Lecture 19 Overview Ch. 4-5. List of topics Heat engines Phase transformations of pure substances, Clausius-Clapeyron Eq. van der Waals gases Thermodynamic potentials, chemical reactions. P. 2. 1. T H. 3. T C. V 1. V 2. V. Problem 1 (heat engine).
VProblem 1 (heat engine)
Working substance for the cycle shown in the Figure is 1 mole of ideal gas. Find the efficiency of this heat engine (in terms of TH and TC).
1 – 2
2 – 3
3 – 1
An insulating membrane (does not conduct heat) divides an insulated tank into two equal volumes. Each volume contains one mole of the same van der Waals gas (the constants a and b are known). The pressure in volume I is P1, in volume II – P2 . The piston has been removed. Find the pressure, explain your reasoning.
10 kg of water at 200C is converted to ice at - 100C by being put in contact with a reservoir at - 100C. This process takes place at constant pressure and the heat capacities at constant pressure of water and ice are 4180 and 2090 J/kg·K respectively. The heat of fusion of ice is 3.34·105 J/kg.Latent heat, dS for phase transformations
(a) Calculate the heat absorbed by the cold reservoir.
(b) Calculate the change in entropy of the closed system “reservoir + water/ice”.
The conversion consists of three processes: (a) water at 200C water at 00C; (b) water at 00C ice at 00C; (c) ice at 00C ice at -100C:
(a) the heat absorbed by the cold reservoir
(b) the change in entropy of the sub-system “water/ice”:
cooling of water
cooling of ice
The increase of entropy of the reservoir:
The total change of entropy of the whole system:
1. Near the triple point:
2. At the solid-gas phase boundary:
Assuming that the H2 vapor can be treated as an ideal gas
The pressure-temperature phase diagram of carbon is shown below. For simplicity, assume that the molar volumes of graphite and diamond are independent of temperature and pressure at 5.3×10-6 and 3.4×10-6 m3, respectively. 1 kbar = 108 N/m2 .
(a) Determine the latent heat per mole of transformation at T = 1000 K.
(b) Sketch the graph of Gibbs free energy of carbon at the constant temperature T = 2000 K as a function of pressure between P = 50 and 70 kbar. Mark the transition pressure. Explain the changes, if any, of the slope of G.
(c) Sketch the graph of entropy per mole of the material at the constant pressure P = 90 kbar as a function of temperature between 4000 and 5500 K. State any assumptions you make and explain your graph.
the slope of G changes at the phase transition because of the change in volume
The vapor pressure of solid ammonia is given by the relation:
where units –mm of Hg, T – absolute temperature.
The vapor pressure of liquid ammonia is given by the relation:
Ttr= 195 K
Ttr – from the equation
From the equation for liq. ammonia
The Clausius-Clapeyron eq. gives
Denote Sg, Sl, and Ss as the entropy for vapor, liquid and solid at triple point.
The latent heat of vaporization -
The latent heat of sublimation -
The latent heat of melting -
A cylinder closed with a piston is filled with the saturated water vapor at T = 1000C. The vapor is heated up by 10C, and, at the same time, the piston is moved to prevent condensation and to keep the vapor saturated (the system is “moving” along the coexistence curve). Find the relative change in the vapor volume, V/V. Assume that the vapor is an ideal gas, the latent heat of vaporization Lvap=40.7 kJ/mol, and the vapor density is negligible in comparison with the water density.
The Clausius-Clapeyron equation for the coexistence curve “liquid-gas”:
For an ideal gas:
For a small temperature changes :
The triple point for water corresponds to Ttr=0.010C and Ptr=0.006 bar. At T~Ttr,,the latent heat of melting is 335 kJ/kg, and the slope of the solid-vapor phase coexistence curve is dP/dT=50 Pa/K . Assume that water vapor behaves as an ideal gas near its triple point. Find the latent heat of vaporization.phase transformations
Along the solid-gas phase equilibrium curve:
- the latent heat of sublimation at the triple point
Consider the following equilibrium at 500 K:
CO(g) + 2 H2 (g) CH3OH(g)
The equilibrium concentrations are: [CO] = 0.0911 M, [H2] = 0.0822 M, [CH3OH] = 0.00892 M, what is the value of the equilibrium constant? Is G positive or negative when the reactants are transformed into products?
Since the value of the equilibrium constant is greater than one, G <0.
Consider the reaction:
Suppose we mix the following initial concentrations:
In which direction will the reaction go?
Thus, the reaction will go to the left (some ammonia molecules will be transformed into hydrogen and nitrogen).
0.1 - x
xProblem (chem. equilibrium)
We start withn0CO and n0H20 moles of the reacting gases and define as the yieldx the number of moles of CO2 and H2 that the reaction will produce at equilibrium:
The mass action law requires:
This is a quadratic equation with respect to x :
Thus, in equilibrium,
1 - 2x