Chemistry 431 Problem Set 9 Fall 2018 Solutions

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Juliet Sabrina Rodgers
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1 Chemistry 43 rolem Set 9 Fll 8 Solutions. A certin gs oeys the eqution of stte Vm 4 where nd re numericl constnts. Derive n expression for the criticl volume of the gs in terms of. = ( ) + 4 Vm 5 ( V m ) = ( ) 3 V 6 m At the criticl point oth the first nd second derivtives vnish or R (V c ) 3 c (V c ) = 4 Vc 5 ( ) ( 4 (Vc ) ) c =. Vc 5 R ( ) ( 4 (Vc ) ) V 5 c R 8 V c = V c V c = = V 6 c V c = 5 3. Derive expressions for the criticl pressure, temperture, volume nd compression fctor for the Berthelot eqution of stte given y n V n n V

2 From Eq.() nd From Eq.() or Equting Eqs.(3) nd (4) or From Eqs.(3) nd (5) Using the eqution of stte V ( ) V V V = (V ) + V 3 () = (V ) 3 6 V 4 () V 3 = = (V ) (V ) RV 3 (3) (V ) = 6 3 V 4 = (V ) RV 3 = = c = 3(V )3 RV 4 (4) 3(V )3 RV 4 3(V ) V V c = 3 (5) (3 ) R(3) 3 ( ) 8 / = 7R = ( ) / (6) 3 3R c = R ( ) / 3 3 3R 9 ( ) / 3R

3 hen z c = cv c c = ( ) R / (7) 3 3 = (/)(R/33 ) / 3 R(/3R) / (/3) = 3 8 = Use the result of prolem to find the reduced eqution of stte for Berthelot gs. V V r c V = V r V c = r c hen r ( ) R / = r(/3)(/3r) / 3 3 3V r r = t R (3V r ) 3 ( 3R 3 Vr 4. A certin gs oeys the eqution of stte 3 9 Vr ( ) / ( 3 3 3R R ( 3 3 ) / 9 = 8 r 3V r 3 r V r R ( + ) ) / ( ) / 3R r where nd re numericl constnts. Derive n expression for the critil volume of the gs in terms of. = ( ) + ( + ) 3 = t = t c = ( ( ) V m R (V c ) 3 ) ( ( (V c + ) 3 ) / ) ( (Vc ) ) R = ( ) 6 3 ( + ) 4 ) ( (Vc ) ) = (V c + ) 3 R V c = 3 V c + V c = (V c + ) 4

4 5. Derive n expression for the criticl volume of gs tht oeys the eqution of stte where nd re constnts. ( ( / ) = t ( / V m) = t hen ) ( ) V m c (V c ) = 6 Vc 7 Vm 6 = ( ) + 6 V 7 m = ( ) 3 4 V 8 m or c = 6 (V c ) Vc 7 R c (V c ) = 4 3 Vc 8 R 6 (V c ) (V c ) 3 Vc 7 R V c = 4 V c = 4 V 8 c or V c = Expnd the vn der Wls eqution of stte s viril expnsion in powers of /V using the geometric series x = + x + x + x for x <. You my terminte the series fter the third viril coefficient. V V = [ ] V /V V V = + ( ) V V V = + / V + ( )... V 4

5 hen B( ) = C( ) = 7. A gs oeys the Berthelot eqution of stte. V m where nd re numericl constnts. By expnding the eqution of stte in viril form using powers of /, determine n expression for the second viril coefficient of the Berthelot gs. = = = ( ( + so tht the second viril coefficient is + / + B( ) = ) V m ( ) +... Vm ) +... ( ) he Boyle temperture of gs is defined to e the temperture t which the second viril coefficient in the inverse volume expnsion vnishes. A certin gs oeys the eqution of stte Vm where nd re numericl constnts. Expnd the eqution of stte in viril form, nd determine the Boyle temperture of the gs in terms of, nd R. Vm 5

6 = + ( ) = + ( ) ( ) hen the second viril coefficient is given y when B( ) = 9. A certin gs oeys the eqution of stte 3 = ( /3 = R) ( + ) where nd re numericl constnts. Given the criticl volume of the gs is V c = 5, derive n expression for the compression fctor t the criticl point. = ( ) + ( + ) 3 = t c = c = ( ) ( + ) 3 R c V c (V c + ) = R 4 4 7R z c = cv c c =. A certin gs oeys the eqution of stte = (6) 3 (4) R = 4 7R 5(7R) 4(7) R(4) V A V + B V 3 36 = = 5 6 4(7) where A nd B re positive constnts. he criticl volume is found to e V c = 3B/A. Derive n expression for the compression fctor z c t the criticl point therey showing z c to e independent of A nd B. Answer = V V + A V 3B 3 V = 4 6

7 t the criticl point. hen or c = R c = c A + B V c Vc Vc 3 c = A V c ( A B 3 = RA 3RB. A certin gs oeys the eqution of stte 3B V c A ) = A B 3RB A 3B A3 9B + z c = cv c = A3 3B 3RB = c 7B AR A 3 = + α + α A3 7B = A3 7B where α is function of temperture only. Determine the fugcity of the gs s function of pressure. { } z(, ) f = exp d Now Let Let I = z(, ) = + z(, ) d = α y = + α α + α d + α α + α d α dy d hen hen I = (+α ) dy y = ln( + α ) f = exp{ln( + α )} = ( + α ) = + α. Derive n expression for the fugcity of gs tht oeys the eqution of stte ( ) = 7

8 where is numericl constnt. Determine the ehvior of the fugcity s nd s. ( ) z f = exp d z = z = z = ( ) = z( ) = z = + p d d y = dy = d d dy z 3. Use the viril expnsion d f = lim f = dy y = ln( ) = ln lim f = [ + ( ) + c( ) + d( ) ] to derive n expression for the fugcity coefficient γ of gs in terms of the viril coefficients. Use the result to find the vlue of γ in the limit tht. z = = + ( ) + c( ) + d( ) 3... z d [( ) + c( ) + d( ) +...] c( ) = ( ) + + d( ) { } z γ = exp d { c( ) = exp ( ) + + d( ) } lim γ = e = 8

2. My instructor s name is T. Snee (1 pt)

2. My instructor s name is T. Snee (1 pt) Chemistry 342 Exm #1, Feb. 15, 2019 Version 1 MY NAME IS: Extr Credit#1 1. At prissy Hrvrd, E. J. Corey is Nobel Prize (1990 winning chemist whom ll students cll (two letters 2. My instructor s nme is