Thermodynamics

658 | ThermodynamicsEXAMPLE 12–4Verification of the Maxwell RelationsVerify the validity of the last Maxwell relation (Eq. 12–19) for steam at250°C and 300 kPa.Solution The validity of the last Maxwell relation is to be verified for steamat a specified state.Analysis The last Maxwell relation states that for a simple compressiblesubstance, the change in entropy with pressure at constant temperature isequal to the negative of the change in specific volume with temperature atconstant pressure.If we had explicit analytical relations for the entropy and specific volumeof steam in terms of other properties, we could easily verify this by performingthe indicated derivations. However, all we have for steam are tables ofproperties listed at certain intervals. Therefore, the only course we can taketo solve this problem is to replace the differential quantities in Eq. 12–19with corresponding finite quantities, using property values from the tables(Table A–6 in this case) at or about the specified state.Ps s 400 kPa s 200 (400 200) kPaPs? T? T 250°C? T 250°C(7.3804 7.7100) kJkg # K(400 200) kPa? a 0v0T b Pa 0v0T bP 300 kPac v 300°C v 200°C(300 200)°C d0.00165 m 3 /kg K 0.00159 m 3 /kg KP 300 kPa(0.87535 0.71643) m 3 kg(300 200)°Csince kJ kPa · m 3 and K °C for temperature differences. The two valuesare within 4 percent of each other. This difference is due to replacing thedifferential quantities by relatively large finite quantities. Based on the closeagreement between the two values, the steam seems to satisfy Eq. 12–19 atthe specified state.Discussion This example shows that the entropy change of a simple compressiblesystem during an isothermal process can be determined from aknowledge of the easily measurable properties P, v, and T alone.12–3 ■ THE CLAPEYRON EQUATIONThe Maxwell relations have far-reaching implications in thermodynamicsand are frequently used to derive useful thermodynamic relations. TheClapeyron equation is one such relation, and it enables us to determine theenthalpy change associated with a phase change (such as the enthalpy ofvaporization h fg ) from a knowledge of P, v, and T data alone.Consider the third Maxwell relation, Eq. 12–18:a 0P0T b a 0sv 0v b TDuring a phase-change process, the pressure is the saturation pressure,which depends on the temperature only and is independent of the specific