课程名称︰物理化学一-热力学
课程性质︰必修
课程教师︰郑原忠
开课学院：理学院
开课系所︰化学系
考试日期（年月日）︰103/12/5
考试时限（分钟）：110分钟
试题 :
Physical Chemistry I: Thermodynamics
Mid-term Exam #2 Date: 12/5/2014
1. (25%) Consider a one-component system dU = TdS - PdV + μdn and
U = TS - PV + μn. Answer the following questions.
(a) Use Legendre transform to derive a thermodynamic potential that has
(1/T, V, μ/T) as its natural variables. Let's call this function
Y = Y (1/T, V, μ/T). Give the definition of Y and its total differential.
(hint: start from the entropy).
┌ ┐2
╭ ∂p ╮ │╭ ∂V ╮ │
(b) Show that C - C = -T(d)│───│ ││───│ │
P V ╰ ∂V ╯T,n│╰ ∂T ╯p,n│
└ ┘
2╭ ∂(ΔG/T) ╮
(c) Show that ΔH = -T │──────│
╰ ∂T ╯P_i,P_f;n_i,n_f
_ ╭ ∂^2μ╮
(d) Show that C = -T│────│
P ╰ ∂T^2 ╯P
(e) Show that within a small temperature range the natural logarithm of an
equilibrium constant (ln K) is a linear function of the inverse temperature
(1/T). Sketch a plot of ln K vs. 1/T for an endothermic reaction to explain
how this can be used to measure the reaction enthalpy and entropy. Clearly
state the approximations involved.
_2 _
2. (20%) Consider a van der Waals gas (P + a/V )(V - b) = RT.
╭ ∂S ╮
(a) Calculate │───│ for a mole of van der Waals gas.
╰ ∂P ╯T
╭ ∂U ╮
(b) Calculate │───│ for a mole of van der Waals gas.
╰ ∂V ╯T
3. (25%) At 2000 ℃ water is 2% dissociated into oxygen and hydrogen at a total
pressure of 1 bar. In the following, you must use Gibbs free energy to
explain your answers when applicable. (R = 8.314 x 10^-2 L bar K^-1 mol^-1)
(a) Give the total balanced chemical equation. Calculate K for the
equilibrium.
(b) Calculate the standard Gibbs energy of formation for water at 2000 ℃.
(c) Assuming that all gases are ideal, calculate the standard reaction
enthalpy (Δ_r, H^0) and entropy (Δ_r, S^0) for the water dissociation
reaction.
(d)(10%) Predict the percentage of dissociated water at a total pressure of
1 bar at 2500 ℃.
4. (20%) In this problem, we consider vaporization of pure liquids.
http://ppt.cc/KXDB
(a) Drive the Clapeyron equation and show that the Clapeyron equation
reduces to
dP P(Δ_vap)H
── = ───────
dT RT^2
for vaporization and sublimation.
(b) On the right we show experimental vapor pressures of several pure liquids
as a function of temperature. Explain why the lines are linear. What does
the slopes tell you?
(c) Estimate the molar enthalpy of vaporization for water from this plot.
(d) How do you expect the curve to change at a wider temperature range?
Sketch a qualitative curve of ln P as a function of inverse temperature.
5. (20%) Let's consider liquid-vapor phase transition for water . At 100 ℃ and
1 atm, the specific heat capacity of liquid water and steam is 75.3
Jmol^-1K^-1 and 37.5 Jmol^-1K^-1, respectively. The heat of vaporization of
water is 40.7 kJ/mol. The critical point of water is (217.7 atm, 374 ℃).
(a) Sketch a plot of the molar enthalpy of water as a function of temperature
from T = 90 ℃ to T = 110 ℃ at 1 atm. Sketch the same plot for the heat
capacity. What is the order of the phase transition here? Explain.
(b) Give the equation that would allow you to calculate the chemical
potential from T = 90 ℃ to T = 110 ℃ at 1 atm. Sketch a plot of μ vs. T for
water around 100 ℃.
(c) Now qualitatively sketch a plot of the molar enthalpy of water as a
function of temperature from T = 364 ℃ to T = 384 ℃ at 217.7 atm. Sketch
the same plot for the heat capacity. What is the order of the phase
transition here? Explain.
(d) Sketch a plot of μ vs. T for water at 217.7 atm and around 374 ℃.
6. (10%) For a pure system, its chemical potential at the gas phase (μ_G),
liquid phase (μ_L), and solid phase (μ_S) are functions of the
temperature. Consider water at 1 atm in this question.
(a) Sketch the three functions, μ_G(T), μ_L(T), and μ_S(T), for water on
the same plot from T = -10 ℃ to T = 110 ℃ to explain the ice → liquid
water → steam transition. You only need to qualitatively sketch the curves,
but a clear explanation of the features of the curves (e.g. the signs of the
slopes) is required.
(b) Now consider that salt is added to the water and we further assume that
the ionic compound does not go into either the gas phase or the solid phase.
What happens to the chemical potentials at 1 atm? What physical phenomena
are predicted by the chemical potential plot (μ_G(T), μ_L(T), and μ_S(T))
of water with salt added?