Calculate the mean free path for $35$ spherical jelly beans in a jar that is vigorously shaken. The volume of the jar is $1.0$ $\mathrm{L}$ and the diameter of a jelly bean is $1.0$ cm.

Let $\rho_n$ be the number of molecules per unit volume in a gas. If we define $p_n$ for a very small volume in a gas $—$ say, one equal to $10$ times the volume of an atom $—$ then $\rho_n$ fluctuates with time through the range of values zero to some maximum value. How then can we justify a statement that $\rho_n$ has a defi- nite value at every point in a gas?

Estimate the van der Waals constant $b$ for $\mathrm{H}_2{O}$ knowing that one kilogram of water has a volume of $0.001$ m$^3$. The molar mass of water is $18$ g/mol.

Find the fraction of particles in a gas having translational ki- netic energies within a range $0.02kT$ centered on the most probable energy $E_p$. (Hint: In this region, $N(E)\thickapprox$ constant. Why?)

A golf ball is suspended from the ceiling by a long thread. Explain in detail why its Brownian motion is not readily ap- parent.

We have assumed that the force exerted by molecules on the wall of a container is steady in time. How is this justi- fied?

A steel tank contains $315$ g of ammonia gas $(\mathrm{NH}_3)$ at an ab- solute pressure of $1.35\times10^6$ $\mathrm{Pa}$ and temperature $77.0\degree\mathrm{C}$. (a) What is the volume of the tank? (b) The tank is checked later when the temperature has dropped to $20.0\degree\mathrm{C}$ and the absolute pressure has fallen to $8.68\times10^5$ $\mathrm{Pa}$. How many grams of gas leaked out of the tank?

A container encloses two ideal gases. Two moles of the first gas are present, with molar mass $M_1$. Molecules of the second gas have a molar mass $M_2=3M_1$, and $0.5$ mol of this gas is present. What fraction of the total pressure on the container wall is attributable to the second gas? (Hint: See earlier Problem.)

List examples of the Brownian motion in physical phenom- ena.

Calculate the fraction of particles in a gas moving with trans- lational kinetic energy between $0.01kT$ and $0.03kT$. $($Hint: For $E\ll$ $kT$, the term $e^{-E/kT}$ in earlier Eq. can be replaced with $1-E/kT$. Why?$)$

An atom of germanium (diameter=246 pm) escapes from a furnace $(T=4220$ $\mathrm{K})$ with the root-mean-square speed into a chamber containing atoms of cold argon (diameter$=$ $300$ pm) at a density of $4.13\times10^{19}$ atoms/cm$^3$. (a) What is the speed of the germanium atom? (b) If the germanium atom and an argon atom collide, what is the closest distance be- tween their centers, considering each as spherical? (c) Find the initial collision frequency experienced by the germanium atom.

The slit system in earlier Figure selects only those molecules moving in the $+x$ direction. Does this destroy the validity of the experiment as a measure of the distribution of speeds of molecules moving in all directions?

Show that, for atoms of mass $m$ emerging as a beam from a small opening in an oven of temperature $T$, the most probable speed is $v_p=\sqrt{3kT/m}$.

The fraction of molecules within a given range $\delta$ $v$ of the rms speed decreases as the temperature of a gas rises. Explain.