Assignment 8, solutions

Problem 1:

imageA hollow ball with a volume V is held in place in a tank under water by a wire under a sloped plank as shown in the figure.  The water density is ρ and average ball density is ρ/5.  The plank makes an angle α with the horizontal, with tan(α) = 1/3.   What is the tension in the wire if the whole system is accelerating horizontally with acceleration a = g/6.

Solution:

Problem 2:

Particles evaporating in the z-direction from a hole in a container with an ideal gas at temperature T.
(a)  Compute the speed distribution, f*(v), of the evaporating particles.  
(b)  Compute the mean z-component of the velocity, <vz>* of the escaping particles.
(c)  Compute the mean energy of the particles leaving the container.
Assume that the equilibrium in the interior of the container is not disturbed by the evaporating particles.

Maxwell-Boltzmann speed distribution:  f(v) = (m/(2πkT))3/2 4πv2exp(-mv2/(2kT)

Solution:

Problem 3:

(a)  One mole of ideal gas with constant heat capacity CV is placed inside a cylinder.  Inside the cylinder there is a piston which can move without friction along the vertical axis.  Pressure P1 is applied to the piston and the gas temperature is T1
At some point, P1 is abruptly changed to P2 (e.g. by adding or removing a weight from the piston).  As a result, the gas volume changes adiabatically.  Find the temperature T2 and the volume V2 after the thermodynamic equilibrium has been reached in terms of CV, P1, T1, and P2.  Use the relation between heat capacities CV and CP to simplify the formulas.
Definition of CV:  dU = CVdT, CP = CV + R

(b)  After the thermodynamic equilibrium has been established in part (a), the pressure is abruptly reset to its original value P1.  Compute final values of the temperature Tf and the volume Vf after the thermodynamic equilibrium has been reached again. 
Compute the difference in temperatures (Tf  - T1) and show that it is quadratic in (P2 - P1).
Comment on the sign of the temperature difference.

Solution:

Problem 4:

The ground state of the neutral lithium atom is doubly degenerate.  The first excited state is 6-fold degenerate, and it is at an energy 1.2 eV above the ground level.
(a)  In the outer atmosphere of the sun, which is at a temperature of 6000 K, what fraction of the neutral lithium is in the first excited level? 
Since all the other levels of Li are at much higher energy, it is safe to assume that they are not significantly occupied.
(b)  Find the average energy of a lithium atom at temperature T.  (Again, consider only the ground and first excited level.)
(c)  Find the contribution of these levels to the specific heat per mole, CV, and sketch CV as a function of 1/T.  Discuss the curve.

Solution:

Problem 5:

A glowing incandescent lamp filament may be regarded as a black body.  If the filament is heated to 2400 K with a power of 100 W, find the surface area of the filament in cm2.

Solution: