課程名稱：物理化學一-熱力學
課程性質：化學系大三必修
課程教師：陸駿逸
開課學院：理學院
開課系所：化學系
考試日期（年月日）：106/11/7
考試時限（分鐘）：130
試題：
(1) (20 pts) Given the van der Waals' equation of state
nRT n^2
P = ——— - a ———
V-nb V^2
(a) Obtain the expression for the second virial coefficient B (where the
pressure is a power series of n/V).
(b) In terms of a, and b, what is the Boyle temperature for the gas?
(c) Derive the formula for the critical temperature and the critical pressure
in terms of the parameters a and b.
(d) The CO2 has the parameters a = 3.6 atm dm^6 mol^(-2) and
b = 4.3 x 10^(-2) dm^3 mol^(-1). Estimate its Boyle temperature, critical
temperature and the critical pressure. ( R = 8.3 JK^(-1)mol^(-1) )
(2) (15 pts) Estimate the collision frequency between one hydrogen molecule and
the oxygen molecules in this room. From the textbook, the measured
collision cross-sections are σ_H2 = 0.27 nm^2, and σ_O2 = 0.40 nm^2.
( Hint : ∫[0,∞] x^2 e^(-x^2) dx = sqrt(π)/4,
∫[0,∞] x^3 e^(-x^2) dx = 1/2,
k_B = 1.38 x 10^(-23) JK^(-1) )
(3) (10 pts) Calculate the final pressure of a sample of carbon dioxide that
expands reversibly and adiabatically from 67.4 kPa and 0.50 dm^3 to a final
volume of 2.00 dm^3. Take γ= 1.4.
(4) (10 pts) Calculate ΔS (for the system) when the state of 3.00 mol of
perfect gas atoms,for which C_p,m = 2.5R, is changed from 25°C and
1.00 atm to 125°C and 5.00 atm.
(5) (10 pts) Someone claims that he has a wonderful engine which, in each
cycle, exchanges heat Q_1 = 60J at T_1 = 300K, Q_2 = -10J at T_2 = 200K,
Q_3 = -10J at T_3 = 100K. Combine suitable Carnot engine(s), show that the
existence of this engine contradicts the Kelvin's version of the second
law of thermodynamics.
∂G/T
(6) (15 pts) Show that ( ———— )_p,n = H.
∂1/T
(7) (10 pts) Show that ( ∂T/∂p )_s = ( ∂V/∂S )_p.
(8) (20 pts) Consider N molecules which are confined in a 2D box of length L in
the both directions.
(a) Count the number of the microstates (Ω) which have the energy within the
given range (E,E+ΔE) using the quantum numbers (n_1x,n_1y,...,n_Ny). Use
the Boltzmann entropy formula S = k_B lnΩ, to derive the expression for
the entropy.
(b) Calculate the chemical potential μ = -T ( ∂S/∂N )_E,L.
π^(n/2)
( Hint : n dimensional sphere has the volume ————— radius^n. )
(n/2)!