Hw b 2
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This homework assignment covers several topics in thermodynamics and statistical mechanics. It includes 6 problems related to thermal equilibrium of isolated systems (problem 4), deriving the escape velocity of Earth (problem 7), applying the equipartition theorem to model thermal fluctuations in a spring balance (problem 8), and reviewing probability distributions and average energy as a function of temperature (problems 9-10). Key steps are outlined for some problems, such as using a Taylor expansion to approximate a probability distribution as Gaussian near its maximum for problem 4.
Hw b 2
- 1. Homework B PHY3513 Due: Friday, Feb. 10, 2012 1. (4.2) 2. (4.4) Note that there is a factor of two error in the equations stated in the text for this problem. Also, recognize that dT/dE is related to an important thermodynamic quantity. ∑α e α− x ∂ ⎛ ⎞ 3. (4.7) Hint, use the relation, α =− ln ⎜ ∑ e −α x ⎟ ∑e α α − x ∂x ⎝ α ⎠ 4. Thermal Equilibrium of an Isolated System: Consider two macroscopic systems, A and B, in thermal contact an in equilibrium with each other but isolated from the surroundings. P(EA)dE is the probability that the system A has an energy between EA and EA+dE. Therefore, P ( E A ) = CΩ A ( E A )Ω B ( E B ) (E = EA + EB = constant). We are interested in the behavior of P(EA) near E A = E where P (E ) is maximum. Show that { P ( E ) ≈ P( E ) exp − γ ( E − E ) 2 / 2 } ∂ 2 nΩ A ( E A ) where γ = γ A + γ B and γ A = − ( γ B similarly defined). Use the Taylor ∂E A 2 expansion, not around E = 0 but around E = E . Do you see that γ has to be positive? This problem shows that you can approximate a probability distribution near the maximum as a Gaussian distribution. 5. (4.8) 6. (5.1) Get used to this type of integration! 7. (5.2) The escape velocity is 2GM where G is gravitational constant, M is the mass R of the earth, and R is the radius of the earth. Derive this expression first and then use constants in Appendix A.. 8. Equipartition Theorem A very sensitive spring balance consists of a quartz spring suspended vertically from a fixed support. The spring constant is k. The balance is at temperature T in a location where the gravitational acceleration is g. (a) When a small object of mass m is suspended from the spring, the spring will be stretched out by an amount x and will reach an equilibrium position. However, thermal
- 2. energy will cause fluctuations in x. Ignoring the mass of the spring, what is the average value of x, <x>? (x − x ) 2 1/2 (b) What is the magnitude of thermal fluctuations of the object, ? (c) When the fluctuations become comparable to the average value, the measurement becomes impractical. How small a mass can this balance measure? 9. (5.4) 10. A particle can have an energy ε ranging continuously from 0 to infinity. The particle is subjected to temperature T. (a) What is the probability of having energy ε, P(ε) (with the correct normalization factor)? (b) Show that <ε> = kBT. Compare this result with your answer for #3 of this HW assignment (i.e., problem 4.7 (b) of the text). In what condition does the result of 4.7 (b) reduce to this result?