PES 3950/PHYS 6950: Homework Assignment 5

Transcription

1 PES 3950/PHYS 6950: Homwork ssignmnt 5 Handd out: Wdnsday pril 8 Du in: Wdnsday pril, at t start of class at 3:05 pm sarp Sow all working and rasoning to rciv full points. Qustion [5 points] a)[0 points] Find t partition function for an idal gas in two dimnsions. You sould gt Z = [ ( )] mπ,! β wr is t ra tat t gas is confind to. b)[5 points] Find t avrag nrgy < E > in two dimnsions. c)[5 points] Find t avrag prssur < p > in two dimnsions. d)[5 points] Find t cmical potntial in two dimnsions. Using Stirling s approximation (!) = (), you sould gt µ = kt ( [ mπ β ]). Qustion [5 points] a)[.5 points] You ar making strawbrry sortcak. You cut up t strawbrris, tn sprinkl on som powdrd sugar. fw momnts latr t strawbrris look juicy. Wat appnd? Wr did tis watr com from? b)[.5 points] You ar trying to mak artificial blood clls. You av managd to gt pur lipid bilayrs to form sprical bags of radius 0 µm, filld wit moglobin. T first tim you did tis, you transfrrd t clls into pur watr and ty promptly burst, spilling t contnts. Evntually you found tat transfrring tm to a millimolar salt solution prvnts bursting, laving t clls sprical and full of moglobin and watr. If millimolar is good tn would millimolar would b twic as good? Wat appns wn you try tis? Qustion 3 [0 points] Considr a 3 stat systm problm wr O can b in on of tr stats: - In solution floating around - On rcptor of typ - binding nrgy E b - On rcptor of typ - binding nrgy E b a) Find an xprssion for t probability for t O to b attacd to rcptor. b) Find an xprssion for t probability for t O to b attacd to rcptor. c) Find t ratio of ts two probabilitis. Dos it nd on t cmical potntial?

2 Qustion 4 [0 points] GRDUTE STUDETS OLY Lasr ligt can b usd to trap micron-sizd bads. T dynamics of suc bads can b tougt of as t Brownian motion of a particl in a quadratic nrgy wll. Comput t man-squard xcursion < x > of suc a bad in a on-dimnsional quadratic wll wit a potntial-nrgy profil U(x) = kx and sow tat w can tn dtrmin t trap stiffnss as k = k BT < x >.

3 Homwork 5 SOLUTIOS Qustion ) a) Partition function for an idal gas in two dimnsions W nd to sum ovr all stats but wat is a stat? stat for on particl is know wn w know t position and momntum of t particl. So to sum ovr all stats w sum (intgrat) ovr all positions and all momnta. Our nrgy is t kintic nrgy only (no intractions) E = p x p y m T partition function is tn givn by Z E x y dxdy T intgration ovr coordinats givs a ra. Brak up t intgral on momnta p Z x / m But ac intgral is idntical, so p x y / m y p Z / m W can look up t intgral Bp B So t partition function is m Z If w ad particls instad of just, w would av to do our "sum" as

4 Z E x y dx dy x y dx dy sum ovr particl stats sum ovr particl stats m Z For particls w would xpct m Z W av mad two mistaks ) W av tratd t particls as if ty ar ac distinguisabl w can tll particl is at (x, y ) and particl is at (x, y ). But tis isn't rigt. W rally can't tll wic particl is wr. So w av to rduc t numbr of stats to account for tis. (Gibb's paradox). To gt rid of tis problm w av to divid by! ) W didn't count t stats corrctly. From our original dfinition w nd to sum ovr stats! Instad w did intgrals ovr position and momntum. But ow many stats dos tis corrspond to? From quantum mcanics w know tat tr is a fundamntal limit. Essntially stat (in pas spac) is givn by t uncrtainty principl dx x and dx dy x y = So w nd to divid t answr by to gt t rigt numbr of stats! T corrct answr for particls is Z m!

5 3 b) Find avrag nrgy Z E ) m (!) (/ m! Or, gtting rid of t otr parts wic giv no contribution kt E ) ( ) ( E ot: Tis answr is consistnt wit t quipartition torm wr tr ar two dgrs of frdom (two dimnsions) and ac dgr of frdom is associatd wit ½ kt c) Find t avrag prssur ormally on as V (Z) p But in two dimnsions, w sould tak t drivativ wit rspct to ara, not volum. So (Z) p Hr w only car about t ndnc of t partition function. T (Z) allows us to isolat tis kt p m! (Z) p Tis is t idal gas law in two-dimnsions 0

6 4 d) Find t cmical potntial m! l (Z) m! Look at t first trm (/!) = -(!) ow us Stirling's approximation (!) = () W gt m () Or m () Or m () kt ow, combining t trms m kt T minus sign indicats tat t cmical potntial is usually ngativ for typical valus.

7 Qustion answr a) T watr did not com from t atmospr. Som of t sugar dissolvd in t surfac watr layr, crating a layr of concntratd sugar solution. Osmotic flow tn pulld watr out of t clls of t fruit to dilut tis xtrior solution. b) From t txt of t problm w can conclud tat t osmotic prssur du to t various soluts on t insid of t bag is balancd by osmotic prssur du to t mm solution of salt on t outsid. Tus mm solution of t salt would xrt largr osmotic prssur inward and collaps t bags. Qustion 3 Tr ar 3 important nrgis in tis problm, E, E and t cmical potntial. Tak t rlativ nrgy for bing in solution as zro. E o = 0 T nrgy for t oxygn to b attacd to rcptor involvs bot t binding nrgy and t cmical potntial E = E b - Similarly E = E b - T partition function is Z 0 E E s = stats s E T probability to b in stat is P(stat ) = (Eb) (Eb) (Eb ) ( Eb) (Eb ) P(stat ) = (Eb) (Eb) (Eb ) T ratio (Eb) (Eb ) Eb It is inndnt of t cmical potntial Eb 5

8 Qustion 4 answr 6