat' tables (valid only at low pressure) and 'real fluid property tables' or 'enthalpy/entropy departure functions.' The departure functions quantify exactly how much h, s, and u deviate from the ideal gas predictions, computed from the equation of state via the fundamental relations. Process simulators like Aspen apply these departure functions automatically when users select a real-fluid thermodynamic model.