Thermodynamics, Statistical Physics, and Quantum Mechanics

THERMODYNAMICS AND STATISTICAL PHYSICS 43

4.90 Gas Fluctuations (Moscow Phys-Tech)

A high-vacuum chamber is evacuated to a pressure of atm. Inside

the chamber there is a thin-walled ballast volume filled with helium gas

at a pressure atm and a temperature On one wall

of this ballast volume, there is a small hole of area A

detector counts the number of particles leaving the ballast volume during

timeintervals

a)

b)

c)

Find the average number of molecules counted by the detector.

Find the meansquarefluctuation ofthisnumber.

What is the probability of not counting any particles in one of the

measurements?

4.91 Quivering Mirror (MIT, Rutgers, Stony Brook)

a)A very small mirror is suspended from a quartz strandwhose elas-

tic constant is D. (Hooke’s law for the torsional twist of the strand

is where is the angle of the twist.) In a real-life ex-

periment the mirror reflects a beam of light in such a way that the

angular fluctuations caused by the impact of surrounding molecules

(Brownian motion) can be read on a suitable scale. The position of

for a strand with dyn.cm, it was found

that You may also use the universal gas constant

Calculate Avogadro’s number.

Can the amplitude of these fluctuations be reduced by reducing gas

density? Explain your answer.

b)

4.92 Isothermal Compressibility and Mean Square
Fluctuation (Stony Brook)

a) Derive the relation

where is the isothermal compressibility:

the equilibrium is One observes theaverage value and

the goal is to find Avogadro’s number (or, what is the same thing,

determine the Boltzmann constant). The following are the data: At