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2.59. (a) T (V — b)Ricv = const; (b) Cp —CV — I-2a (V — RTV3 •
2.60. AT=
vaV2 (y-1) (^) 3.0 K. Rv, (V„-{-v,) 2.61. Q = Oa (V2— V1)1VIV2= 0.33 kJ. 2.62. n = p/kT = 1.105 cm-3;^ (1)^ =^ 0.2 mm. 2.63. p = (1 mRTI MV =^ 1.9 atm, where^ M^ is the mass of an N2 mole. 2.64. n = (p/kT — p/m2)/(1 — ml/m2) = 1.6.1019 cm-3, where mland m2are the masses of helium and nitrogen molecules.
2.65. p = 2nmv2cost 0 = 1.0 atm, where^ m^ is the mass of^ a
nitrogen molecule. 2.66. i = 2/(pv2/p — 1) = 5. 2.67. v/vn + 2)/3i; (a) 0.75; (b) 0.68. (3N — 3) kT^ for volume molecules. 2.68. (8)= (3N — 5/2) kT for linear molecules. 1/2(N-1) and 1/(2N-5/3) respectively. 2.69. (a) CV =7/2R, y = 9/7; (b)^ CV = (3N — 5/2)^ R, y = (6N — 3)/(6N — 5); (c) Cr = 3 (N — 1) R, y = = (N — 213)/(N — 1). 1/(3N — 2) for volume molecules, 2.70. A/Q= 1/(3N — 3/2) for linear molecules. For monoatomic molecules A/Q = 2/5. 2.71. M = RI(cp— cv.) = 32 g/mol. i = 2/(cp/cv — 1) = 5.
2.72. (a) i = 2 (CpIR — 1) = 5; (b) i = 2^ [C/R^ 1/(n —^ 1)] =
= 3, where n = 1/2 is the polytropic index. 2.73. y = (5v1 7v2)/(3v1 5v2). 2.74. Increases by Aplp = Mv2IiRT =^ 2.2%, where^ i^ = 5.
2.75. (a) vn=-113RTIM= 0.47 km/s, (8)= 3/2kT = 6.0-10-21 J;
(b) 3 1/2kTlapd3= 0.15 m/s. 2.76. 11= 7.6 times. 2.77. (^) Q = 112 — 1) imRTIM = 10 kJ. 2.78. Ws q = 172 kTa^ =^ 6.3 • 1012rad/sec.
2.79. (€),..,t = kToi2/i = 0.7.10-2°^ J.
2.80. Decreases 11(11-1)/1times, where i = 5. 2.81. Decreases 11(i-10-2) = 2.5 times. 2.82. C = 1/2R (i 1) = 3R. 2.83. vp, = 112p/p = 0.45 km/s, (v) = 0.51 km/s, = 0.55 km/s. 2.84. (a) 6NIN = (8/1/n) e-161 = 1.66%; (b) ON IN = 12 1/-3/2n e-3/28i = 1.85%. ) 2 )
MV 2.85. (a) T = k
m (Av -v-
—380 K; (b) T =^ 2k =340 K.
2.86. (a) T — —330 K; (b) v=
3kTo •ri 4k ln (v2/vi
i) ) V m 1-1 •
2.87. T — 2k (i mAvN)m 0)2— 0.37 kK.
2.88. v =
3kT 1n (m2/m1)=1.61 km/s. M2 — (^) 1 2.89. T = 113mv21k. 2.90. dN/N = (-72akT2' ) 3/2 e-mv212hT 2nvldvldvx.
2.91. (vx)= (^) 0, (I vx ) =1/.2kT I nm• 2.92. (v1) = kT/m. 2.93. v =1/4r1 (v), where (v) = li8kT/am. Co 2.94. p = 2mvx• vxdn (vx) = nkT , where dn (vx) =
(m12nkT) 1/2n • e-m4/2hT^ dvx. 2.95. (1/v) =1/2mInkT =^ 4n (v). 2.96. dN/N= 21s (nkT)-3/2e-ena^ dc; ep, =1/2kT;^ no. 2.97. SNIN = 3 6n e-3126i= 0.9%. CO 2.90 — = Q^ AN 231 (nkT)312 J
The principal contribution to the value of the integral is provided by the smallest values of a, namely a x ao. The slowly varying factor lii-can be taken from under the radical sign if ascribed the constant value 1/- so. Then
AN 1 N = 2 V eolnkT e-sona 2.99. (a) vp,. = 1/3kT/m; (b) apr = kT. 00 2.100. dv = dn (42143-t)v cos 0 = n (2kT/am)I/2sin 0 cos 0 de. v=- n/ 2.101. dv = dn (d52/4n) v cos 0 = n (m/2akT)3/2 e-mv2/211Tv3 dv. e=o 2.102. F = (kT I Ah) ln = 0.9.10-0 N. 2.103. NA = (6RT Ind3Apgh) ln 6.4.1023mo1-1. 2.104. 11/10 = ec.m2-m1vninT^ = 1.39. 2.105. h — kT (m2^ In (n21%)— mi) g • 2.106. Will not change. 2.107. (U) = kT. Does not depend. 2.108. w riliT I M1 N 70 g. 2.109. M — 2RTp In 1 (P— Po) (r2— r?) 0)2^ • 2.110. co = V (2RT M12) ln rl = 280 rad/s. 2.111. (a) dN = no e-ar2/hT4ar2 dr; (b) kT/a; (c) dN = = (a/akT)3/2 e-ar2/14T4TEr2^ dr; (d) Will increase 13/2-fold.
1/^8 e-e/kT de.
2.135. (^) 152-S1=-• vR (In a yin^ =1.0 J/K.
2.136. AS = (n^ ' (n-1) (
OR
v —1) In^ T. 2.137. AS —v(+1).R^ v ln a 46 J/K. 71 2.138. V7nVo/a (1 +1').
2.139. T = To+ (R/a) In (V/Vo)•
2.140. AS = R In [(V2— b)/(Vi — b)].
2.141. AS = Cv In (T2/T1) + R In [(V2— b)/(Vi— b)].
2.142. S = aT3/3.
2.143. AS = m [a In (T2/T1) + b (T 2— T
Ti)] = 2.0 kJ/K.
2.144. C = Sin; C <0 for n <0.
2.145. T = T oes-sox.^ See Fig. 15.
2.146. (a) C= —air; (b) Q.- a In (T1/T2);
(c) A = a In (Ti/T2) + Cv T2)• C<
2.147. (a) = (n — 1)/2n; (b) = (n —
—1)/(n + 1).
2.148. AS = vR In n = 20 J/K. so
2.149. AU = (2Y-1— 1) RT0/(y —1), AS Fig. 15.
= R In 2.
2.150. The pressure will be higher after the fast expansion. 2.151. AS = v1R In (1 + n) + v2R In (1 + 1/n) = 5.1 J/K.
2.152. AS = m1clIn^ (T/Ti) + m2c2 In^ (T/T 2) =^ 4.4 J/K, where
7' = (^) + m2c2T2)/(m1c1 + m2c2), c1and c2are the specific heat capacities of copper and water.
2.153. AS = Cv In (T 4T
2.154. (a) P .112N; (b) N — hrog(t2IT) 80, where 10-2 s is
the mean time which takes a helium atom to cover distances of the order of the vessel's dimensions.
2.155. Op,. = Ar1/1(N/2)!P = 252. Pnia = 52p7.12N =^ 24.6%.
2.156. Po,—
N! '
n! (N—n)! 2N respectively.
2.157. Pn=
n1ovNI where
p.V /V°.
2.158. d= )376/Icn0re= 0.4 fun, where no is Loschmidt's num-
ber; (n) = 1.0 .106. 2.159. Will increase S2/00= (1 + AT/To)iNA/2= 101.31.10" times. 2.160. (a) Ap = 4a/d = 13 atm; (b) Ap = 8a/d = 1.2.10-3atm.
2.161. h = 4alpgd = 21 cm.
2.162. a = 1/8pod (1. — re/n)/(112— 1).
2.163. p = po+ p gh 4a/d 2.2 atm.
2.164. h= [Po (n3— 1) + 4a (n2— 1)/d1/pg = 5 m.
2.165. Ah = 4a I cos 0 I (d2— dOldid2pg = 11 mm.
C>
2.166. R = 2a/pgh = 0.6 mm. 2.167. (^) x = //(1 pod/4a) = 1.4 cm. 2.168. a = [pgh poll(1 — h)] d/4 cos 0. 2.169. h = 4a/pg (d2— d1) = 6 cm. 2.170. h = 2a cos 0/pgx&p.
2.171. V1= 1/4ld^
n
(n 1
— 1)/pd 0.9 ems/s. 2.172. R^2 R 1 %.t'.• 1/8pgh2/ct = 0.20 mm. 2.173. m 2nR2ccl cos 0 l(n2— 1)Igh = 0.7 kg. 2.174. (^) F 2am/ph2= 1.0 N. 2.175. F = 2nR2alh = 0.6 kN. 2.176. F = 2a21/pgd2= 13 N. 2.177. t = 21T1R4/ar4. 2.178. Q = 2na2lpg. 2.179. (a) F = nad2 = 3μJ; (b)^ F = 2nad2 = 10 P. 2.180. AF = 2nad2(2-h/3— 1) = —1.5 la. 2.181. A' = F pV In (plp,),^ where^ F =^ 8nR2a,^ p = 4aIR, V = 413nR3. 2.182. C — Cp =1/2R/(1 2/2por/a). 2.184. (a) AS = —2 (da/dT)^ Au; (b) A^ U^ = 2 (a —^ T daldT) X
X Aa.
2.185. A = AmRTIM =^ 1.2 J. 2.186. m, = (V — inVi)/(V; — VD = 20 g, V, = 1.0^ 1. Here 17; is the specific volume of water. 2.187. m1 z Mpo (V, — V)IRT =^ 2.0 g, where Pois the stan- dard atmospheric pressure. 2.188. (n — 1)I(N — 1); = 1/(N^ 1). 2.189. AS = mq/T = 6.0 kJ/K; AU = m(q — RT/M) = (^) 2.1 MJ, where T = 373 K.
2.190. h x
(Q —mcAT) = 20 cm, where c is the specific poS(1+ qM RT) heat capacity of water, AT = 100 K,^ q^ is the specific heat of vapo- rization of water, T is its boiling temperature. 2.191. A = me (T — T o) RTIqM = 25 1, where c is the specific heat capacity of water, T is the initial vapour temperature equal to the water boiling temperature, as is seen from the hypothesis, q is the specific heat of vapour condensation. 2.192. d 4aMITOT = 0.2 am, where p is the density of water. 2.193. a = 71poYM/25TRT = 0.35 g/(s•cm2), where pc, is the standard atmospheric pressure. 2.194. p = pr2nRTIM = 0.9 nPa. 2.195. Ap = a/V2M = 1.7.104 atm. 2.196. pi pq.^ About 2.104 atm. 2.198. a = 27 /64R2T2r/Per (^) = 3.6 atm•12/mo12, b=118RT„Ip„. = 0.043 1/mol. 2.199. T7;,. = 3181 :1TcrIMper = 4.7 cm3/g. 2.200. (n 3/v2) (3v — 1) = 8T, T = 1.5.
