Physics H133: Problem Set #17

Two-Minute Problems

Remember to give a good explanation, no longer than two sentences.

1. T7T.1: A. What is the density of states for a particle with velocity v? B. Sketch the Boltzmann distribution to check! C. Look at the definition of v_P and check the validity of this claim!
2. T7T.5: What is the typical value of thermal kinetic energy at room temperature?

Chapter T Problems

• T6B.5: This is an application of Eq. (T6.20), but watch out! How many states with n=2 and n=1 are there to choose from?
• T6S.3: How is temperature defined? We are given an expression for the multiplicity - how can we relate this to entropy?
• T6S.8: (a) This is a Unit Q problem! How is the photon wavelength related to the energy difference? (b) Compute first the ratio of probabilities for the two states.
• T7B.1: Room temperature is T=300 K. What is v_P for this temperature? What is the probability for a molecule having a speed v different from v_P, in the range from 495 to 505 m/s?
• T7S.3: This problem requires the integration of the Boltzmann distribution from the escape velocity to infinity. You can use the MBoltz program from the H133 web page, or Mathematica, or ... to do this. To use MBoltz you need to feed it the unitless ratio x_esc = v_esc/v_P. How is the escape velocity v_esc related to the given escape kinetic energy? Where does the temperature enter in this exercise?
• T6R.2: (a) Use the definition of temperature in terms of entropy. Solve for U in terms of T. (b) Calculate Omega_{AB} as a function of U_A (remember that U_B = U - U_A). Where does it peak? How does this correspond to coming into equilibrium? (c) How does the shape of the Omega_{AB} curve depend on the size of the system (N_A and N_B)? Does it get narrow as we saw for the Einstein solid? (d) How does the amount of thermal energy depend on the number of particles in this system and in other ones we've considered? What is the consequence for slicing in half? How easily will it be to change the temperature of such an object?