Sample problems chapter 21 20. A 1.00-mol sample of a diatomic ideal gas has pressure P and volume V. When the gas is heated, its pressure triples and its volume doubles. This heating process includes two steps, the first at constant pressure and the second at constant volume. Determine the amount of energy transferred to the gas by heat. P21.20 Q = ( nC P ∆T )isobaric + ( nCV ∆T )isovolum etric In the isobaric process, V doubles so T must double, to 2Ti. In the isovolumetric process, P triples so T changes from 2Ti to 6Ti. 7 5 Q = n R ( 2Ti − Ti) + n R ( 6Ti − 2Ti) = 13.5nRTi = 13.5PV 2 2 25. A 2.00-mol sample of a diatomic ideal gas expands slowly and adiabatically from a pressure of 5.00 atm and a volume of 12.0 L to a final volume of 30.0 L. (a) What is the final pressure of the gas? (b) What are the initial and final temperatures? (c) Find Q, W, and ∆Eint. P21.25 (a) γ γ PV i i = PfV f γ 1.40 V 12.0 = 1.39 atm Pf = Pi i = 5.00 atm 30.0 Vf (b) ( Tf = (c) )( 5.00 1.013 × 105 Pa 12.0 × 10−3 m PV i i = Ti = nR 2.00 m ol( 8.314 J m ol⋅ K ) PfV f nR = ( )( 3 1.39 1.013 × 105 Pa 30.0 × 10−3 m 2.00 m ol( 8.314 J m ol⋅ K ) )= 3 )= 365 K 253 K The process is adiabatic: Q = 0 γ = 1.40 = C P R + CV 5 = , CV = R CV CV 2 5 ∆Eint = nC V ∆T = 2.00 m ol ( 8.314 J m ol⋅ K ) ( 253 K − 365 K ) = −4.66 kJ 2 W = ∆Eint − Q = −4.66 kJ− 0 = −4.66 kJ 51. The function Eint = 3.50nRT describes the internal energy of a certain ideal gas. A sample comprising 2.00 mol of the gas always starts at pressure 100 kPa and temperature 300 K. For each one of the following processes, determine the final pressure, volume, and temperature; the change in internal energy of the gas; the energy added to the gas by heat; and the work done on the gas. (a) The gas is heated at constant pressure to 400 K. (b) The gas is heated at constant volume to 400 K. (c) The gas is compressed at constant temperature to 120 kPa. (d) The gas is compressed adiabatically to 120 kPa. P21.51 (a) Pf = 100 kPa Vf = nRT f Pf = T f = 400 K 2.00 m ol( 8.314 J m ol⋅ K ) ( 400 K ) 100 × 103 Pa = 0.066 5 m 3 = 66.5 L ∆Eint = ( 3.50) nR ∆T = 3.50( 2.00 m ol) ( 8.314 J m ol⋅ K ) ( 100 K ) = 5.82 kJ W = − P∆V = − nR ∆T = − ( 2.00 m ol) ( 8.314 J m ol⋅ K ) ( 100 K ) = −1.66 kJ Q = ∆Eint − W = 5.82 kJ+ 1.66 kJ= 7.48 kJ (b) T f = 400 K V f = Vi = nRTi 2.00 m ol( 8.314 J m ol⋅ K ) ( 300 K ) = = 0.049 9 m Pi 100 × 103 Pa Tf 400 K Pf = Pi = 100 kPa = 133 kPa 300 K T i V = constant 3 = 49.9 L W = − ∫ PdV = 0 since ∆Eint = 5.82 kJ as in part (a) Q = ∆Eint − W = 5.82 kJ− 0 = 5.82 kJ (c) Pf = 120 kPa T f = 300 K P 100 kPa V f = V i i = 49.9 L = 41.6 L 120 kPa P f ∆Eint = ( 3.50) nR ∆T = 0 since T = constant Vf W = − ∫ PdV = − nRTi ∫ Vi P V f dV = − nRTiln = − nRTiln i V Vi Pf 100 kPa W = − ( 2.00 m ol) ( 8.314 J m ol⋅ K ) ( 300 K ) ln = +909 J 120 kPa Q = ∆Eint − W = 0 − 910 J= −909 J (d) γ = Pf = 120 kPa γ C P CV + R 3.50R + R 4.50 9 = = = = CV CV 3.50R 3.50 7 P V f = V i i Pf γ PfV f = PiV i : so 1γ 100 kPa = 49.9 L 120 kPa 79 = 43.3 L PfV f 120 kPa 43.3 L = 300 K = 312 K T f = Ti 100 kPa 49.9 L PV i i ∆Eint = ( 3.50) nR ∆T = 3.50( 2.00 m ol) ( 8.314 J m ol⋅ K ) ( 12.4 K ) = 722 J Q = 0 ( adiabatic process) W = −Q + ∆Eint = 0 + 722 J= +722 J Sample Problems Chapter 22 3. A particular heat engine has a useful power output of 5.00 kW and an efficiency of 25.0%. The engine expels 8 000 J of exhaust energy in each cycle. Find (a) the energy taken in during each cycle and (b) the time interval for each cycle. P22.3 (a) We have e = W eng Qh = Qh − Qc Qh = 1− Qc Qh = 0.250 with Q c = 8 000 J, we have Q h = 10.7 kJ (b) W eng = Q h − Q c = 2 667 J and from P = W eng ∆t , we have ∆t= W eng P = 2 667 J = 0.533 s . 5 000 J s 13. An ideal gas is taken through a Carnot cycle. The isothermal expansion occurs at 250°C, and the isothermal compression takes place at 50.0°C. The gas takes in 1 200 J of energy from the hot reservoir during the isothermal expansion. Find (a) the energy expelled to the cold reservoir in each cycle and (b) the net work done by the gas in each cycle. P22.13 Isothermal expansion at Th = 523 K Isothermal compression at Tc = 323 K Gas absorbs 1 200 J during expansion. (a) T 323 Q c = Q h c = 1200 J = 741 J 523 Th (b) W eng = Q h − Q c = (1200 − 741) J= 459 J
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