(a) If the molecular attraction can be neglected, derive the expression of PV as a function of P and T, and plot PV vs. P.

(a)Solve the Schrodinger equation by using A sin(kx) as a trial solution to determine k and total energy E in terms of quantum number n.

(b)What's the probability of finding the particle between L/3 to L/2 if n = 3?

(a)Derive the expression of In ([B]/ [A]) as a function of a, k, and t.

(b)Calculate the time required for half of A is reacted.

(b)If the excluded volume effect can be neglected, derive the expression of PV as a function of P and T, and plot PV vs. P. (Hint: use PV=nRT as approximation if needed.)

(c) (5%) What is the Gibbs free energy of freezing water at -10°C? (Molar melting enthalpy of ice: △mH° = 5619 J mol -1at–10°C, △mH° = 6008 J mol-¹ at 0°C; molar heat capacity: ice Cp,m=37.66 J mol-¹K-¹, water Cp,m-75.38 J mol¹K¹)

(b) (5%) The entropy of change for the isolated system upon freezing includes the entropy change of the surrounding as the entropy change of the water. Since the heat capacity of surrounding is large, the heat evolved by the water on freezing us absorbed by the surrounding with only an infinitesimal change in temperature. Calculate the entropy change of the surrounding! What is the change in the entropy of the universe?

(a) (5%) Calculate the change in the entropy when 1 mol of liquid water at -10°C changes to ice at -10°C. Explain your result!

(d) (4%) Sketch the graph ofthe lineweaver-Nurk equation. Use a coordinate system in which (1/[S]0) is on the horizontal axis and (1/V0) is on the vertical axis. Show the resulting graph is a line that intersects the horizontal axis at (-1/Km) and the vertical axis at (1/Vmax)

(c) (4%) Show that the Michaelis-Menten equation can be written in the form (1/V0)=(Km/Vmax)(1/[S]0) + (1/Vmax) (This formula is know as the Lineweaver-Burk equation and shows that there is a linear relationship between (1/V0) and (1/[S]0).)

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