Chapter Questions
Gold has a molar (atomic) mass of $197 \mathrm{~g} / \mathrm{mol}$. Consider a 2.56$g$ sample of pure gold vapor. (a) Calculate the number of moles of gold present. (b) How many atoms of gold are present?
(a) Find the number of molecules in $1.00 \mathrm{~m}^{3}$ of air at $20.0^{\circ} \mathrm{C}$ and at a pressure of $1.00 \mathrm{~atm} .(b)$ What is the mass of this volume of air? Assume that $75 \%$ of the molecules are nitrogen $\left(\mathrm{N}_{2}\right)$ and $25 \%$ are oxygen $\left(\mathrm{O}_{2}\right)$.
A steel tank contains $315 \mathrm{~g}$ of ammonia gas $\left(\mathrm{NH}_{3}\right)$ at an absolute pressure of $1.35 \times 10^{6} \mathrm{~Pa}$ and temperature $77.0^{\circ} \mathrm{C} .(a)$ What is the volume of the tank? $(b)$ The tank is checked later when the temperature has dropped to $22.0^{\circ} \mathrm{C}$ and the absolute pressure has fallen to $8.68 \times 10^{5} \mathrm{~Pa}$. How many grams of gas leaked out of the tank?
(a) Consider $1.00 \mathrm{~mol}$ of an ideal gas at $285 \mathrm{~K}$ and $1.00 \mathrm{~atm}$ pressure. Imagine that the molecules are for the most part evenly spaced at the centers of identical cubes. Using Avogadro's constant and taking the diameter of a molecule to be $3.00 \times 10^{-8} \mathrm{~cm}$, find the length of an edge of such a cube and calculate the ratio of this length to the diameter of a molecule. The edge length is an estimate of the distance between molecules in the gas. (b) Now consider a mole of water having a volume of $18 \mathrm{~cm}^{3}$. Again imagine the molecules to be evenly spaced at the centers of identical cubes and repeat the calculation in $(a)$.
Consider a sample of argon gas at $35.0^{\circ} \mathrm{C}$ and $1.22 \mathrm{~atm}$ pressure. Suppose that the radius of a (spherical) argon atom is $0.710 \times 10^{-10} \mathrm{~m} .$ Calculate the fraction of the container volume actually occupied by atoms.
The mass of the $\mathrm{H}_{2}$ molecule is $3.3 \times 10^{-24} \mathrm{~g}$. If $1.6 \times 10^{23}$ hydrogen molecules per second strike $2.0 \mathrm{~cm}^{2}$ of wall at an angle of $55^{\circ}$ with the normal when moving with a speed of $1.0 \times 10^{5} \mathrm{~cm} / \mathrm{s}$, what pressure do they exert on the wall?
At $44.0^{\circ} \mathrm{C}$ and $1.23 \times 10^{-2}$ atm the density of a gas is $1.32 \times 10^{-5} \mathrm{~g} / \mathrm{cm}^{3} .(a)$ Find $v_{\mathrm{rms}}$ for the gas molecules. (b)Using the ideal gas law, find the number of moles per unit volume (molar density) of the gas. ( $c$ ) By combining the results of(a) and $(b)$, find the molar mass of the gas and identify it.
A cylindrical container of length $56.0 \mathrm{~cm}$ and diameter $12.5 \mathrm{~cm}$ holds $0.350$ moles of nitrogen gas at a pressure of $2.05$ atm. Find the rms speed of the nitrogen molecules.
At standard temperature and pressure $\left(0^{\circ} \mathrm{C}\right.$ and $\left.1.00 \mathrm{~atm}\right)$ the mean free path in helium gas is $285 \mathrm{~nm}$. Determine $(a)$ the number of molecules per cubic meter and $(b)$ the effective diameter of the helium atoms.
At $2500 \mathrm{~km}$ above the Earth's surface the density is about $1.0$ molecule/cm $^{3}$. ( $a$ ) What mean free path is predicted by Eq. $22-13$ and $(b)$ what is its significance under these conditions? Assume a molecular diameter of $2.0 \times 10^{-8} \mathrm{~cm}$.
At what frequency would the wavelength of sound be on the order of the mean free path in nitrogen at $1.02$ atm pressure and $18.0^{\circ} \mathrm{C} ?$ Take the diameter of the nitrogen molecule to be $315 \mathrm{pm}$
In a certain particle accelerator the protons travel around a circular path of diameter $23.5 \mathrm{~m}$ in a chamber of $1.10 \times 10^{-6}$ $\mathrm{mm}$ Hg pressure and $295 \mathrm{~K}$ temperature. ( $a$ ) Calculate the number of gas molecules per cubic meter at this pressure. (b) What is the mean free path of the gas molecules under these conditions if the molecular diameter is $2.20 \times 10^{-8} \mathrm{~cm} ?$
. In Sample Problem $22-4$, at what temperature is the average rate of collision equal to $6.0 \times 10^{9} \mathrm{~s}^{-1} ?$ The pressure remains unchanged.
The speeds of a group of ten molecules are $2.0,3.0,4.0, \ldots$, $11 \mathrm{~km} / \mathrm{s}$. ( $a$ ) Find the average speed of the group. (b) Calculate the root-mean-square speed of the group.
(a) Ten particles are moving with the following speeds: four at $200 \mathrm{~m} / \mathrm{s}$, two at $500 \mathrm{~m} / \mathrm{s}$, and four at $600 \mathrm{~m} / \mathrm{s}$. Calculate the average and root-mean-square speeds. Is $v_{\mathrm{rms}}>v_{\mathrm{av}} ?(b)$ Make up your own speed distribution for the ten particles and show that $v_{\mathrm{rms}} \geq v_{\mathrm{av}}$ for your distribution. ( $c$ ) Under what condition (if any) does $v_{\mathrm{rms}}=v_{\mathrm{av}} ?$
Calculate the root-mean-square speed of ammonia $\left(\mathrm{NH}_{3}\right)$ molecules at $56.0^{\circ} \mathrm{C}$. An atom of nitrogen has a mass of $2.33 \times 10^{-26} \mathrm{~kg}$ and an atom of hydrogen has a mass of $1.67 \times 10^{-27} \mathrm{~kg}$
The temperature in interstellar space is $2.7 \mathrm{~K}$. Find the rootmean-square speed of hydrogen molecules at this temperature. (See Table 22-1.)
Verify Eq. $22-16$ by evaluating $d N(v) / d v=0$ and solving for $v$.
Evaluate the integral in Eq. $22-17$ to verify Eq. $22-18$.
Evaluate the integral in Eq. $22-19$ to verify that $\left(v^{2}\right)_{\mathrm{av}}=$ $3 \mathrm{kT} / \mathrm{m} .$
Calculate the root-mean-square speed of smoke particles of mass $5.2 \times 10^{-14} \mathrm{~g}$ in air at $14^{\circ} \mathrm{C}$ and $1.07$ atm pressure.
At what temperature do the atoms of helium gas have the same rms speed as the molecules of hydrogen gas at $26.0^{\circ} \mathrm{C} ?$
(a) Compute the temperatures at which the rms speed is equal to the speed of escape from the surface of the Earth for molecular hydrogen and for molecular oxygen. (b) Do the same for the Moon, assuming the gravitational acceleration on its surface to be $0.16 g .(c)$ The temperature high in the Earth's upper atmosphere is about $1000 \mathrm{~K}$. Would you expect to find much hydrogen there? Much oxygen?
You are given the following group of particles $\left(N_{n}\right.$ represents the number of particles that have a speed $v_{n}$ ): $$\begin{array}{lc}N_{n} & v_{n}(\mathrm{~km} / \mathrm{s}) \\\hline 2 & 1.0 \\4 & 2.0 \\6 & 3.0 \\8 & 4.0 \\2 & 5.0\end{array}$$(a) Compute the average speed $v_{\mathrm{av}} .(b)$ Compute the rootmean-square speed $v_{\text {rms. }}$ (c) Among the five speeds shown, which is the most probable speed $v_{\mathrm{p}}$ for the entire group?
In the apparatus of Miller and Kusch (see Fig. $22-8$ ) the length $L$ of the rotating cylinder is $20.4 \mathrm{~cm}$ and the angle $\phi$ is $0.0841$ rad. What rotational speed corresponds to a selected speed $v$ of $212 \mathrm{~m} / \mathrm{s} ?$
. It is found that the most probable speed of molecules in a gas at temperature $T_{2}$ is the same as the rms speed of the molecules in this gas when its temperature is $T_{1}$. Calculate $T_{2} / T_{1}$.
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 $v_{\mathrm{p}}=\sqrt{3 \mathrm{kT} / \mathrm{m}}$.
An atom of germanium (diameter $=246 \mathrm{pm}$ ) escapes from a furnace $(T=4220 \mathrm{~K})$ with the root-mean-square speed into a chamber containing atoms of cold argon (diameter = $300 \mathrm{pm}$ ) at a density of $4.13 \times 10^{19}$ 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 between their centers, considering each as spherical? ( $c$ ) Find the initial collision frequency experienced by the germanium atom. -
Calculate the fraction of particles in a gas moving with translational kinetic energy between $0.01 k T$ and $0.03 k T$. (Hint: For $E<<k T$, the term $e^{-E / k T}$ in Eq. $22-25$ can be replaced with $1-E / k T$. Why?)
Find the fraction of particles in a gas having translational kinetic energies within a range $0.02 k T$ centered on the most probable energy $E_{\mathrm{p} \cdot}$ (Hint: In this region, $N(E) \approx$ constant. Why?)
Estimate the van der Waals constant $b$ for $\mathrm{H}_{2} \mathrm{O}$ knowing that one kilogram of water has a volume of $0.001 \mathrm{~m}^{3}$. The molar mass of water is $18 \mathrm{~g} / \mathrm{mol}$.
The value of the van der Waals constant $b$ for oxygen is $32 \mathrm{~cm}^{3} / \mathrm{mol}$. Compute the diameter of an $\mathrm{O}_{2}$ molecule.
Show that the constant $a$ in the van der Waals equation can be written in units of$\frac{\text { energy per particle }}{\text { particle density }}$