Research: Science and Education
Conversion of Chemical Reaction Energy into Useful Work in the Van’t Hoff Equilibrium Box N. M. Bazhin* Institute of Chemical Kinetics and Combustion, Novosibirsk State University, Novosibirsk 630090, Russia; *
[email protected] V. N. Parmon Boreskov Institute of Catalysis, Novosibirsk State University, Novosibirsk 630090, Russia
A detailed consideration of the peculiarities in the conversion of chemical reaction energy into useful work (nonexpansion work) using the ideal “van’t Hoff equilibrium box” (VHEB) is instructive for teaching undergraduate students. The VHEB illustrates work production in isothermal conditions when the quantity of work is limited by the change in the Gibbs function rather than by the Carnot coefficient. The operation of VHEB has been considered in a few textbooks (e.g., refs 1–3). In the present article, we draw attention to two aspects of the VHEB: 1. The VHEB allows the use of the energy from chemical reactions to produce useful work with the theoretical efficiency of 100%, even at uniform temperature of the system, in agreement with the second law of thermodynamics. 2. The ideal VHEB makes possible the total apparent efficiency of the utilization of the reaction energy much more than 100% of the enthalpy change during the chemical reaction. By “the total apparent efficiency” of the utilization of reaction energy we mean the ratio of the useful work done and reaction heat to the reaction heat released.
An Ideal VHEB and Production of Useful Work Let us consider the production of useful work using the well-known exothermic reaction of the slow, complete oxidation of carbon with oxygen
C + O2
CO2
with the help of the VHEB as done in ref 1, p 73. Although the equilibrium of this reaction is shifted to the right at common temperatures, we shall consider this reaction to be reversible. Note that the reaction is the same as in ref 1 only for sake of its simplicity; other reactions in a general form can be considered as done in ref 3. To produce work, the VHEB (Figure 1) is designed from a large chemical reactor, two portable cylinders AC and BC, as well as two stationary vessels AV and BV that contain oxygen (gas A with pressure pA ) and carbon dioxide (gas B with pressure pB ). AV and BV are cylinders with pistons to provide constant pressure in the vessels during introduction and removal of the gases. The gases are considered to be ideal. Cylinders AC and BC are used to introduce (remove) gases into (from) the reactor. The portability of AC and BC is important for the discussion of different thermodynamic processes occurring in a spatial separation. The reactor is supplied with two semipermeable membranes.
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Figure 1. A scheme of the van’t Hoff equilibrium box under consideration.
Appropriate quantities of carbon, oxygen, and carbon dioxide are situated in the reactor at the equilibrium partial pressures pAeq and pBeq of the reacting gases. The reactor also has a cylinder with a piston (at the top of the reactor) to provide a constant pressure in the reactor when introducing and removing a substance through the semipermeable membranes. The total pressure in the reactor is pr. The reactor, portable cylinders, and stationary vessels are located in a large thermostat that has constant temperature T. We assume ideal thermal contact between the reactor and the thermostat, as well as between the portable cylinders and the thermostat that allows the transfer of heat between the thermostat and the reactor, as well as the thermostat and the portable cylinders. The initial reactants together with the reaction products and contents of the reactor will be considered as the “thermodynamic system”. The reactor walls, pistons, and cylinders will be attributed to the thermostat. The value of reversible useful work that can be produced during one cycle by transporting 1 mol of oxygen from vessel AV to the reactor and 1 mol of carbon dioxide from the reactor to vessel BV can be calculated. The pressures in the reactor and AV and BV are constants, which allow use the Gibbs function for calculating the useful work. This work is produced by the pistons that move in the portable cylinders, and thus this work is not the work of expansion of the entire reacting system.
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Research: Science and Education
The transport and chemical transformation of the substances in the VHEB include the following operations. 1. First, one has to disconnect the portable cylinder AC from the reactor with the piston fully inserted. Movable valves should be used to avoid the contact of the inner parts of both the reactor and the portable cylinders with the atmosphere.
wB = RT ln
3. We expand gas A isothermally and reversibly in AC from pressure pA to equilibrium partial pressure pAeq. The volume occupied by gas A changes from V = RT兾pA to VA = RT兾pAeq. In this case, work wA is done on the gas (4, p 43). eq
pA
(1)
pA
The change in the Gibbs function of the system is also wA. Also, in this case, the thermostat energy changes owing to the heat transfer between the thermostat and gas A inside AC as ∆Hthermostat = wA. It is worth noting that the work produced by the gas expansion in AC is fully useful since it can increase the potential energy in the surroundings of the system under consideration, that is, by lifting an external weight. Thus, we do not include this potential energy in the internal energy of the system.
6. We remove gas B (1 mol) from the reactor to BC at constant pressure pBeq; the useful work is zero for the same reasons as in item 5. The change in the Gibbs function is zero. 7. We disconnect portable cylinder BC from the reactor. 8. We change pressure in BC isothermally and reversibly from pBeq to pB, which is the pressure in BV. The work
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(2)
eq
pB
9. We connect portable cylinder BC to vessel BV. The portion of gas B taken from the reactor is transported into BV by pistons moving at constant pressure. No work is performed in this case, and the change in the Gibbs function is zero. 10. Connect the portable cylinder BC to the reactor to prepare the VHEB for a new cycle.
The work done on the gases is the maximum useful work w = w max′ = wA + wB eq
= −RT ln
pB
eq pA
+ RT ln
pB pA
p = ∆ rG ° + RT ln B pA
4. Connect portable cylinder AC to the reactor. 5. Then we open the valves between the reactor and the portable cylinder to introduce gas A to the reactor through the semipermeable membrane at constant pressure exceeding pAeq by an infinitesimal value. The useful work, in this case, is zero because the reactor is equipped with the cylinder with a piston to keep the reaction pressure constant. Indeed, the volume occupied by gas A (1 mol of oxygen) in AC will vary from VA to zero. The work done to introduce 1 mol of gas A into the reactor is pAeqVA = RT. The volume occupied by the mixture of the reaction gases in the reactor increases with introducing 1 mol of gas A by value ∆Vr = RT兾pr owing to the piston movement in the cylinder that supports the constant total pressure pr. The work produced on the gas in the reactor by this cylinder is ᎑pr ∆Vr = ᎑RT. Thus, the total work to introduce gas A into the reactor is zero. The change in the Gibbs function of the system for this step is also zero.
pB
The heat transferred during such transformation is exchanged with the thermostat and thus the change of the thermostat internal energy is ∆Hthermostat = wB. The change in the Gibbs function at this stage is also wB. Note that in the system under consideration the reversible useful work is produced at stages 3 and 8 by the gas expansion and compression in the portable cylinders in contrast to an erroneous assertion (1, p 73) according to which this work is produced by introducing (removing) gases with the equilibrium pressure into the reactor (from reactor) in stages 5 and 6.
2. Second, one has to connect the AC to the vessel AV and introduce 1 mol of oxygen from AV to AC at constant pressure. At this stage of the procedure, the change in the Gibbs function of the system is zero.
w A = RT ln
done on gas B is
(3)
since ∆rG ⬚ = ᎑RT ln( pAeq兾pBeq). In a particular case of 1 bar pressures in vessels AV and BV, the maximum useful work is equal to
wmax′ = ∆ rG ° = ∆ r H ° − T ∆ r S °
(4)
For the particular reaction under consideration (the carbon oxidation at 25 ⬚C), thermodynamic parameters in eq 4 are ∆rG ⬚ = ᎑395 kJ兾mol, ∆rH ⬚ = ᎑394 kJ兾mol, and T∆rS ⬚ = 1 kJ兾mol (1, p 73). Note that a direct source of energy for producing the mechanical work of the pistons in the portable cylinders is a change in the thermal energy of the thermostat. In this case, the thermal energy of the thermostat decreases by value ∆rG ⬚ (at pA = pB = 1 bar). These losses are largely compensated by the exothermic reaction in the reactor. During one cycle, the thermostat gets heat equal to ᎑∆rH ⬚ from the reaction inside the reactor. Overall, during one cycle, the energy of the thermostat varies by the value
∆H thermostat = ∆ rG ° − ∆ r H ° = −T ∆ r S °
(5)
The extent to which the energy evolved (᎑∆rH ⬚) by the reaction produces useful work depends on the result of the summation of the two values, ∆rG ⬚ and ᎑∆rH ⬚, the sum being equal, according to eq 5, to ᎑T∆rS ⬚. In the case of the
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carbon oxidation T∆rS ⬚ > 0, and the energy change (᎑∆rH ⬚) during one cycle is less than the quantity of useful work. In this case, additional energy T∆rS ⬚ will be taken from the thermostat, which will be cooled, losing the internal energy ᎑T∆rS ⬚ during one cycle. If in an exothermic reaction ∆rS ⬚ < 0, then T∆rS ⬚ < 0. The evolved reaction energy (᎑∆rH ⬚) is more than production of useful work. In this case, the thermostat will be heated, and the change in its thermal energy will be positive and equal, according to eq 5, to (᎑T∆rS ⬚). So far we have considered to the reaction of slow oxidation of carbon for which ∆rG ⬚ < 0 and ∆rH ⬚ < 0. Let us consider another reaction for which ∆rG ⬚ > 0 and ∆rH ⬚ > 0. This reaction cannot occur spontaneously. However, it can be carried out under reversible conditions with the consumption of the work of the pistons operating in the portable cylinders. In this case, the net heat equal to the work done, ∆rG ⬚, will pass from the portable cylinders to the thermostat. After introducing the reagents through semipermeable membranes into the reactor, the reaction occurs under equilibrium conditions. This evolution of the system is accompanied by the absorption of heat supplied by the thermostat because the reaction is endothermic. The thermostat provides heat in quantity of ∆rH ⬚ to the reactor. Therefore, the internal thermostat energy will change, also according to eq 5, by ∆rG ⬚ − ∆rH ⬚ = ᎑T∆rS ⬚. Thus, if T∆rS ⬚ > 0, the thermostat is cooled because the energy given by it to the reactor is greater than that obtained from the portable cylinders. If T∆rS ⬚ < 0, the thermostat is heated. The total energy spent on the reversible process consists of the energy spent on making the reversible work, ∆rG ⬚, and T∆rS ⬚ that is used to maintain the constant thermostat temperature. This gives ∆rH ⬚. Thus, the thermostat is an active participant in the reversible process under consideration. In this case, the change in the internal energy of the thermostat is always ᎑T∆rS ⬚. The production of work using the VHEB at a uniform temperature does not contradict the second law of thermodynamics. Indeed, in terms of this law (1, p 26), “It is impossible both to transfer heat from a heat bath at a uniform temperature and to produce an equivalent quantity of work causing no changes in the thermodynamic state of another body.” In the case of carbon oxidation, the initial reagents are converted into the products, mass is transferred from stationary vessel AV and the reactor to stationary vessel BV, and also the internal energy of the thermostat decreases. Thus, the production of work at constant temperature is always accompanied by some changes of thermodynamic parameters in other bodies. A Simultaneous Production of Useful Work and Heat To produce work and heat simultaneously, the VHEB is divided spatially (Figure 2): the reactor is located “indoors”, while the stationary vessels AV and BV are located “outdoors”. Thus, the portable cylinders will be located in different conditions depending on the stage of the process. The process is
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Figure 2. A spatially separated van’t Hoff equilibrium box.
carried out as described in the previous section. However, procedures 2, 3, 8, and 9 will be performed outdoors. The portable cylinders will work due to cooling the environment but not due to utilization the reaction heat and the cooling of the thermostat. Thus, the work is produced outdoors, and the heat is evolved indoors. The quantity of the evolved heat is ᎑∆rH ⬚, while that of the work produced is ᎑∆rG ⬚. Unfortunately, it is impossible to realize such VHEB in practice (1). Summary The production of useful work by means of the ideal “van’t Hoff equilibrium box” is considered in detail. It is shown that useful work arises according to the scheme “reaction energy → heat → useful work” without violation of the second law of thermodynamics, even at constant temperature, using the heat evolved by a reaction. Van’t Hoff equilibrium box divided in two parts can simultaneously produce heat (in volume 100% from theoretical one) and useful work (in volume 100% from theoretical one) without violation of the first law of thermodynamics. Literature Cited 1. Denbigh, K. The Principles of Chemical Equilibrium, 3rd ed.; The University Press: Cambridge, 1971. 2. Physical Chemistry; Gerasimov Ya. I., Ed.; Chimiya: Moscow, 1980; (in Russian). 3. Steiner L. E. Introduction to Chemical Thermodynamics, 2nd ed.; McGraw-Hill: New York, 1948. 4. Atkins, P. The Elements of Physical Chemistry, 3rd ed.; Oxford University Press: Oxford, 2001.
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