Problem Set 10 - SLIDEBLAST.COM
Massachusetts Institute of Technology Physics Department Physics 8.044 Statistical Physics I
Spring 2014 April 23, 2014
Assignment 9
Due Friday May 2, 2014
Reading Assignment 10 • On thermal (blackbody) radiation Schroeder §7.4 (though he uses the grand canonical ensemble) Jaffe & Taylor §22.2 & 22.3 posted on the 8.044 website • On the grand canonical ensemble: • On the Einstein theory of solids: • On the Debye theory of solids:
Schroeder §7.1 and 7.2 J&T §8.7
Schroeder §7.5
Problem Set 10 1. Adsorption on a stepped surface
Fig.1 Cut through crystal
Fig.2 Looking down on the
perpendicular to the surface
surface, showing adsorption sites
If a perfect crystal is cleaved along a symmetry direction, the resulting surface could expose a single geometrically flat plane of atoms. Alternatively, if the crystal is cut at a slight angle with respect to this direction, the resulting surface might take the form of a series of terraces of fixed width separated by steps of height corresponding to one atomic layer. This situation is illustrated in figure 1. The steps themselves may not be straight lines, but may have kinks when viewed from above as shown in figure 2. If impurity atoms were adsorbed on such a surface, their energy could depend on where they reside relative to the steps. Consider N identical xenon atoms adsorbed on a silicon surface which has a total of M possible adsorption sites. There are three different types of site that the xenon could occupy. 1
Physics 8.044, Statistical Physics I, Spring 2014
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They would prefer to snuggle into a corner site at a kink in a step. Two percent of the sites are corner sites, and their energy defines the zero of the energy scale for adsorbed atoms. Next in preference are edge sites. Eighteen percent of the M sites are edge sites with an energy above that of a corner site. The majority (80%) of the adsorption sites are face sites, but they have an energy which is 2 above that of the corner sites. M is so large compared to N competition for a given site can be neglected; the xenon atoms can be considered completely independent. You may neglect the kinetic energy of the adsorbed atoms. (a) Find the partition function, Z(N, T ), for the xenon atoms in terms of the parameters M and . (b) Find the ratio of xenon atoms on face sites to the number on corner sites. (c) Find an expression for the heat capacity of the adsorbed xenon atoms in the limit kT ⌧ . (d) What is the probability that a given xenon atom will be found on a face site in the limit kT ? (e) What is the limit of the entropy of this system as T ! 1 ?
(f) Should this model for adsorbed atoms exhibit energy gap behavior? Why?
2. A two dimensional atomic trap It is possible to trap neutral atoms between two solid surfaces in a potential of the form V (x, y, z) = ax2 + by 2 where a and b are parameters. The allowed space for the gas extends to infinity in the x and y directions but is limited in the z-direction by the two solid surfaces, which are located at z = ±L/2. The gas is initially in thermal equilibrium at temperature T0 . The values of a and b can be changed slowly (reversibly) and adiabatically (without changing the entropy of the gas). Find the final temperature of the gas in terms of T0 and the initial and final values of a and b. [You should assume throughout that the atoms do not interact with each other. And, you should treat the entire problem classically. That is, assume throughout that kB T is much greater than the spacing between quantum energy levels and assume throughout that the density of the gas is such that the translational motion of the atoms may be treated classically.] 3. Solar energy The sun’s average power output is 384 ⇥ 1024 W. The sun’s radius is R = 6.95 ⇥ 108 m. Earth’s radius is R = 6.38 ⇥ 106 m, and its distance from the sun is 1AU = d = 1.496 ⇥ 1011 m. (a) Treating it as a black body, what is temperature that characterizes the sun’s radiation? (b) What fraction of the sun’s radiation is in the window of visible light (wavelengths between 400 and 750 nanometers)? (c) What is the total solar power hitting Earth?
Physics 8.044, Statistical Physics I, Spring 2014
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(d) The solar constant is defined as the flux of solar energy (watts/meter2 ) incident on the top of Earth’s atmosphere when the sun is directly overhead. What is the solar constant? (e) About 70% of the solar energy incident on Earth is absorbed. The remainder is reflected directly back into space. The fraction reflected, 0.3, is known as Earth’s albedo. If Earth had no atmosphere (but the same albedo as now), assuming it is in radiative equilibrium, what would its surface temperature be? Greenhouse gases, primarily water vapor, maintain Earth’s average surface temperature far about the value it would have in the absence of an atmosphere. Note, however, that the temperature that you computed in the last part of this problem provides a good estimate of the temperature at the top of Earth’s atmosphere, from which thermal energy is radiated back into space. 4. Efficiency of an incandescent light bulb The incandescent light bulb is a notoriously inefficient way to convert electric power into visible light. The tungsten filament emits blackbody radiation at a temperature that is limited by its melting point. Define the lighting efficiency as the ratio of the power emitted in the visible range (400 – 750 nm) to the total power emitted. What is the theoretical maximum efficiency of a tungsten light bulb, given that tungsten melts at 3 422 C? 5. Properties of black body radiation Given the free energy for the radiation gas at temperature T , ✓ Z 1 kB 2 1 F = 2 3TV d!! ~! + ln(1 ⇡ c 2 0
e
~!
)
◆
(a) Compute the entropy, the heat capacity (at constant volume), and the pressure p of the radiation gas. (b) At what temperature is the pressure of a radiation gas equal to 1 atm? (c) Consider a mole of oxygen confined in a volume of one liter at room temperature. In thermal equilibrium what fraction of its heat capacity is due to the heat capacity of the radiation in equilibrium with the oxygen? At what temperature is the heat capacity of the radiation in a one liter volume equal to the gas constant R = 8.3 J/K? 6. More properties of black body radiation In the last problem you computed the entropy and heat capacity of a radiation gas. (a) Compute the number of photons in a radiation gas with volume V and temperature T . (Simplify your answer by expressing numerical constants in terms of values of the Riemann Zeta function, ⇣(3) and ⇣(4)). Notice that the entropy, heat capacity, and number all have the same dependence on temperature and volume. In particular, the entropy per photon is a universal constant independent of T and V .
Physics 8.044, Statistical Physics I, Spring 2014
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(b) Compute the Shannon entropy per photon SShannon = S/(N kB ln 2) of a radiation gas. This is the number of bits on average necessary to completely specify the state of a photon chosen at random from a radiation gas. (c) Rewrite the Sackur-Tetrode formula for the entropy of an ideal monatomic gas as a function of temperature and pressure. Evaluate it numerically for helium at p = 1 atm, and show Shelium ⇡ N kB (1 + 52 ln T ) (T in Kelvin). What is the (Shannon) entropy of helium at room temperature and p = 1 atm? The simple expressions for the energy U , entropy S, and number of photons N in a radiation gas as a function of V and T gives us a chance to check some of the fundamental relations of thermodynamics. (d) Express S as a function of U and V by eliminating T . Then show that the relations @S/@U |V = 1/T and @S/@V |U = p/T lead to the expected formulas for U (T, V ) and p(T, V ). 7. Extra credit 20 points: Earth’s entropy balance Earth is in radiative equilibrium, absorbing solar thermal radiation at an effective black body temperature of T ⇠ = 5, 800 K and emitting thermal radiation at an effective black body temperature of T ⇠ = 255 K. (a) Explain why Earth’s entropy S is decreasing in time. (b) Show that
✓ ◆ T ) dS R R 2 T 3 (T = a⇡ , dt d T where a is Earth’s albedo (a ⇠ = 0.7), and R , R , and d are the radius of the sun, radius of Earth, and distance from the sun to Earth, respectively.
(c) Evaluate dS /dt in W/K. When water freezes its entropy decreases by ⇡ H/T ⇡ 1.22 kJ/kg K. Express Earth’s entropy loss in units of tons of ice frozen per second.
Spring 2014 April 23, 2014
Assignment 9
Due Friday May 2, 2014
Reading Assignment 10 • On thermal (blackbody) radiation Schroeder §7.4 (though he uses the grand canonical ensemble) Jaffe & Taylor §22.2 & 22.3 posted on the 8.044 website • On the grand canonical ensemble: • On the Einstein theory of solids: • On the Debye theory of solids:
Schroeder §7.1 and 7.2 J&T §8.7
Schroeder §7.5
Problem Set 10 1. Adsorption on a stepped surface
Fig.1 Cut through crystal
Fig.2 Looking down on the
perpendicular to the surface
surface, showing adsorption sites
If a perfect crystal is cleaved along a symmetry direction, the resulting surface could expose a single geometrically flat plane of atoms. Alternatively, if the crystal is cut at a slight angle with respect to this direction, the resulting surface might take the form of a series of terraces of fixed width separated by steps of height corresponding to one atomic layer. This situation is illustrated in figure 1. The steps themselves may not be straight lines, but may have kinks when viewed from above as shown in figure 2. If impurity atoms were adsorbed on such a surface, their energy could depend on where they reside relative to the steps. Consider N identical xenon atoms adsorbed on a silicon surface which has a total of M possible adsorption sites. There are three different types of site that the xenon could occupy. 1
Physics 8.044, Statistical Physics I, Spring 2014
2
They would prefer to snuggle into a corner site at a kink in a step. Two percent of the sites are corner sites, and their energy defines the zero of the energy scale for adsorbed atoms. Next in preference are edge sites. Eighteen percent of the M sites are edge sites with an energy above that of a corner site. The majority (80%) of the adsorption sites are face sites, but they have an energy which is 2 above that of the corner sites. M is so large compared to N competition for a given site can be neglected; the xenon atoms can be considered completely independent. You may neglect the kinetic energy of the adsorbed atoms. (a) Find the partition function, Z(N, T ), for the xenon atoms in terms of the parameters M and . (b) Find the ratio of xenon atoms on face sites to the number on corner sites. (c) Find an expression for the heat capacity of the adsorbed xenon atoms in the limit kT ⌧ . (d) What is the probability that a given xenon atom will be found on a face site in the limit kT ? (e) What is the limit of the entropy of this system as T ! 1 ?
(f) Should this model for adsorbed atoms exhibit energy gap behavior? Why?
2. A two dimensional atomic trap It is possible to trap neutral atoms between two solid surfaces in a potential of the form V (x, y, z) = ax2 + by 2 where a and b are parameters. The allowed space for the gas extends to infinity in the x and y directions but is limited in the z-direction by the two solid surfaces, which are located at z = ±L/2. The gas is initially in thermal equilibrium at temperature T0 . The values of a and b can be changed slowly (reversibly) and adiabatically (without changing the entropy of the gas). Find the final temperature of the gas in terms of T0 and the initial and final values of a and b. [You should assume throughout that the atoms do not interact with each other. And, you should treat the entire problem classically. That is, assume throughout that kB T is much greater than the spacing between quantum energy levels and assume throughout that the density of the gas is such that the translational motion of the atoms may be treated classically.] 3. Solar energy The sun’s average power output is 384 ⇥ 1024 W. The sun’s radius is R = 6.95 ⇥ 108 m. Earth’s radius is R = 6.38 ⇥ 106 m, and its distance from the sun is 1AU = d = 1.496 ⇥ 1011 m. (a) Treating it as a black body, what is temperature that characterizes the sun’s radiation? (b) What fraction of the sun’s radiation is in the window of visible light (wavelengths between 400 and 750 nanometers)? (c) What is the total solar power hitting Earth?
Physics 8.044, Statistical Physics I, Spring 2014
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(d) The solar constant is defined as the flux of solar energy (watts/meter2 ) incident on the top of Earth’s atmosphere when the sun is directly overhead. What is the solar constant? (e) About 70% of the solar energy incident on Earth is absorbed. The remainder is reflected directly back into space. The fraction reflected, 0.3, is known as Earth’s albedo. If Earth had no atmosphere (but the same albedo as now), assuming it is in radiative equilibrium, what would its surface temperature be? Greenhouse gases, primarily water vapor, maintain Earth’s average surface temperature far about the value it would have in the absence of an atmosphere. Note, however, that the temperature that you computed in the last part of this problem provides a good estimate of the temperature at the top of Earth’s atmosphere, from which thermal energy is radiated back into space. 4. Efficiency of an incandescent light bulb The incandescent light bulb is a notoriously inefficient way to convert electric power into visible light. The tungsten filament emits blackbody radiation at a temperature that is limited by its melting point. Define the lighting efficiency as the ratio of the power emitted in the visible range (400 – 750 nm) to the total power emitted. What is the theoretical maximum efficiency of a tungsten light bulb, given that tungsten melts at 3 422 C? 5. Properties of black body radiation Given the free energy for the radiation gas at temperature T , ✓ Z 1 kB 2 1 F = 2 3TV d!! ~! + ln(1 ⇡ c 2 0
e
~!
)
◆
(a) Compute the entropy, the heat capacity (at constant volume), and the pressure p of the radiation gas. (b) At what temperature is the pressure of a radiation gas equal to 1 atm? (c) Consider a mole of oxygen confined in a volume of one liter at room temperature. In thermal equilibrium what fraction of its heat capacity is due to the heat capacity of the radiation in equilibrium with the oxygen? At what temperature is the heat capacity of the radiation in a one liter volume equal to the gas constant R = 8.3 J/K? 6. More properties of black body radiation In the last problem you computed the entropy and heat capacity of a radiation gas. (a) Compute the number of photons in a radiation gas with volume V and temperature T . (Simplify your answer by expressing numerical constants in terms of values of the Riemann Zeta function, ⇣(3) and ⇣(4)). Notice that the entropy, heat capacity, and number all have the same dependence on temperature and volume. In particular, the entropy per photon is a universal constant independent of T and V .
Physics 8.044, Statistical Physics I, Spring 2014
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(b) Compute the Shannon entropy per photon SShannon = S/(N kB ln 2) of a radiation gas. This is the number of bits on average necessary to completely specify the state of a photon chosen at random from a radiation gas. (c) Rewrite the Sackur-Tetrode formula for the entropy of an ideal monatomic gas as a function of temperature and pressure. Evaluate it numerically for helium at p = 1 atm, and show Shelium ⇡ N kB (1 + 52 ln T ) (T in Kelvin). What is the (Shannon) entropy of helium at room temperature and p = 1 atm? The simple expressions for the energy U , entropy S, and number of photons N in a radiation gas as a function of V and T gives us a chance to check some of the fundamental relations of thermodynamics. (d) Express S as a function of U and V by eliminating T . Then show that the relations @S/@U |V = 1/T and @S/@V |U = p/T lead to the expected formulas for U (T, V ) and p(T, V ). 7. Extra credit 20 points: Earth’s entropy balance Earth is in radiative equilibrium, absorbing solar thermal radiation at an effective black body temperature of T ⇠ = 5, 800 K and emitting thermal radiation at an effective black body temperature of T ⇠ = 255 K. (a) Explain why Earth’s entropy S is decreasing in time. (b) Show that
✓ ◆ T ) dS R R 2 T 3 (T = a⇡ , dt d T where a is Earth’s albedo (a ⇠ = 0.7), and R , R , and d are the radius of the sun, radius of Earth, and distance from the sun to Earth, respectively.
(c) Evaluate dS /dt in W/K. When water freezes its entropy decreases by ⇡ H/T ⇡ 1.22 kJ/kg K. Express Earth’s entropy loss in units of tons of ice frozen per second.
