Hafez a radi, john o rasmussen auth principles of physics for scientists and engineers 19

13.2 Molar Specific Heat Capacity of an Ideal Gas 439 Freely moving piston Fig 13.6 Insulation P P P P P Vf Tf QP P V i Ti Heat reservoir at T i Initial Heat reservoir at Tf Final Solution: (a) Treating the helium as an ideal monatomic gas undergoing a constant-pressure process, we use CP = 20.8 J/mol.K and T = 20 C◦ = 20 K in Eq 13.20 to get: QP = nCP T = n(5R/2) T = (5 mol)(20.8 J/mol.K)(20 K) = 2,080 J (b) Even though the temperature of the helium increases at a constant pressure (not at constant volume), we use Eq 13.23 to calculate the change in internal energy when CV = 12.5 J/mol.K as follows: Eint = nCV T = n(3R/2) T = (5 mol)(12.5 J/mol.K)(20 K) = 1,250 J (c) From the first law of thermodynamics, Eint = QP − W, we calculate the work done by the gas in this process as follows: W = QP − Eint = 2,080 J − 1,250 J = 830 J Of all the heat energy QP = 2,080 J that is transferred to the helium during the increase in temperature, only 1,250 J goes to increasing the helium’s internal energy and hence its temperature The remaining 830 J is transferred out of the helium as work done during the expansion Internal Energy of a Diatomic Ideal Gas In a diatomic ideal-gas model, a molecule can rotate about two different axes, while the rotation about the third axis passing through the two atoms gives very little energy because the moment of inertia about this axis is very small, see Fig 13.7 Therefore, a diatomic gas is said to have five energy degrees of freedom: three translational and two rotational According to the principle of equipartition of energy, 440 13 Kinetic Theory of Gases each active degree of freedom of a molecule has on average an energy equal to 21 kB T Thus, the average energy for a molecule in a diatomic gas is: E = 25 kB T (Diatomic ideal gas) (13.34) Very small moment of inertia Axis Axis Axis Fig 13.7 A diatomic molecule can rotate about two perpendicular axes with appreciable rotational energy while the rotation about the third axis gives very little rotational energy (i.e only two degrees of freedom) Hence, the internal energy Eint of a diatomic ideal gas of N (or n kmol) at pressure P, volume V, and temperature T will be: Eint = NkB T nRT (Diatomic ideal gas) (13.35) In general, the internal energy of an ideal gas is a function of T only, and the exact relationship depends on the type of gas The vibrational (kinetic and potential) degrees of freedom have only a tiny effect on Eqs 13.34 and 13.35 unless the temperature is extremely high Quantum mechanical study (which is not our aim) predicts discrete vibrational levels with spacing generally much larger than kB T We can use the above results and Eq 13.25 to find CV and CP as follows: d dEint = n dT n dT CP = CV + R = 27 R CV = nRT = 25 R (Diatomic ideal gas) (13.36) Table 13.2 displays the measured molar specific heats of some gases These results are in good agreement with the predicted CV and CP The small deviations from the predicted values are due to the fact that real gases are not ideal gases Real gases experience weak intermolecular interactions, which are not addressed in the presented ideal gas model 13.3 Distribution of Molecular Speeds 441 Table 13.2 Some molar specific heats of various gases at 15 ◦ C Molar specific heat (J/mol C◦ ) Cp CV CP − C V γ = CP /CV He 20.8 12.5 8.33 1.67 Ar 20.8 12.5 8.33 1.67 Ne 20.8 12.7 8.12 1.64 Kr 20.8 12.3 8.49 1.69 H2 20.8 20.4 8.33 1.41 N2 29.1 20.8 8.33 1.40 O2 29.4 21.1 8.33 1.40 CO 29.3 21.0 8.33 1.40 Cl2 34.7 25.7 8.96 1.35 CO2 37.0 28.5 8.50 1.30 SO2 40.4 31.4 9.00 1.29 H2 O 35.4 27.0 8.37 1.30 Gas Monatomic gases Diatomic gases Triatomic gases 13.3 Distribution of Molecular Speeds Molecules in a gas at thermal equilibrium are assumed to be in random motion, i.e they have a wide range of molecular speeds In 1859, James Clerk Maxwell derived an expression that describes the distribution of speeds in a gas containing N molecules in thermal equilibrium at temperature T The number of molecules dN with speeds in the range v and v + dv is defined by the distribution function f (v) (known as Maxwell-Boltzmann distribution) through the following relation: dN = f (v) dv = 4π N m 2π kB T 3/2 v e− mv /k BT dv (13.37) where m is the mass of a gas molecule, kB is Boltzmann’s constant, and T is the absolute gas temperature A sketch of the distribution function f (v) is shown in Fig 13.8 at a certain temperature T The average speed v¯ can be obtained by integrating the product of the speed v with dN and dividing by the total number N In addition, one can find vrms and the most probable speed vp as follows: 442 13 Kinetic Theory of Gases v¯ = vrms = N ∞ kB T π m vf (v) dv = N v2 = vp = ∞ v f (v) dv = (13.38) 3kB T m 2kB T m (13.39) (13.40) From these results, we see that vrms > v¯ > vp as displayed in Fig 13.8 f( ) T p rms Fig 13.8 Distribution of speeds of an ideal gas, f (v) The area under the curve gives the total number of gas molecules The speed at the peak vp is less that v¯ and vrms because f (v) is skewed to the right of the peak 13.4 Non-ideal Gases and Phases of Matter The isothermal PV-diagram presented in Sect 12.4 and Sect 13.2 can help us grasp the overall behavior of a gas described by the gas law PV = nRT The PV-isotherm for a constant amount of an ideal gas is displayed for temperatures T4 > T3 > T2 > T1 in Fig 13.9a Notice that the pressure P is inversely proportional to V and that the isotherms are hyperbolic curves Fig 13.9b displays a PV-diagram for a substance that does not obey the ideal gas law for temperatures T4 > T3 > Tc > T2 > T1 The solid curve at T4 represents the behavior of the non-ideal gas, while the dashed curve represents the behavior predicted by the ideal gas law at the same temperature The curves at T3 and Tc deviate more from the dashed curves predicted by the ideal gas law At successively lower temperatures T2 and T1 (both below Tc ), the behavior deviates even more from the curves of part a, and the isotherms develop flat regions in which the substance can be compressed without an increase in its pressure Observation shows that the non-ideal gas is condensing from vapor (gas) to a liquid state The colored region represents isotherms in their liquid-vapor phase equilibrium A transition from one phase to 13.4 Non-ideal Gases and Phases of Matter 443 another requires phase equilibrium between the two phases This occurs at only one definite temperature for a given pressure value P P Ideal gas Non-ideal gas T4 > T3 > Tc > T2 > T1 T4 > T3 > T2 > T1 Isotherms Isotherms T4 T3 T4 c b T2 T1 V Liquid T3 Tc a Vapor Liquid-vapor (a) T2 T1 V (b) Fig 13.9 (a) An isothermal P-V diagram of an ideal gas for temperatures T4 > T3 > T2 > T1 (b) An isothermal P-V diagram for a non-ideal gas for temperatures T4 > T3 > Tc > T2 > T1 Kinetic theory can help us understand this behavior if we note that at higher pressures, and particularly at lower temperatures, the attractive potential energy due to attractive forces between molecules cannot be ignored as in the case of an ideal gas These attractive forces at lower temperatures tend to pull the molecules and cause liquefaction The curve at Tc in Fig 13.9b represents the substance at its critical temperature, and point c on the curve is called the critical point When we compress the gas at constant temperature T2 (