Physics 43 Final Examination May 7 In each problem, be sure to give the reasoning for your answer and define any variables you create. If you use a general formula, state that formula clearly before manipulating it..[4 pts] The figure below shows the van der Waals prediction of p vs V for a gas/liquid at temperature T. Find the approximate volume and pressure at which the liquid has been completely transformed to gas. Briefly describe the method by which you determined your answer..9.8.7.6.5.4.3.. 3 4 P(atm) V (cm 3 ). [5 pts] The figure below shows the calculated Gibbs Free energy G at T =, as a function of the concentration of B in the A+B binary solution. Are A and B miscible at T =? Describe the reasoning behind your answer. Also, describe the behavior of the mixture as T increases, ignoring any possible phase transformations or any possible temperature dependence of the energy of mixing. G(x B,T) 9 8 7 6 5 4 3..4.6.8 x B
3. [5 pts] The diagram shows the temperature dependence of a binary mixture of substances A and B as a function of the concentration of B, as they transform from only gas to only liquid. Which relations determine the curves labeled () and ()? At what temperature does a mixture of 5% B begin to condense? 4. [4 pts] Pure water boils at pressure and temperature p and T, respectively. A dilute solution of sugar in water is formed. Is the chemical potential of the (pure) water vapor greater, less than, or equal to the chemical potential of the dilute solution at p and T? Explain your reasoning. 5. [4 pts] The K value of HCl (hydrochloric acid) in water at 98 K is 7. If. mole of HCl is added to kg of water, find the ph (= log m H+ ) of the solution.
6. The hyperfine structure of hydrogen atoms is caused by the magnetic dipole interaction between the electron and the proton. This causes the s-orbital ground state (no orbital excitation) of spin / electrons and spin / protons (degeneracy 4), to split into a state of total angular momentum F = (degeneracy ) and a state of total angular momentum F = (degeneracy 3). The energy difference between the states, ɛ HF S = E(F = ) E(F = ), is roughly 6 6 ev ( ev =.6 9 J). (a) [7 pts] Assuming the (monatomic) H atom is in thermal equilibrium at temperature T, derive an expression for the probability of finding the H atom in the F = state as a function of T and ɛ HF S. You may neglect the (higher) states of electronic excitation (this is a good approximation, since the next state is roughly ev higher). (b) [ pts] Evaluate this probability at room temperature, T = 98 K. 7.[6 pts] A gas of atoms of mass m is in a volume V at thermal equilibrium with a reservoir of temperature T. Derive an expression for the most probable speed of the atoms. 8.[7 pts] A biological molecule known as a cytokine activates white blood cells to respond to infection. The cytokine does this by binding to the surface of the white blood cell. However, two cytokine molecules must bind to the surface before the white blood cell will actually be activated. Derive an expression for the probability of finding two bound molecules as a function of temperature T and cytokine chemical potentual µ c. The binding energy for each cytokine molecule is ɛ. You may assume that the molecules and binding sites are distinguishable. 9.[6 pts] At low temperatures atoms of 3 He may be treated as spin / particles. If one mole of 3 He gas were held in a volume of 6 cm 3, what would be the Fermi energy and Fermi temperature of the gas? The mass of 3 He is roughly 8 MeV/c, or 5 7 kg. Also, if you use atomic units, you may wish to know that hc = MeV-fm. 3
Physical Constants and Formulae k =.38 3 J/K R = 8.35 J/mol K C V = U T V α = L/L T B = p V/V T = S U N,V p = T S V N,U N A = 6. 3 h = 6.66 34 J s C p = U T p β = V/V T S = k ln Ω ds = C V dt T µ = T S N For adiabatic ideal gas processes: V T f/ =constant and pv (f+)/f =constant. Ω osc (N, q) = (q + N )! q!(n )! Ω ideal gas = N! Multiplicities V N Ω para (N, n ) = U,V h 3N π 3N/ ( 3N )!m( mu) 3N δu Paramagnetism ( B points up ) N! n!(n n )! U = µb(n n ) M = µ(n n ) Sacker-Tetrode Equation [ ( ) V S = Nk ln N (4πmU 3Nh )3/ + 5 ] Thermodynamic Potentials and Identity H = U + pv F = U T S G = U + pv T S G = H T S du = T ds p dv + µ dn 4
Clausius-Clapeyron Relation dp dt = S V van der Waals Equation (p + an )(V Nb) = NkT V Gibbs Free Energy for an Ideal Mixture (x = N B /N total ) G = ( x)g A + xg B + RT [x ln x + ( x) ln ( x)] Dilute Solution of B in A µ A (p, T ) = µ A (p, T ) N BkT N A µ B (p, T ) = µ B(p, T ) + kt ln m B moles of solute m B = kg of solvent Canonical Ensemble P (state i) = Z e E i/kt Z all states i e E i/kt Ideal Gas Relations and Maxwell-Boltzmann Speed Distribution m D(v) = ( πkt )3/ 4πv e mv /kt µ = kt ln V Z int Nv Q h v Q = ( πmkt )3/ 5
Grand Canonical Ensemble and Degenerate Fermi Gas P (state i) = Z e (E i N i µ)/kt n F D = e (ɛ µ)/kt + ɛ F = h 8m (3N p = U 3 V πv )/3 Z n BE = E(n x, n y, n z ) = all states i e (ɛ µ)/kt e (E i N i µ)/kt h 8mL (n x + n y + n z) U = 3 5 Nɛ F Binomial Distribution P N (n) = N! n!(n n)! pn q N n Mathematics N! N N e N πn πd/ A d (r) = ( d )!rd Γ(n + ) = n! Γ(n + ) = nγ(n) Γ( 3 ) = π! = sinh x = ex e x cosh x = ex + e x tanh x = ex e x e x + e x Gaussian Integrals (n =,, 3,...) e ax dx = I = π a x n e ax dx = ( ) n dn da n I xe ax dx = I = a x n+ e ax dx = ( ) n dn da n I 6