This preview has intentionally blurred parts. Sign up to view the full document

View Full Document

Unformatted Document Excerpt

When ules. the kinetic energy of the molecules is sufficient to overcome the attractive energy that holds them together the liquid vaporizes. The enthalpy of vaporization is the heat energy (enthalpy) required to accomplish this at constant pressure. It seems reasonable that the greater the enthalpy of vaporization, the greater the kinetic energy required, and the greater the temperature needed to achieve this kinetic energy. Hence, we expect that vap H Tb , which implies that their ratio is a constant. The device proposed uses geothermal heat (energy) and appears to be similar to devices currently in existence for heating and lighting homes. As long as the amount of heat extracted from the hot source (the ground) is not less than the sum of the amount of heat discarded to the surroundings (by heating the home and operating the steam engine) and of the amount of work done by the engine to operate the heat pump, this device is possible; at least, it does not violate the first law of thermodynamics. However, the feasability of the device needs to be tested from the point of view of the second law as well. There are various equivalent versions of the second law, some are more directly useful in this case than others. Upon first analysis, it might seem that the net result of the operation of this device is the complete conversion of heat into the work done by the heat pump. This work is the difference between the heat absorbed from the surroundings and the heat discharged to the surroundings, and all of that difference has been converted to work. We might, then, conclude that this device violates the second law in the form stated in the introduction to Chapter 4; and therefore, that it cannot operate as described. However, we must carefully examine the exact wording of the second law. The key words are "sole result." Another slightly different, though equivalent, wording of Kelvin's statement is the following: "It is impossible by a cyclic process to take heat from a reservoir and convert it into work without at the same time transferring heat from a hot to a cold reservoir." So as long as some heat is discharged to surroundings colder than the geothermal source during its operation, there is no reason why this device should not work. A detailed analysis of the entropy changes associated with this device follows. E4.2(b) Environment at Tc Pump Flow Flow "ground" water at Th Figure 4.1 CV and Cp are the temperature dependent heat capacities of water 58 INSTRUCTOR'S MANUAL Three things must be considered in an analysis of the geothermal heat pump: Is it forbidden by the first law? Is it forbidden by the second law? Is it efficient? Etot = Ewater + Eground + Eenvironment Ewater = 0 Eground = -CV (Th ){Th - Tc } Eenvironment = -CV (Th ){Th - Tc } adding terms, we find that and Tc . Stot = Swater + Swater = 0 Sground = qground -Cp (Th ){Th - Tc } = Th Th Cp (Tc ){Th - Tc } qenvironment Senvironment = = Tc Tc Cp (Tc ) = Cp , we find that 1 1 - Tc Th Etot = 0 which means that the first law is satisfied for any value of Th Senvironment Sground + adding terms and estimating that Cp (Th ) Stot = Cp {Th - Tc } This expression satisfies the second law ( Stot > 0) only when Th > Tc . We can conclude that, if the proposal involves collecting heat from environmentally cool ground water and using the energy to heat a home or to perform work, the proposal cannot succeed no matter what level of sophisticated technology is applied. Should the "ground" water be collected from deep within the Earth so that Th > Tc , the resultant geothermal pump is feasible. However, the efficiency, given by eqn 4.11, must be high to compete with fossil fuels because high installation costs must be recovered during the lifetime of the apparatus. Erev = 1 - Tc Th with Tc 273 K and Th = 373 K (the highest value possible at 1 bar), Erev = 0.268. At most, about 27% of the extracted heat is available to do work, including driving the heat pump. The concept works especially well in Iceland where geothermal springs bring boilin kPa when xmethanol = 0.2, P = 10.00 kPa. (d) The vapor pressure plot shows positive deviations from ideality. The escaping tendency is stronger than that of an ideal solution. To get the Henry's law constants, estimate values for the targets of Pmethanol at xmethanol = 0 and PTAME at xmethanol = 1. 126 INSTRUCTOR'S MANUAL 12 Total Vapor Pressure / kPa Soln composition (xmethanol) 10 8 Vapor composition (ymethanol) 6 0 0.2 0.4 0.6 0.8 1 Methanol Mole Fraction (xmethanol or ymethanol) 15 Figure 8.18(c) P = Pmethanol + PTAME 10 Vapor Pressure / kPa Pmethanol 5 PTAME 0 0 0.2 0.4 xmethanol 0.6 0.8 1 Figure 8.18(d) For methanol in TAME (eqn 7.26): Kmethanol = dPmethanol = 45.1 kPa dxmethanol xmethanol =0 dPTAME dPTAME =- = 25.3 kPa dxTAME xTAME =0 dxmethanol xmethanol =1 For TAME in methanol: KTAME = (e) According to eqn 6.3, the vapor pressure should increase when the applied pressure is increased. For TAME: P = P eVm P /RT = 6.16 kPa The applied pressure increases the vapor pressure by about 1%, molecules have been "squeezed" out of the liquid phase and into the gas phase but only to a slight extent. -1 -1 -1 cm3 mol bar)/[(83.1451 3 )(288.15 cm bar K mol K)] = (6.09 kPa) e(131.78 )(2.0 9 Chemical equilibrium Solutions to exercises Discussion questions E9.1(b) The thermodynamic equilibrium constant involves activities rather than pressures. See eqn 9.18 and Example 9.1. For systems involving gases, the activities are the dimensionless fugacities. At low pressures, the fugacity may be replaced with pressures with little error, but at high pressures that is not a good approximation. The difference between the equilibrium constant expressed in activities and the constant expressed in pressures is dependent upon two factors: the stoichiometry of the reaction and the magnitude of the partial pressures. Thus there is no one answer to this question. For the example of the ammonia synthesis reaction, in a range of pressures where the fugacity coefficients are greater than one, an increase in pressure results in a greater shift to the product side than would be predicted by the constant expressed in partial pressures. For an exothermic reaction, such as the ammonia synthesis, an increase in temperature will shift the reaction to the reactant side, but the relative shift is independent of the fugacity coefficients. The ratio ln(K2 /K1 ) depends only on r H . See eqn 6.26. The physical basis of the dependence of the equilibrium constant on temperature as predicted by the - - - van't Hoff equation can be seen when the expression r G- = r H - - T r S - is written in the - - - - form R ln K = - r H /T + r S . When the reaction is exothermic and the temperature is raised, ln K and hence K decrease, since T occurs in the denominator, and the reaction shifts to favor the reactants. When the reaction is endothermic, increasing T makes ln K less negative, or K more positive, and products are favored. Another factor of importance when the reaction is endothermic is the increasing entropy of the reacting system resulting in a more positive ln K, favoring products. A typical pH curve for the titration of a weak base with a strong acid is shown in Figure 9.1. The stoichiometric point S occurs on the acidic side of pH = 7 because the salt formed by the neutralization reaction has an acid cation. Buffers work best when S A , that is when the concentrations of the salt and acid are not widely different. An abundant supply of A- ions can remove by reaction any H3 O+ supplied by the addition of an acid; likewise an abundant supply of HA can remove by reaction any OH- supplied by addition of base. Indicators are weak acids which in their undissociated acid form have one colour, and in their dissociated anion form, another. In acidic solution, the indicator exists in the predominantly acid form (one colour), in basic solution in the predominantly anion form (the other colour). The ratio of the two forms is very pH sensitive of because the smalposite face. When calculating the interactions of a molecule in a box, it interacts with all the molecules in the box and all the periodic replications of those molecules and itself in the other boxes. Once g(r) is known it can be used to calculate the thermodynamic properties of liquids. (a) Monte Carlo methods In the Monte Carlo method, the particles in the box are moved through small but otherwise random distances, and the change in total potential energy of the N particles in the box, VN , is calculated E21.2(b) E21.3(b) 338 INSTRUCTOR'S MANUAL using one of the intermolecular potentials discussed in Sections 21.5 and 21.6. Whether or not this new configuration is accepted is then judged from the following rules: 1 If the potential energy is not greater than before the change, then the configuration is accepted. 2 If the potential energy is greater than before change, the Boltzmann factor e- VN /kT is compared with a random number between 0 and 1; if the factor is larger than the random number, the configuration is accepted; if the factor is not larger, the configuration is rejected. This procedure ensures that at equilibrium the probability of occurrence of any configuration is proportional to the Boltzmann factor. The configurations generated in this way can then be used to construct g(r) simply by counting the number of pairs of particles with a separation r and averaging the result over the whole collection of configurations. (b) Molecular dynamics In the molecular dynamics approach, the history of an initial arrangement is followed by calculating the trajectories of all the particles under the influence of the intermolecular potentials. Newton's laws are used to predict where each particle will be after a short time interval (about 1 fs. which is shorter than the average time between collisions), and then the calculation is repeated for tens of thousands of such steps. The time-consuming part of the calculation is the evaluation of the net force on the molecule arising from all the other molecules present in the system. A molecular dynamics calculation gives a series of snapshots of the liquid, and g(r) can be calculated as before. The temperature of the system is inferred by computing the mean kinetic energy 2 of the particles and using the equipartition result that 1/2 mvq = 1/2 kT for each coordinate q. E21.4(b) Describe how molecular beams are used to investigate intermolecular potentials. A molecular beam is a narrow stream of molecules with a narrow spread of velocities and, in some cases, in specific internal states or orientations. Molecular beam studies of non-reactive collisions are used to explore the details of intermolecular interactions with a view to determining the shape of the intermolecular potential. The primary experimental information from a molecular beam experiment is the fraction of the molecules in the incident beam that are scattered into a particular direction. The fraction is normally expressed in terms of dI , the rate at which molecules are scattered into a cone that represents the area covered by the "eye" of the detector (Fig. 21.21 of the text). This rate is reported as the differential scattering cross-section, , the constant of proportionality between the value of dI and the intensity, I , of the incident beam, the number density of target molecules, N , and the infinitesimal path length dx through the sample: dI = I Ndx. The value of (which has the dimensions of area) depends on the impact parameter, b, the initial perpendicular separation of the paths of the colliding molecules (Fig. 21.22), and the details of the intermolecular potential. The scattering pattern of real molecules, which are not hard spheres, depends on the details of the intermolecular potential, including the anisotropy that is present when the molecules are nonspherical. The scattering also depends on the relative speed of approach of the two particles: a very fast particle might pass through the interaction region without much deflection, whereas a slower one on the same path might be temd reverse current maxima bracket - E - (Ox, Red), so the species present can be identified. Furthermore, the forward and reverse peak currents are proportional to the concentration of the couple in the solution, and vary with the sweep rate. If the electron transfer at the electrode is rapid, so that the ratio of the concentrations of Ox and Red at the electrode surface have their equilibrium values for the applied potential (that is, their relative concentrations are given by the Nernst equation), the voltammetry is said to be reversible. In this case, the peak separation is independent of the sweep rate and equal to (59 mV)/n at room temperature, where n is the number of electrons transferred. If the rate of electron transfer is low, the voltammetry is said to be irreversible. Now, the peak separation is greater than (59 mV)/n and increases with increasing sweep rate. If homogeneous chemical reactions accompany the oxidation or reduction of the couple at the electrode, the shape of the voltammogram changes, and the observed changes give valuable information about the kinetics of the reactions as well as the identities of the species present. Corrosion is an electrochemical process. We will illustrate it with the example of the rusting of iron, but the same principles apply to other corrosive processes. The electrochemical basis of corrosion in the presence of water and oxygen is revealed by comparing the standard potentials of the metal reduction, such as Fe2+ (aq) + 2e- Fe(s) - E - = -0.44 V E29.4(b) with the values for one of the following half-reactions In acidic solution (a) 2 H+ (aq) + 2 e- H2 (g) - E- = 0 V - E - = +1.23 V (b) 4 H+ (aq) + O2 (g) + 4 e- 2H2 O(l) In basic solution: (c) 2H2 O(l) + O2 (g) + 4 e- 4OH- (aq) - E - = +0.40 V DYNAMICS OF ELECTRON TRANSFER 467 - Because all three redox couples have standard potentials more positive than E - (Fe2+ /Fe), all three can drive the oxidation of iron to iron(II). The electrode potentials we have quoted are standard values, and they change with the pH of the medium. For the first two - E(a) = E - (a) + (RT /F ) ln a(H+ ) = -(0.059 V)pH - E(b) = E - (b) + (RT /F ) ln a(H+ ) = 1.23 V - (0.059 V)pH These expressions let us judge at what pH the iron will have a tendency to oxidize (see Chapter 10). A thermodynamic discussion of corrosion, however, only indicates whether a tendency to corrode exists. If there is a thermodynamic tendency, we must examine the kinetics of the processes involved to see whether the process occurs at a significant rate. The effect of the exchange current density on the corrosion rate can be seen by considering the specific case of iron in contact with acidified water. Thermodynamically, either the hydrogen or oxygen reduction reaction (a) or (b) is effective. However, the exchange current density of reaction (b) on iron is only about 10-14 A cm-2 , whereas for (a) it is 10-6 A cm-2 . The latter therefore dominates kinetically, and iron corrodes by hydrogen evolution in acidic solution. For corrosion reactions with similar exchange current densities, eqn 29.62 predicts that the rate of corrosion is high when E is large. That is, rapid corrosion can be expected when the oxidizing and reducing couples have widely differing electrode potentials. Several techniques for inhibiting corrosion are available. First, from eqn 62 we see that the rate of corrosion depends on the surfaces exposed: if either A or A is zero, then the corrosion current is zero. This interpretation points to a trivial, yet often effective, method of slowing corrosion: cover the surface with some impermeable layer, such as paint, which prevents access of damp air. Paint also increases the effective solution resistance between the cathode and anode patches on the surface. Another form of surface coating is provided by galvanizing, the coating of an iron object with zinc. Because the latter's standard potential is -0.76 V, which is more negative than that of the iron couple, the corrosion of zinc is thermodynamically favoured and the iron survives ... View Full Document

End of Preview

Sign up now to access the rest of the document