THE IDEAL-GAS EQUATION OF STATE • Equation of state: Any equation that relates the pressure, temperature, and specific volume of a substance. • The simplest : ideal-gas equation of state. This equation predicts the P-v-T behavior of a gas quite accurately within some properly selected region. Ideal gas equation of state Note , P is absolute pressure T is absolute temperature R: gas constant M: molar mass (kg/kmol) Different substances have different Ru: universal gas constant gas constants. 1 Mass = Molar mass Mole number Real gases behave as an ideal Different forms of equation of state gas at low densities (i.e., low Pv= RT pressure, high PV=mRT temperature). PV= NRuT Ideal gas equation at two states for a fixed mass P1V1=mRT1 P2V2=mRT2 The ideal-gas relation often is not applicable to real gases; thus, care should be exercised when using it. 2 When the Ideal gas equation is applicable • The pressure is small compared to the critical pressure P< Pcr • The temperature is twice the critical temperature and pressure is less than 10 times of the critical pressure. T=2Tcr and P<10Pcr 3 Is Water Vapor an Ideal Gas? • YES: at pressures below 10 kPa. • NO: in actual application (higher pressure environment) such as Steam Power Plant 4 4-7 COMPRESSIBILITY FACTOR—A MEASURE OF DEVIATION FROM IDEAL-GAS BEHAVIOR Compressibility factor Z Z bigger than unity, the more the gas deviates A factor that accounts for from ideal-gas behavior. the deviation of real gases Gases behave as an ideal gas at low densities from ideal-gas behavior at (i.e., low pressure, high temperature). a given temperature and pressure. Pvreal =ZRT Pvideal =RT vreal/videal = Z % error= [(videal-vreal)/vreal]*100 5 Generalized compressibility chart. We can determine compressibility factor Z for normalized pressure and temperature from Generalized Compressibility Chart Here PR is called the reduced pressure and TR the reduced temperature. We can find Pcr and Tcr from Table A1. Z values for PR and TR from Generalized Compressibility Chart A-28 like below: FIGURE A–28 Nelson–Obert generalized compressibility chart. 6 1. Van der Waals Equation of State Van der Waals includes two effects not considered in the ideal-gas Eq. a/v2 is intermolecular forces, and b is volume occupied by the gas molecules. The constants a and b can be determined for any substance from the critical point data alone (Table A–1). 2. Beattie-Bridgeman Equation of State The Beattie-Bridgeman equation of state based on five experimentally determined constants. The values of the constants appearing in this equation are given in Table 3–4. 7 3. Benedict-Webb-Rubin Equation of State Benedict, Webb, and Rubin extended the Beattie-Bridgeman equation by raising the number of constants to eight. The values of the constants appearing in this equation are given in Table 3–4. 8 • 3–84 Determine the specific volume of superheated water • vapor at 15 MPa and 350°C, using (a) the ideal-gas equation, • (b) the generalized compressibility chart, and (c) the steam • tables. Also determine the error involved in the first two cases. 9 • 3–84 Determine the specific volume of superheated water • vapor at 15 MPa and 350°C, using (a) the ideal-gas equation, • (b) the generalized compressibility chart, and (c) the steam • tables. Also determine the error involved in the first two cases. 15 0.68 623 0.96 % error= [(videal-vreal)/vreal]*100 10 .