The Physics of Energy, page 40
9.9 [T] Approximating the interatomic potential as a harmonic oscillator with angular frequency , the vibrational-rotational energy levels of a diatomic molecule are given by . Here is the equilibrium interatomic separation, μ is the reduced mass, and is the rotational quantum number. The CO bond in carbon monoxide has m and eV. Ignoring rotations, what is the wavelength (in microns) of the radiation absorbed when the CO molecule makes a transition from vibrational level n to ? Vibrational transitions are always accompanied by a transition from rotational level J to . The result is a sequence of equally spaced absorption lines centered at . What is the spacing between these vibrational-rotational absorption lines?
9.10 [T] Show that the data in Table 9.6 satisfy the definitions of Gibbs and Helmholtz free energy, (1) and (2)
9.11 [T] Suppose reaction data, and (and therefore ), on a chemical reaction are all known at a temperature and pressure p, but you want to know the reaction enthalpy and free energy at a different temperature . Suppose the heat capacities of all the reactants (at pressure p) are known throughout the range . Find expressions for and . Express your answers in terms of , the sum of the heat capacities of the products minus the heat capacities of the reactants, and the reaction data at .
9.12 Estimate the amount of CO produced per kilogram of CaCO in the calcination reaction (9.10). In addition to the CO released in the reaction, include the CO emitted if coal is burned (at 100% efficiency) to supply the enthalpy of reaction and the energy necessary to heat the reactants to 890℃. You can assume that the heat capacity of CaCO is kJ/mol K, and assume that the coal is pure carbon so .
9.13 Look up the standard enthalpies of formation and the entropies of solid ice and liquid water and verify that ice may spontaneously melt at NTP (20℃, 1 atm).
9.14 The convention we have used for LHV differs from the one offered by the US Department of Energy, which defines LHV as “the amount of heat released by combusting a specified quantity (initially at 25℃) and returning the temperature of the combustion products to 150℃, which assumes the latent heat of vaporization of water in the reaction products is not recovered” [53]. According to the DOE convention, the LHV of methane is 789 kJ/mol, compared with 802 kJ/mol from the convention we use in Example 9.2. Account for this difference.
9.15 Find the enthalpy of reaction for the two pathways of decomposition of TNT mentioned in §9.5.2.
9.16 Wood contains organic polymers such as long chains of cellulose (CHO), and is commonly used as a biofuel.15 Write a balanced equation for the complete combustion of one unit of the cellulose polymer to water and CO. Note that hydrogen and oxygen are already present in the ratio 2:1. Make a crude estimate of the enthalpy of combustion of cellulose by assuming that the conversion of the H and O to water does not contribute significantly, so that the energy content can be attributed entirely to the combustion of carbon. Compare your estimate to the true enthalpy of combustion of cellulose, 14.9 MJ/kg.
9.17 Estimate the lower heating value of a typical cord of wood by first estimating the mass of the wood and then assuming that this mass is completely composed of cellulose (see Problem 9.16). Compare your answer with the standard value of 26 GJ. Real “dried” firewood (density 670 kg/m) contains a significant amount of water in addition to cellulose (and lignin and hemicellulose). When it burns, the water must be driven off as vapor. Compute the fraction y of water by mass in firewood that would account for the discrepancy you found in the first part of this problem.
9.18 Acetylene (used in welding torches) CH, sucrose (cane sugar) CHO, and caffeine CHON, are all popular energy sources. Their heats of combustion are 310.6 kcal/mol, 1348.2 kcal/mol, and 1014.2 kcal/mol, respectively. Which substance has the highest specific energy density (kJ/kg)?
9.19 Thermite is a mixture of powdered metallic aluminum and iron oxide (usually FeO). Although stable at room temperature, the reaction 2 Al FeO AlO 2 Fe proceeds quickly when thermite is heated to its ignition temperature. The reaction itself generates much heat. Thermite is used as an intense heat source for welding and is particularly useful because it does not need an oxygen supply. What is the energy density in megajoules per kilogram of thermite?
9.20 An inexpensive hand warmer uses an exothermic chemical reaction to produce heat: iron reacts with oxygen to form ferric oxide, Fe. Write a balanced chemical reaction for this oxidation process. Compute the energy liberated per kilogram of iron. Assuming that your hands each weigh about 0.4 kilogram and assuming that the specific heat of your hands is the same as water, what mass of iron would be needed to warm your hands from 10℃ to body temperature, ~38℃?
9.21 Show that roasting lead (2 PbS 3 O 2 PbO 2 SO) is an exothermic reaction and compute the free energy of this reaction under standard conditions. See Table 9.10 for data.
Table 9.10 Data for Problem 9.21.
Compound (kJ/mol) (kJ/mol)
PbO(s)
PbS(s)
SO(g)
9.22 Show that the method of refining tin explained in Example 9.3 will not work for alumina.
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1 There is a further complication in this story from the fact that there are actually many different (16 known) solid phases of water ice, some of which can form at low temperatures at atmospheric pressure. We also ignore this subtlety here, which is irrelevant for the rest of the discussion in this chapter.
2 Einstein’s model, presented in the previous chapter, is based on the idea that all bonds are characterized by a single frequency . It predicts that heat capacities of solids vanish exponentially at small T. Instead, they are observed to vanish like . A more sophisticated model with a range of frequencies is required to describe this behavior. Soon after Einstein, the Dutch physicist Peter Debye constructed such a model, which includes a range of vibrational frequencies. Like the Einstein model, the Debye model depends on a single parameter, the Debye temperature . The Debye temperature corresponds to the temperature at which the most energetic (shortest wavelength) modes of the lattice are excited. At temperatures below the Debye temperature, the solid acts as a phonon gas, and is described in a very similar fashion to a photon gas, which is analyzed in §22.3. For solids where the Einstein model is good, and agree up to a constant factor of order one.
3 The conversion factor 1 eV/molecule = 96.49 kJ/mol is useful here.
4 Actually diatomic molecules made of the same atoms, like O or N, are exceptional: because they are symmetric upon interchanging the atoms, they have no permanent electric dipole moment. This means that they cannot emit or absorb single photons and make a transition from one vibrational energy level to another. HO and CO, which do not have this symmetry, do not have this restriction. See §34 for further discussion.
5 The fraction ionized x is determined by the Saha–Langmuir equation, which in this case gives , where , where is in eV and p is in atmospheres. Note that the large fraction of ionized hydrogen atoms at roughly 10% of arises for a similar reason to the breakdown of the Debye scaling of the specific heat at a temperature of roughly .
6 There is a vigorous ongoing discussion of the possible health risks of non-ionizing radiation.
7 Here we anticipate the notation introduced in §17 and §18, where a neutral atom is labeled by the chemical species (which depends on the number of protons) and the total number of protons and neutrons A, in the form , and the fully ionized nucleus of the atom is labeled by .
8 Light nuclei with fewer than ~15–20 nucleons (neutrons protons) are more variable.
9 As discussed in §8.8, a negative reaction free energy is a necessary, but not sufficient, condition for a reaction to proceed. Activation barriers often inhibit reactions that are thermodynamically allowed.
10 Although originally named for the reaction of eq. (9.10), calcination generally refers to any process in which volatile material is driven off from an ore by heating it.
11 Note that some fraction of the CO emitted through calcination in concrete production is reabsorbed in the lifetime of concrete. This fraction can be a half or more with proper recycling techniques; variations in manufacturing technique can also increase the reabsorption of CO in the setting process.
12 Notice that the numbers quoted in Table 9.6 are all referred to the standard temperature of 25. To apply these ideas quantitatively in practice these values must be determined for different temperatures.
13 For example, the term combustion is used to describe sulfur burning in the powerful oxidant fluorine, F2 SF.
14 Our convention differs, however, from one offered by the US Department of Energy [53] (Problem 9.14).
15 Wood also contains other organic polymers such as lignin (see §26 (Biofuels)), but the enthalpy of combustion of the other polymers is close enough to that of cellulose for the purpose of this problem.
CHAPTER 10
Thermal Energy Conversion
Heat engines – devices that transform thermal energy into mechanical work – played a fundamental role in the development of industrial society. In the early eighteenth century, English ironmonger Thomas Newcomen developed the first practical steam engine. Newcomen's design, shown schematically in Figure 10.1, employed a reciprocating piston, i.e. a piston that is forced to move back and forth as heat energy is converted to mechanical energy. Later improvements by Scottish engineer James Watt led tothe widespread use of the reciprocating-piston steam engine in a variety of applications, from pumping water out of mines and driving factory machinery to powering steamboats and locomotives. The ability to easily and relatively effectively convert heat from combustion of coal or other fuels into usable mechanical energy via steam engines provided the power needed for the Industrial Revolution that transformed society in the early nineteenth century. Many new types of engines followed in the late nineteenth and twentieth centuries. Internal combustion engines were developed through the latter half of the nineteenth century, and the modern steam turbine was invented and implemented in the1880s.
Figure 10.1 Thomas Newcomen's eighteenth-century steam engine was originally designed to pump water out of coal mines that were subject to frequent flooding. (Credit: Redrawn from Newton Henry Black, Harvey Nathaniel Davis, via Wikimedia Commons)
In the mid twentieth century, heat extraction devices such as refrigerators, which use mechanical work to move thermal energy from a colder material to a warmer material, came into widespread use. The history and importance of refrigerators, air conditioners, and heat pumps are surveyed in §13, where these devices are discussed in detail. Heat extraction devices rely on the same principles as heat engines, and are essentially heat engines run in reverse.
Modern society depends critically upon the dual uses of heat engines in electric power generation and transport. As can be seen from Figure 3.1(a), over two-thirds of the world's electric power in 2013 came from thermal energy produced by combustion of coal, oil, and natural gas. And most vehicles used currently for transport, from automobiles to jet airplanes, rely on fossil fuel powered internal combustion engines. Even non-fossil fuel energy sources such as nuclear and solar thermal power plants depend on heat engines for conversion of thermal energy to mechanical and electrical energy.
Reader’s Guide
This chapter and the three that follow describe the physics of heat engines and heat extraction devices, which convert thermal energy into work or move thermal energy from low to high temperature. In this chapter we describe the basic thermodynamics behind these devices in the context of gas phase closed-cycle systems. We focus on the idealized Carnot and Stirling engines that have simple theoretical models. In the following chapters we turn to the more complex devices that dominate modern applications. Internal combustion engines are described in §11. Phase change and its utility in engines, refrigerators, air conditioners, and power plant turbines is described in §12, and the corresponding engine cycles are surveyed in §13.
Prerequisites: §5 (Thermal energy), §8 (Entropy and temperature). In particular, this chapter relies heavily on the first and second laws of thermodynamics, the notion of heat capacity, and the Carnot limit on efficiency.
Aside from their role as a foundation for §11–§13, the ideas developed in this chapter are relevant to the practical implementation of any thermal energy source, including §19 (Nuclear reactors), §24 (Solar thermal energy), §32 (Geothermal energy), and §33 (Fossil fuels).
In this chapter we focus on simple heat engines that employ a confined gas, known as the working fluid, to absorb and release heat in the process of doing work. In such closed-cycle engines, the working fluid is kept within the containment vessel and is not replaced between cycles. By contrast, in open-cycle engines (such as those in most automobile engines, described in §11) some or all of the working fluid is replaced during each cycle. Closed-cycle engines employ a sequence of processes such that after a single complete cycle, the working fluid and containment vessel return to their initial state. In an idealized implementation without friction or other losses, the only effects of the engine cycle are to transform some heat energy into mechanical energy, and to transfer some heat energy from a region of higher temperature to a region of lower temperature (generally the surrounding environment). In the engines we consider in this chapter, the working fluid is assumed to be a gas that does not change phase through the engine cycle. Engines and heat transfer devices in which the fluid changes phase, including steam engines and practical refrigerators and air conditioners, are described in §12 and §13.
Figure 10.2 In a simple heat engine, a working fluid is contained within a cylindrical vessel. A reciprocating piston moves back and forth as the working fluid expands and contracts, converting heat energy into mechanical energy.
Any heat engine cycle incorporates some basic component processes:
Heat input Heat is transferred into the working fluid from a substance at a higher temperature ().
Expansion Thermal energy within the fluid is transformed into mechanical energy through work as the working fluid expands.
Heat output Entropy, associated with waste heat, must be dumped to a substance at a lower temperature () to satisfy the second law.
Contraction To close the cycle, the containment vessel contracts to its original size.
These processes need not be completely distinct. For example, the first two processes can be combined by allowing the working fluid to expand while adding heat to maintain a constant temperature. This process is known as isothermal expansion (§10.2.1). We can describe the sequence of processes involved in a heat engine by following the state of the working fluid using state variables such as the pressure p, volume V, and temperature T.
While in this chapter we focus on the conversion of thermal energy to mechanical energy through heat engines, which can in turn be used to power electrical generators, there are also mechanisms that can convert thermal energy directly into electrical energy. The process of thermoelectric energy conversion is described in Box 10.1. Such mechanisms can be useful in niche applications where size and reliability are primary considerations; due to the relatively low conversion efficiency of existing technologies, however, they are not suitable for large-scale use in electric power production.
Box10.1 Thermoelectric Energy Conversion
An electric current passing through a resistor results in Joule heating, transforming low-entropy electromagnetic energy into higher-entropy thermal energy. This is only the simplest example, however, of a more complex set of thermoelectric relations between electric and thermal energy transfer. One of these, the Seebeck effect, which transforms thermal into electrical energy, forms the basis for exceptionally stable and long-lived thermoelectric generators that find application in circumstances where the engines described in §10–§13 are not sufficiently reliable.
The Seebeck effect, named after the nineteenth-century German physicist Thomas Seebeck, is the appearance of an electric field within a bar of metal when the ends of the bar are held at different temperatures as shown in the figure below. The Seebeck effect can be characterized by the equation
where the Seebeck coefficient S can be either positive or negative. This relation is somewhat of an over-simplification – the Seebeck coefficient depends strongly on temperature – but it captures the essence of the physical relationship between temperature gradient and current. For most common materials, the Seebeck coefficient at room temperature is very small, with of order a few μV/K, though it is higher for some substances, including, bismuth (V/K), silicon (440 μV/K), tellurium (500 μV/K), and selenium (900 μV/K), and for doped semiconductors (see §25).
The physics underlying the Seebeck effect relies on quantum aspects of the physics of solid materials, which we develop in §25. In brief, the idea is that when a material is at a higher temperature, the Boltzmann distribution (8.32) increases the number of electrons in excited higher-energy states, while leaving empty orbitals (holes) in lower-energy states. The rate at which excited electrons diffuse away from the higher-temperature region can differ from the rate at which electrons diffuse back in to fill the vacated low-energy orbitals, and the difference between these rates can lead to a net current characterized by the field .
A simple thermoelectric generator can be constructed by connecting two materials with different Seebeck coefficients (for example, and ) to heat reservoirs at temperatures and as shown in the figure below. The difference in the electric fields in the two materials generates a potential difference between the terminals A and B that can drive a current through an external circuit. Ideally a thermoelectric generator should have a large difference of Seebeck coefficients, , good electrical conductivity (to minimize resistive losses), and poor thermal conductivity (to minimize thermal losses).
