Solutions to Haynie Ch. 1
a solution for biological thermodynamic...
ANSWERS TO EXERCISES Chapter 1 1. Encyclopædia Britannica CD98 defines energy as the “the capacity to produce an effect.” The 1968 edition of the Shorter Oxford English Dictionary gives a variety of definitions of energy. These include “ability or capacity to produce an effect” (1677) and “the power of doing work possessed by a body or system of bodies” (late 19th c.). We can see where the Britannica definition comes from! Most dictionaries and encyclopedias of physics define energy as “the capacity to do work.” Work has a rather special meaning in this context; we shall learn more about this in Chapter 2. 2. Stars (more specifically, supernovae) are the factories in which “heavy” elements like carbon, nitrogen, oxygen and phosphorus are made. A large fraction of the dry mass of the human body is made of these elements. In this sense we are made of “stardust.” We owe our lives to stars, even if they are not gods. Many concepts inherited from the Greeks are now known or believed to be right in broad outline but wrong in detail. We see this in thought of Thales, Anaximenes and other notable Greek thinkers, including especially (Socrates,) Plato and Aristotle. 3. Eqn. 1 provides an extremely good description of the relationship between energy and wavelength, and we are therefore inclined to take it very seriously. This, however, implies that there is no fundamental limit to the magnitude of the energy of photon: it can be as large or small as it pleases. The same is true for other fundamental particles, e.g. the electron, if they are “free” to move in a field of uniform potential. An electron “bound” to an atomic nucleus is not “free:” its energy depends on its distance from the nucleus. This raises some interesting “philosophical” questions. If the total energy of the universe is finite, then why does Eqn. 1 seem to allow photons of arbitrarily large energy? Is there something wrong with the equation? Is it possible for the energy of the universe to be infinite? The imposition of “boundary conditions,” e.g. positively charged nuclei in a chemical group, places restrictions on the energy of an electron. The energy spectrum changes from continuous to discrete, each energy level being separated from the others by a gap. When a visible wavelength photon is absorbed by a pigment molecule, an electron “jumps” to a higher energy (“excited”) state by an amount nearly identical to but slightly less than the energy of the photon (the energy conversion process is not arbitrarily efficient). The immediate biological consequences of the return of the excited electron returns to its ground state are the subject of photosynthesis. If a plant is exposed to relatively high frequency photons (e.g. UV), so much energy is imparted to the bound state electrons of the pigment molecules that the electrons exceed the energy of the excited state and “escape.” These high-energy electrons bombard the plants protein and DNA, breaking covalent bonds. In this way high frequency photons “sunburn” plants. The part of the biosphere that plays a key role in shielding the surface of Earth from health-hazardous UV radiation is the ozone layer. This occurs “naturally” in the atmosphere of Earth and is damaged by man-made chlorofluorocarbons (CFCs). 4. Chlorophylls absorb red and blue light well, but not green light. Plants reflect green light, and that is why they appear green to the human eye. As long as leaves are alive and producing pigment molecules, they are green. In the autumn, when the © 2001-2007 by D.T. Haynie. All rights reserved.
temperature decreases, leaves die. They also change of color, reflecting an underlying change in the distribution of pigment molecules. In late autumn tree leaves are brown, when no pigment molecules are being produced. 5. Energy of a 470 nm photon = E470 = (6.63 × 10–34 J s) × (2.998 × 108 m s–1) / (470 × 10–9 m) = 4.2 × 10–19 J. E700 = E470 × 470 / 700 = 2.8 × 10–19 J. 3.5 × 10–19 J photon–1 × (6.02 × 1023 photons mol–1) = 2.1 × 105 J mol–1. 1 cal = 4.184 J. 2.1 × 105 J mol–1 / (7 × 103 cal mol–1 × 4.184 J cal–1) = 7.2. That is, a single visible wavelength photon corresponds to an energy about seven times greater than the energy released upon hydrolysis of one molecule ATP to ADP under standard state conditions. 6. 1000 µE s–1 = 103 × 10–6 E s–1 × 6.02 × 1023 photons E–1 × 36 × 102 s hr–1 = 217 × 1022 photons hr–1. 0.4 efficiency × 217 × 1022 photons hr–1 = 86.7 × 1022 effective photons hr–1. 86.7 × 1016 photons hr–1 / (8 photons / molecule of CO2 fixed) = 10.8 × 1022 molecules of CO2 fixed hr–1. 10.8 × 1022 molecules of CO2 hr–1 × (1 mol CH2O produced / 1 mol CO2 consumed) × 30 g CH2O mol–1 / (6.02 × 1023 mol–1) = 1.8 × 10–1 mol of CH2O hr–1 = 5.4 g of CH2O produced. 7. At least 2870 kJ mol–1 are needed to synthesize glucose. The energy of a 700 nm photon is E = hc/λ = 6.63 × 10–34 J s × 2.998 × 108 m s–1 / (700 × 10–9 m) = 0.0284 × 10–17 J One mole of such photons has an energy of 0.02839 × 10–17 J × 6.02 × 1023 = 171 kJ. The energy required to fix one mole of CO2 is 2870 kJ / 6 = 478 kJ. The number of 700 nm photons required is 478 kJ mol–1 / 171 kJ mol–1 = 2.80. Because 3–4 times that many photons are required, the efficiency of the process is 1/4–1/3, or 25–33%. In comparison with industrial chemical processes, the level of efficiency of biotic glucose synthesis is very good. 8. Be creative!
© 2001-2007 by D.T. Haynie. All rights reserved.
9. Spatial: dry land on the surface of Earth (the continents); homes of persons with a particular surname (the surnames ‘Davenport’ and ‘Tyrer’ are today common in the Liverpool–Manchester region of England but rare in Scotland); sports facilities (stadium and gym on campus, stadium in the city, private health clubs throughout the city, etc.). Temporal: appearance of dry land on the surface of Earth (this certainly changes, owing largely to the melting and freezing of polar ice and continental drift); arrival times of persons with particular surnames (permanent English settlement of North America began in the early 17th century; Spanish exploration and settlement of the Americas had begun earlier); access to one of a number of nearby sports facilities (different opening and closing times and so on). All these can also be thought of spatio-temporal distributions. Other temporal distributions: the number of cars passing a given point of roadway per unit time (high at rush hour, low at 2 am); the number of students arriving to class per minute (low until about one minute before class is scheduled to begin, very high at the time class is scheduled to begin, moderately high about one minute after class has begun …); home energy consumption (high in the morning, practically nil during the day when no one is present, high in the evening, practically nil at night). Spatio-temporal: where people are standing or sitting during a party; where all the cars of a city are located; where all ants of an anthill are located during the cycle of digging a tunnel, having this rained out, foraging for food, “calling” other ants to help carry a freshly smashed cockroach (lots of nutritious protein) back to the nest, and so on. 10. Closed: an unopened bottle of delicious Cabernet Sauvignon from vineyards in the vicinity of La Charité, France; a lava lamp, its volatile fluid, and “lava;” a thermometer and all its poisonous mercury. Open: any living organism; an automobile with its vents open or windows down; a bookstore. 11. Electrical or gas energy is used to heat the water. Electrical energy is used to keep the milk cool in the fridge. Chemical energy is converted to mechanical energy in pouring the hot water into the teapot and in opening of the fridge to get the milk. 12. The astronaut in a spaceship is a closed system (ignoring space walks!). In fact, the spaceship is nearly an isolated system, exchanging neither matter nor energy with its surroundings. The astronaut, however, is an open system; she exchanges both matter and energy can be exchanged with the interior of the spacecraft. 13. Reptiles have no internal mechanism for regulating body temperature. When a reptile’s body temperature falls below the optimum, it moves to a region of the environment of higher temperature. If this is not possible, activity drops, movement becomes sluggish, heartbeat slows, and rate of breathing decreases. 14. 11 g glucose × 15.6 kJ (g glucose)–1 = 172 kJ. © 2001-2007 by D.T. Haynie. All rights reserved.
15. Banana skins turn brown as the sugar in them is oxidized. The process occurs very rapidly after the fruit has been peeled, because the skin becomes exposed to a high concentration of oxygen on both its sides. 16. 135 W × (1 J s–1 / W) × 60 s min–1 × 60 min hr–1 × 24 hr day–1 = 11.7 MJ. 17. The energy of catabolism is less than the energy of combustion in a calorimeter because the body does not completely oxidize foodstuffs. Some partial breakdown products are utilized as precursors for biochemical physiological reactions. 18. 8 kg fat × (37 kJ (g fat)–1) / (16 kJ (g carbohydrate)–1) = 18.5 kg carbohydrate She would be heavier by 18.5 kg – 8 kg = 10.5 kg. 19. Student A’s daily energy expenditure/requirement (EA) = (7.5 hr × 5.0 kJ min–1) + (15 hr × 5.9 kJ min–1) + (1.5 hr × 13.4 kJ min–1) × 60 min hr–1 = 8.8 MJ. EA – EB = ((7.5 hr – 7.5 hr) × 5.0 kJ min–1) + ((15 hr – 16 hr) × 5.9 kJ min–1) + ((1.5 hr – 0.5 hr) × 13.4 kJ min–1) = 450 kJ. 8,766 kJ / (1.5 kcal g–1 wet wt of carbohydrate × 4.184 J cal–1) = 1.4 kg carbohydrate. 1,397 g × 1.5 / 8.8 = 240 g fat. 1,397 g × 1.5 / 1.5 = 1.4 kg protein. 1 hr exercise × 13.4 kJ min–1 × 60 min hr–1 / (1.5 kcal g–1 wet wt of carbohydrate × 4.184 J cal–1) = 130 g carbohydrate. The last four calculations assume that the substance is hydrated and that it is burned to completion. Water content will of course depend on the type of food, and the metabolic pathways of the body do not completely oxidize foodstuffs. See legend of Table 1.2. 20. The difference in mass, ∆m, between the products and reactants is ∆m = 2 × (mass of 2H) – (mass of 3He + mass of n) = 2(2.0141 a.m.u.) – (3.0160 a.m.u. + 1.0087 a.m.u.) = 0.0035 a.m.u. The mass difference for one mole of 3He formed is 0.0035 a.m.u. reaction–1 × (1.6605 × 10–27 kg a.m.u.–1) × (6.02 × 1023 reactions) = 3.5 × 10–6 kg, or just a few milligrams. The heat energy released per mole of reactions is E = ∆mc2 = 3.499 × 10–6 kg × (2.998 × 108 m s–1)2 = 3.14 × 108 kJ mol–1, the energy released on cooling the water of several Olympic-sized swimming pools by 1 °C. © 2001-2007 by D.T. Haynie. All rights reserved.
21. WEP1996 = (320 + 55) × 1015 Btu × 1.055 kJ / Btu = 395 × 1015 kJ. ((375 – 320) / 320) × 100 = 17 % 0.1719 / 9 yr = 0.019 yr–1 or 2 % p.a. Contribution of U.S.A. to WEP = 73 / 375 = 19.5 %. 0.00025 captured × 1.7 × 1017 J s–1 = 4.3 × 1010 kJ s–1. 395 × 1015 kJ yr–1 / (365.25 day yr-1 × 24 hr day-1 × 3600 s hr-1) / 4.3 × 1010 kJ s–1 = 0.29. That is, the rate of human energy production is about 1/3 that of photosynthesis. The surface area of a sphere with a radius of 149.6 × 106 km = 4 × π × (149.6 × 106 km)2 = 281,200 × 1012 km2. The area of a circle with the diameter of Earth = π × (12,756 km / 2)2 = 127,800,000 km2. The ratio of these areas is 454.5 × 10–12. The total energy output of the Sun is therefore estimated to be 1.7 × 1017 J s–1 × (454.5 × 10–12)–1 = 3.7 × 1026 J s–1. The number of fusion reactions required per second is 3.7 × 1026 J / (3.14 × 1011 J mol–1) = 1.2 × 1015 mol. The number of moles of 2H consumed is twice as large, because there are 2 moles of 2H in each reaction. Mass energy of Earth = 5.976 × 1024 kg × (2.998 × 108 m s–1)2 = 53.71 × 1040 J. 53.71 × 1040 J / (5 × 1024 J) = 10 × 1016. There is a lot of energy stored in the matter of our planet! 22. There are many ways in which energy is to biology as money is to economics. For instance, as described in the text, the total amount of energy in a closed (or isolated) system does not change; one distribution changes to another. Similarly, the total amount of money does not change; it simply changes hands (ignoring, for example, Treasury-led changes in the money supply). ATP is known as the “energy currency of the cell,” because this one molecule is used to “energize” an amazingly broad variety of processes. Similarly, legal tender can be used to purchase a wide range of things, from objects to services to shares on the stock exchange.
© 2001-2007 by D.T. Haynie. All rights reserved.