Physcs 41 HW Set 11 Chapters 0 and 1 Chapter 0 1 An deal gas ntally at P,, and T s taken through a cycle as shown Fnd the net work done on the gas per cycle What s the net energy added by heat to the system per cycle? (c) Obtan a numercal value or the net work done per cycle or 100 mol o gas ntally at 0 C The work done durng each step o the cycle equals the negatve o the area under that segment o the P curve W W W W W DA AB BC CD W P 0 P 0 4P The ntal and nal values o T or the system are equal Thereore, E nt 0 and Q W 4P (c) W P nrt 4 4 4 100 814 7 908 kj Fnd the work done on a lud that expands as ndcated n the gure W Pd 6 W 60010 Pa 00 100 m W 6 40010 Pa 00 00 m 0010 Pa 400 00 m 10 MJ W 10 MJ 6
A sample o deal gas s expanded to twce ts orgnal volume o 100 m n a quas-statc process or whch P =, wth = 500 atm/m 6, as shown How much work s done on the expandng gas? W Pd The work done on the gas s the negatve o the area under the curve P between and 1 W d W 100 m 00 m 1 500 atm m 6 101 10 5 Pa atm 00 m 100 m 118 MJ 4 An deal gas s carred through a thermodynamc cycle consstng o two sobarc and two sothermal processes as shown Show that the net work done on the gas n the entre cycle s gven by W net P 1 1 ln P P 1 W W AB W BC W CD W D A B C D A W Pd Pd Pd Pd A B C D B C D A d d W nrt P d nrt P d 1 1 A B C D W nrt P nrt P ln B 1 ln C B 1 A D 1 C Now P1 A P B and P C P1 not cancel out Also, B P1 and P 1 C P P 1 D, so only the logarthmc terms do P P P P P W nrt nrt nrt nrt nr T T 1ln 1 ln 1ln ln 1 ln P P1 P1 P1 P1 Moreover P1 nrt and P1 1 nrt1 W P ln P 1 1 P 1
4 The nsde o a hollow cylnder s mantaned at a temperature Ta whle the outsde s at a lower temperature, Tb The wall o the cylnder has a thermal conductvty k Ignorng end eects, show that the rate o energy conducton rom the nner to the outer surace n the radal drecton s dq Lk T T a b ln b / a (Suggestons: The temperature gradent s dt/dr Note that a radal energy current passes through a concentrc cylnder o area rl) For a cylndrcal shell o radus r, heght L, and thckness dr, the equaton or thermal conducton, constant; thereore, dq dt ka becomes dx dq dt k rl dr Under equlbrum conons, dq s dq 1 dr dt and kl r Tb Ta dq 1 bdr dt kl a r dq 1 b Tb Ta ln kl a But T a T b, so dq kl Ta T ln b a b 5 In an nsulated vessel, 50 g o ce at 0 C s added to 600 g o water at 180 C What s the nal temperature o the system? How much ce remans when the system reaches equlbrum? The latent heat o uson s x10 5 J/kg and the specc heat o water s 4186 J/kg C Snce the heat requred to melt 50 g o ce at 0 C exceeds the heat requred to cool 600 g o water rom 18 C to 0 C, the nal temperature o the system (water + ce) must be 0C
Let m represent the mass o ce that melts beore the system reaches equlbrum at 0 C Q Q cold hot ml m c 0C T m w w 5 10 J kg 0600 kg 4 186 J kg C0C 180C m 16 g, so the ce remanng 50 g 16 g 114 g 6 An deal gas ntally at 00 K undergoes an sobarc expanson at 50 kpa I the volume ncreases rom 100 m to 00 m and 15 kj s transerred to the gas by heat, what are the change n ts nternal energy and ts nal temperature? E Q P nt 15 kj 50 kpa 00 100 m 750 kj 1 T 1 T T 00 T 00 K 900 K 100 1 1 Chapter 1 7 A cylnder contans a mxture o helum and argon gas n equlbrum at 150 C What s the average knetc energy or each type o gas molecule? What s the root-mean-square speed o each type o molecule? 18 10 J K 4 K 876 10 1 B J K k T 1 1 K m v rm s 876 10 J so For helum, v rm s 0 175 10 J (1) m 400 g m ol 4 m 664 10 g m olecule 60 10 m olecules m ol 7 m 664 10 kg m olecule Smlarly or argon, 99 g m ol m 66 10 g m olecule 60 10 m olecules m ol Substtutng n (1) above, we nd or helum, and or argon, 6 m 66 10 kg m olecule v 16 km s rm s v 514 m s rm s
8 A 100-mol sample o hydrogen gas s heated at constant pressure rom 00 K to 40 K Calculate the energy transerred to the gas by heat, the ncrease n ts nternal energy, and (c) the work done on the gas We us the tabulated values or C P and C Q nc T 100 m ol 88 J m ol K 40 00 K 46 kj P E nc T nt 100 m ol 04 J m ol K 10 K 45 kj (c) W Q Ent 46 kj 45 kj 101 kj 9 A 00-mol sample o a datomc deal gas expands slowly and adabatcally rom a pressure o 500 atm and a volume o 10 L to a nal volume o 00 L What s the nal pressure o the gas? What are the ntal and nal temperatures? (c) Fnd Q, W, and Ent P P 140 10 P P 500 atm 19 atm 00 5 P 500 101 10 Pa 10 10 m T nr 00 m ol 814 J m ol K T 5 P 19 101 10 Pa 00 10 m nr 00 m ol 814 J m ol K (c) The process s adabatc: Q 0 nt CP R C 140, C C C 5 R 65 K 5 K 5 Ent nc T 00 m ol 814 J m ol K 5 K 65 K 466 kj W E Q 466 kj 0 466 kj
10 Fteen dentcal partcles have varous speeds: one has a speed o 00 m/s; two have speeds o 00 m/s; three have speeds o 500 m/s; our have speeds o 700 m/s; three have speeds o 900 m/s; and two have speeds o 10 m/s Fnd the average speed, the rms speed, and (c) the most probable speed o these partcles nv v av 1 1 5 4 7 9 1 680 m s N 15 v nv av N so vrm s v 549 m s 549 741 m s av (c) vmp 700 m s 11 In an ultrahgh vacuum system, the pressure s measured to be 100 10 10 torr (where 1 torr = 1 Pa) Assumng the molecular dameter s 00 10 10 m, the average molecular speed s 500 m/s, and the temperature s 00 K, nd the number o molecules n a volume o 100 m, the mean ree path o the molecules, and (c) the collson requency P N N A RT and N PN A so that RT 10 100 10 1 100 60 10 N 1 10 m olecules 814 00 1 100 m n d N d 1 10 m olecules 00 10 m 1 10 1 1 1 1 779 km (c) v 64 10 s 4 1