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physical chemistry
Physical Chemistry 3rd edition Thomas Engel, Philip Reid - Solutions
Within what range can you restrict the values of P and/or T if the following information is known about sulfur? Use Figure 8.11 to answer this problem.a. Only the rhombic solid phase is observed for P = 1atm.b. When the pressure on the vapor is increased, the monoclinic solid phase is formed.c.
The vapor pressure of methanol(l) is 16.94 × 103 Pa at 298.15 K. Use this value to calculate ΔGof (CH3OH, g) − ΔGof (CH3OH, l). Compare your result with those in Table 4.1.
Are the two P€“T phase diagrams below likely to be observed for a pure substance? If not, explain all features of the diagram that will not be observed.a.b. Liquid Vapor Solid Solid II Liquid Solid Vapor
Prove that a substance for which the solid–liquid coexistence curve has a negative slope contracts upon melting.
Use the vapor pressures of SO2(l) given in the following table to calculate the enthalpy of vaporization using a graphical method or a least squares fitting routine. T (K) 190. P (Pa) T (K) 230. P (Pa) 17950 32110 54410 87930 824.1 2050. 200. 240. 210. 250. 4591 220. 9421 260.
Calculate the vapor pressure of CH3OH(l) at 298.15 K if He is added to the gas phase at a partial pressure of 200. Bar using the data tables. By what factor does the vapor pressure change?
Calculate the difference in pressure across the liquid–air interface for a (a) Mercury and (b) Methanol droplet of radius 125 nm.
For water, ΔH vaporization is 40.656 kJmol−1, and the normal boiling point is 373.12 K. Calculate the boiling point for water on the top of Mt. Everest of (elevation 8848 m), where the normal barometric pressure is 253 Torr.
The vapor pressure of an unknown solid is approximately given by ln(P/Torr) = 22.413 − 2211 (K/T), and the vapor pressure of the liquid phase of the same substance is approximately given by ln(P/Torr) = 18.352 − 1736 (K/T).a. Calculate ΔH vaporization and ΔH sublimation.b. Calculate ΔH
Calculate the vapor pressure of water droplets of radius 1.00 × 10−8 m at 360. K in equilibrium with water vapor. Use the tabulated value of the density and the surface tension at 298 K from Appendix B for this problem.
Use the values for ΔGof (CCl4, l) and ΔGof (CCl4, g) from Appendix B to calculate the vapor pressure of CCl4 at 298.15 K.
Benzene(l) has a vapor pressure of 0.1269 bar at 298.15 K and an enthalpy of vaporization of 30.72 kJmol−1. The CP,m of the vapor and liquid phases at that temperature are 82.4 and 136.0 J K−1mol−1, respectively. Calculate the vapor pressure of C6H6(l) at 340.0 K assuminga. That the enthalpy
The vapor pressure of liquid benzene is 20,170 Pa at 298.15 K, and Δ Hvaporization=30.72 kJ mol-1at 1 atm pressure. Calculate the normal and standard boiling points. Does your result for the normal boiling point agree with that in Table 8.3? If not, suggest a possible cause. T (K) P
Use the following vapor pressures of propane given here to calculate the enthalpy of vaporization using a graphical method or a least squares fitting routine. P (Torr) T (K) 0.01114 100. 120 2.317 140. 73.91 160. 838.0 5054 180. 2.016 x 104 200. 6.046 x 10* 220.
The vapor pressure of a liquid can be written in the empirical form known as the Antoine equation, where A(1), A(2), and A(3) are constants determined from measurements:Starting with this equation, derive an equation giving ΔH Vaporization m as a function of temperature. A(2) A(1) – -
Calculate the vapor pressure for a mist of spherical water droplets of radius a. 1.95 × 10−8 m b. 2.25 × 10−6 m at 298 K. The vapor pressure of water at this temperature is 25.2 Torr.
Use the vapor pressures of ice given here to calculate the enthalpy of sublimation using a graphical method or a least squares fitting routine. T (K) P (Torr) 200. 0.1676 210. 0.7233 2.732 220. 230. 9.195 240. 27.97 250. 77.82 200.2 260.
The phase diagram of NH3 can be characterized by the following information. The normal melting and boiling temperatures are 195.2 and 239.82 K, respectively; the triple point pressure and temperature are 6077 Pa and 195.41 K, respectively. The critical point parameters are 112.8 × 105 Pa and 405.5
You have collected a tissue specimen that you would like to preserve by freeze drying. To ensure the integrity of the specimen, the temperature should not exceed −5.00°C. The vapor pressure of ice at 273.16 K is 624 Pa. What is the maximum pressure at which the freeze drying can be carried out?
For a van der Waals gas, z = Vm/(Vm − b) − a/RTVm. Expand the first term of this expression in a Taylor series in the limit Vm >> b to obtain z ≈ 1 + (b − a/RT)(1/Vm).
For the Berthelot equation, Vm = (RT/P) + b − a (RT2) find an expression for the Boyle temperature in terms of a, b, and R.
The observed Boyle temperatures of H2, N2, and CH4 are 110, 327, and 510. K, respectively. Compare these values with those calculated for a van der Waals gas with the appropriate parameters.
The experimental critical constants of CH4 are found in Table 7.2. Use the values of Pc and Tc to calculate Vc. Assume that CH4 behaves as a. An ideal gasb. A van der Waals gasc. A Redlich–Kwong gas at the critical point. For parts (b) and (c), use the formulas for the critical
A sample containing 42.1 g of Ar is enclosed in a container of volume 0.0885 L at 375 K. Calculate P using the ideal gas, van der Waals, and Redlich–Kwong equations of state. Based on your results, does the attractive or repulsive contribution to the interaction potential dominate under these
Show that Tβ = 1 + T (∂ In z/∂T)P, and that Pk = 1 – P(∂ In z/∂T)T.
Calculate the density of O2(g) at 480. K and 280. bar using the ideal gas and the van der Waals equations of state. Use a numerical equation solver to solve the van der Waals equation for Vm or use an iterative approach starting with Vm equal to the ideal gas result. Based on your result, does the
At what temperature does the slope of the z versus P curve as P → 0 have its maximum value for a van der Waals gas? What is the value of the maximum slope?
Show that Tβ = 1 + T (∂ In z/∂T)P or a real gas where β is the volumetric thermal expansion coefficient.
Calculate the critical volume for ethane using the data for Tc and Pc in Table 7.2 (see Appendix B, Data Tables) assuming a.The ideal gas equation of stateb. The van der Waals equation of state. Use an iterative approach to obtain Vc from the van der Waals equation, starting with the
Calculate the Redlich€“Kwong parameters of fluorine from the values of the critical constants and compare your results with the values for a and b in Table 7.4.Table 7.4 Redlich–Kwong van der Waals b (dm² mol¬!) a (dmº bar mol-) b (dm² mol-!) a (dmº bar mol~² K) Substance Formula
Calculate the van der Waals parameters of carbon dioxide from the values of the critical constants and compare your results with the values for and b in Table 7.4.Table 7.4 Redlich–Kwong van der Waals b (dm² mol¬!) a (dmº bar mol-) b (dm² mol-!) a (dmº bar mol~² K) Substance Formula NH3
Show that PK = 1 – P(∂ In z/∂P) for a real gas where κ is the isothermal compressibility.
Another equation of state is the Berthelot equation, Vm = (RT/P) + b – a/(RT2). Derive expressions for β = 1/V (∂V /∂T)P and κ = −1/V (∂V /∂P)T from the Berthelot equation in terms of V, T, and P.
Use the law of corresponding states and Figure 7.8 to estimate the molar volume of propane at T = 500.K and P = 75.0 bar. The experimentally determined value is 0.438 mol Lˆ’1. What is the relative error of your estimate?Figure 7.8 1.6 1.4 1.2 4.0 v 1.0 8. 10 2.0 0.8 1.7 1.5 1.4 1.3 0.6
Show that the van der Waals and Redlich–Kwong equations of state reduce to the ideal gas equation of state in the limit of low density.
The volume of a spherical molecule can be estimated as V = b/(4NA), where b is the van der Waals parameter for the excluded molar volume and NA is Avogadro’s number. Justify this relationship by considering a spherical molecule of radius r, with volume V = 4/3π r3. What is the volume centered at
Show that the second virial coefficient for a van der Waals gas is given by az = b B(T) = - RT 1 RT V.
A 1.75 mole sample of Ar undergoes an isothermal reversible expansion from an initial volume of 2.00 L to a final volume of 85.00 L at 310. K. Calculate the work done in this process using the ideal gas and van der Waals equations of state. What percentage of the work done by the van der Waals gas
At 725 K and 280. bar, the experimentally determined density of N2 is 4.13 mol L−1. Compare this with values calculated from the ideal and Redlich–Kwong equations of state. Use a numerical equation solver to solve the Redlich–Kwong equation for Vm or use an iterative approach starting with Vm
The experimentally determined density of O2at 140. bar and 298 K is 192 g Lˆ’1. Calculate z and Vmfrom this information. Compare this result with what you would have estimated from Figure 7.8. What is the relative error in using Figure 7.8 for this case?Figure 7.8 1.6 1.4 1.2 4.0 v 1.0 8.
In Example Problem 6.9, KP for the reaction CO(g) + H2O(l) ⇋ CO2(g) + H2(g) was calculated to be 3.32 × 103 at 298.15 K. At what temperature is KP = 5.50 103? What is the highest value that KP can have by changing the temperature? Assume that ΔHoR is independent of temperature.
A gas mixture with 4.50 mol of Ar, x moles of Ne, and y moles of Xe is prepared at a pressure of 1 bar and a temperature of 298 K. The total number of moles in the mixture is five times that of Ar. Write an expression for ΔG mixing in terms of x. At what value of x does the magnitude of ΔG mixing
If T is increased at constant total pressure, how will the degree of dissociation of HCl(g) change?
If additional Cl2(g) is added to the reaction system at constant V and T, how will the degree of dissociation of HCl(g) change
Show that [∂(A/T)/∂(1/T)]V / UWrite an expression analogous to Equation (6.36) that would allow you to relate ΔA at two temperatures.
If additional Cl2 (g) is added to the reaction system at constant total pressure and temperature, how will the partial pressures of H2 (g) and HCl(g) change?
If the total pressure is increased at constant T, how will the relative amounts of H2 (g) and HCl(g) change?H2(g) + Cl2(g) ⇋ 2HCl(g) at equilibrium. Assume ideal gas behavior.
Is the partial pressure of H2 (g) dependent on T? If so, how will it change as T decreases?H2(g) + Cl2(g) ⇋ 2HCl(g) at equilibrium. Assume ideal gas behavior.
Derive an expression for A(V, T) analogous to that for G(T, P) in Equation (6.33).
If the reaction is carried out at constant V, how does the total pressure change if T increases?H2(g) + Cl2(g) ⇋ 2HCl(g) at equilibrium. Assume ideal gas behavior.
How does the total number of moles in the reaction system change as T increases?H2(g) + Cl2(g) ⇋ 2HCl(g) at equilibrium. Assume ideal gas behavior.
If KP is independent of pressure, why does the degree of dissociation in the reaction Cl2(g) ⇋ 2Cl(g) depend on pressure?
Consider the reaction FeO(s) + CO(g) ‡‹ Fe(s) + CO2(g) for which KPis found to have the following values: a. Calculate ΔGoR , ΔSoR , and ΔHR???? for this reaction at 700.°C. Assume that ΔHoR is independent of
Under what conditions is the distribution of products in an ideal gas reactions system at equilibrium unaffected by an increase in the pressure?
A hard-working horse can lift a 350. lb. weight 100. ft. in one minute. Assuming the horse generates energy to accomplish this work by metabolizing glucose:C6H12O6(s) + 6O2(g) → 6CO2(g) + 6H2O(l)Calculate how much glucose a horse must metabolize to sustain this rate of work for one hour
Under what conditions is Kx > KP?
Is the equation ΔA = ΔU − TΔS applicable to all processes?
A sample containing 2.50 moles of He (1 bar, 350. K) is mixed with 1.75 mol of Ne (1 bar, 350. K) and 1.50 mol of Ar (1 bar, 350. K). Calculate ΔG mixing and ΔS mixing .
Is the equation ΔG = ΔH − TΔS applicable to all processes?
Calculate the maximum expansion work that is available in carrying out the combustion reactions in Example Problems 6.1 and 6.2. Explain both the magnitude and the sign of the work.
Predict the change in the partial pressure of CO2 as O2 is removed from the reaction vessel at constant pressure and temperature.CO(g) + 1/2O2(g) ⇋ CO2(g) at equilibrium for which ΔHoR = −283.0 kJ mol−1.
Predict the change in the partial pressure of CO2 as a platinum catalyst is introduced into the reaction vessel at constant volume and temperature.CO(g) + 1/2O2(g) ⇋ CO2(g) at equilibrium for which ΔHoR = −283.0 kJ mol−1.
You place 3.00 mol of NOCl(g) in a reaction vessel. Equilibrium is established with respect to the decomposition reaction NOCl(g) ⇋ NO(g) + 1/2Cl2 (g).a. Derive an expression for KP in terms of the extent of reaction ξ .b. Simplify your expression for part (a) in the limit that ξ is
Predict the change in the partial pressure of CO2 as Xe gas is introduced into the reaction vessel at constant volume and temperature.CO(g) + 1/2O2(g) ⇋ CO2(g) at equilibrium for which ΔHoR = −283.0 kJ mol−1.
Calculate ΔG for the isothermal expansion of 2.25 mol of an ideal gas at 325 K from an initial pressure of 12.0 bar to a final pressure of 2.5 bar.
You wish to design an effusion source for Br atoms from Br2(g). If the source is to operate at a total pressure of 7.5 Torr, what temperature is required to produce a degree of dissociation of 0.20? What value of the pressure would increase the degree of dissociation to 0.65 at this temperature?
Predict the change in the partial pressure of CO2 as Xe gas is introduced into the reaction vessel at constant pressure and temperature.CO(g) + 1/2O2(g) ⇋ CO2(g) at equilibrium for which ΔHoR = −283.0 kJ mol−1.
Predict the change in the partial pressure of CO2 as the pressure is increased at constant temperature.CO(g) + 1/2O2(g) ⇋ CO2(g) at equilibrium for which ΔHoR = −283.0 kJ mol−1.
Predict the change in the partial pressure of CO2 as the pressure is increased at constant temperature.CO(g) + 1/2O2(g) ⇋ CO2(g) at equilibrium for which ΔHoR = −283.0 kJ mol−1.
Which thermodynamic state function gives a measure of the maximum electric work that can be carried out in a fuel cell?
The reaction A + B ⇋ C + D is at equilibrium for ξ = 0.5. What does this tell you about the variation of G pure with ξ ?
Why is it reasonable to set the chemical potential of a pure liquid or solid substance equal to its standard state chemical potential at that temperature independent of the pressure in considering chemical equilibrium?
Is the equation (∂U/∂V )T = (βT − κP)/κ valid for liquids, solids, and gases?
Can equilibrium with respect to the concentration of Ar and H2be attained in the system shown in Figure 6.2? If so, what can you say about the partial pressure in each part of the system?Figure 6.2 H2- H, +Ar Pure H2 Pd membrane
Under what conditions is dG ≤ 0 a condition that defines the spontaneity of a process?
By invoking the pressure dependence of the chemical potential, show that if a valve separating a vessel of pure A from a vessel containing a mixture of A and B is opened, mixing will occur. Both A and B are ideal gases, and the initial pressure in both vessels is 1 bar.
Under what conditions is dA ≤ 0 a condition that defines the spontaneity of a process?
Calculate the maximum non-expansion work that can be gained from the combustion of benzene(l) and of H2(g) on a per gram and a per mole basis under standard conditions. Is it apparent from this calculation why fuel cells based on H2 oxidation are under development for mobile applications?
What is the relationship between the KP for the two reactions 3/2H2(g) + 1/2N2(g) ⇋ NH3(g) and 3H2(g) + N2(g) ⇋ 2NH3(g)?
Calculate μO2 mixture (298 15 K, 1 bar) , for oxygen in air, assuming that the mole fraction of O2 in air is 0.210. Use the conventional molar Gibbs energy defined in Section 6.17.
Under what condition is KP = Kx ?
The reaction A + B ⇋ C + D is at equilibrium for ξ = 0.1. What does this tell you about the variation of G pure with ξ ?
It is found that KP is independent of T for a particular chemical reaction. What does this tell you about the reaction?
Collagen is the most abundant protein in the mammalian body. It is a fibrous protein that serves to strengthen and support tissues. Suppose a collagen fiber can be stretched reversibly with a force constant of k = 10.0 N mˆ’1and that the force, F (see Table 2.1), is given by F = kl. When
Nitrogen is a vital element for all living systems, but except for a few types of bacteria, blue-green algae, and some soil fungi, most organisms cannot utilize N2from the atmosphere. The formation of €œfixed€ nitrogen is therefore necessary to sustain life, and the simplest
Calculate ΔA for the isothermal compression of 2.95 mol of an ideal gas at 325 K from an initial volume of 60.0 L to a final volume of 20.5 L. Does it matter whether the path is reversible or irreversible?
Ca(HCO3)2(s) decomposes at elevated temperatures according to the stoichiometric equation Ca(HCO3)2(s) ???? CaCO3(s) + H2O(g) + CO2(g).a. If pure Ca(HCO3)2(s) is put into a sealed vessel, the air is pumped out, and the vessel and its contents are heated, the total pressure is 0.290 bar.
For the reaction C(graphite) + H2O(g) ⇋ CO(g) + H2(g), ΔHoR =131.28 kJ mol-1 at 298.15 K. Use the values of CP,m at 298.15 K in the data tables to calculate ΔH°R at 125.0°C.
Consider the equilibrium CO(g) + H2O(g) ???? CO2(g) + H2(g). At 1150. K, the composition of the reaction mixture isa. Calculate KP and ΔGoR at 1150. K.b. Given the answer to part (a), use the ΔHof of the reaction species to calculate ΔGoR at 298.15 K.
Calculate KP at 600.K for the reaction N2O4(l) ⇋ 2NO2(g) assuming that ΔHoR is constant over the interval 298– 725 K.
Assume that a sealed vessel at constant pressure of 1 bar initially contains 2.00 mol of NO2(g). The system is allowed to equilibrate with respect to the reaction 2NO2(g) ⇋ N2O4(g). The number of moles of NO2(g) and N2O4(g) at equilibrium is 2.00 – 2ξ and ξ , respectively, where ξ
Many biological macromolecules undergo a transition called denaturation. Denaturation is a process whereby a structured, biologically active molecule, called the native form, unfolds or becomes unstructured and biologically inactive. The equilibrium is
The pressure dependence of G is quite different for gases and condensed phases. Calculate ΔGm for the processes (C, solid, graphite, 1 bar, 298.15 K) → (C, solid, graphite, 325 bar, 298.15 K) and (He, g, 1 bar, 298.15 K) → (He, g, 325 bar, 298.15 K). By what factor is
Consider the equilibrium C2H6(g) ⇋ C2H4(g) + H2(g). At 1000.K and a constant total pressure of 1.00 bar, C2H6(g) is introduced into a reaction vessel. The total pressure is held constant at 1 bar and at equilibrium the composition of the mixture in mole percent is H2(g): 26.0%, C2H4(g):
Consider the equilibrium NO2(g) ‡‹ NO(g) + 1/2O2(g). One mole of NO2(g) is placed in a vessel and allowed to come to equilibrium at a total pressure of 1 bar. An analysis of the contents of the vessel gives the following results:a. Calculate KP at 700. and 800. K.b. Calculate
The shells of marine organisms contain calcium carbonate CaCO3, largely in a crystalline form known as calcite. There is a second crystalline form of calcium carbonate known as aragonite. Physical and thermodynamic properties of calcite and aragonite are given below.a. Based on the thermodynamic
A sample containing 2.75 moles of N2 and 6.25 mol of H2 is placed in a reaction vessel and brought to equilibrium at 52.0 bar and 690. K in the reaction 1/2N2(g) + 3/2H2 (g) ⇋ NH3(g).a. Calculate KP at this temperature.b. Set up an equation relating KP and the extent of reaction as in
Calculate KP at 298 and 490. K for the reaction NO(g) + 1/2O2(g) → NO2(g) assuming that ΔHoR is constant over the interval 298−600. K. Do you expect Kp to increase or decrease as the temperature is increased to 600. K?
Calculate ΔAoR and ΔGoR for the reaction C6H6(l) + 15/2O2(g) → 6CO2(g) + 3H2O(l) at 298 K from the combustion enthalpy of benzene and the entropies of the reactants and products.
From the following data, derive the absolute entropy of crystalline glycine at T = 300.K. You can perform the integration numerically using either a spreadsheet program or a curve-fitting routine and a graphing calculator (see Example Problem 5.9). CP„(JK-mol-) Temperature (K) 10. 0.3 20. 2.4 30.
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