Physical Chemistry

Robert J. Silbey, Robert A. Alberty, Moungi G. Bawendi

Chapter 24

Surface Dynamics - all with Video Answers

Educators

Chapter Questions

Problem 1

In an ultrahigh vacuum chamber $\left(P=5 \times 10^{-10} \text {torr }\right)$ how many molecules strike $1 \mathrm{cm}^{2}$ of surface in one second at 298 $\mathrm{K}$ if $(a)$ the gas is helium and $(b)$ the gas is mercury vapor?

Madi Sousa

Madi Sousa

Numerade Educator

Problem 2

A readily oxidized metal surface with $10^{15}$ metal atoms per square centimeter is exposed to molecular oxygen at $10^{-5} \mathrm{Pa}$ at $298 \mathrm{K} .$ How long will it take to completely oxidize the surface if the oxide formed is $\mathrm{MO} ?$

Madi Sousa

Numerade Educator

Problem 3

In Problem $17.19,$ we found that at 1 bar and $298 \mathrm{K}$ $1.074 \times 10^{28}$ molecules of molecular hydrogen strike a surface per square meter per second. When the 100 plane of metallic copper is exposed to molecular hydrogen under these conditions, what is the rate of collisions with atoms of copper? Copper forms face-centered cubic crystals with the length of the side of the unit cell equal to $361 \mathrm{pm}$

Madi Sousa

Numerade Educator

Problem 4

For the adsorption of nitrogen molecules on a certain sample of carbon, the pressure required to half-saturate the surface at $298 \mathrm{K}$ is $2 \times 10^{-5}$ Pa. If the enthalpy of adsorption is $-10 \mathrm{kJ} \mathrm{mol}^{-1}$ and the sticking coefficient $s^{*}$ is unity, what is the rate constant of desorption $k_{\mathrm{d}} ?$

Madi Sousa

Numerade Educator

Problem 5

The pressures of nitrogen required for adsorption of $1.0 \mathrm{cm}^{3} \mathrm{g}^{-1}\left(25^{\circ} \mathrm{C}, 1.013 \mathrm{bar}\right)$ of gas on graphitized carbon black
are 24 Pa at $77.5 \mathrm{K}$ and 290 Pa at $90.1 \mathrm{K}$. Calculate the enthalpy of adsorption at this fraction of surface coverage.

Madi Sousa

Numerade Educator

Problem 6

A mixture of A and B is adsorbed on a solid for which the adsorption isotherm follows the Langmuir equation. If the mole fractions in the gas at equilibrium are $y_{\mathrm{A}}$ and $y_{\mathrm{B}},$ what is the equation for the adsorption isotherm in terms of total pressure? What is the expression for the mole fraction of A in the adsorbed gas in terms of $K_{\mathrm{A}}, K_{\mathrm{B}}, y_{\mathrm{A}},$ and $y_{\mathrm{B}} ?$

Narayan Hari

Numerade Educator

Problem 7

The following table gives the volume of nitrogen (reduced to $0^{\circ} \mathrm{C}$ and 1 bar) adsorbed per gram of active carbon at $0^{\circ} \mathrm{C}$ at a series of pressures:
\[
\begin{array}{lccccc}
P / \mathrm{Pa} & 524 & 1731 & 3058 & 4534 & 7497 \\
v / \mathrm{cm}^{3} \mathrm{g}^{-1} & 0.987 & 3.04 & 5.08 & 7.04 & 10.31
\end{array}
\]
Plot the data according to the Langmuir isotherm, and determine the constants.

Hast Aggarwal

Numerade Educator

Problem 8

According to Problem 24.7, the Langmuir constant for the adsorption of molecular nitrogen on active carbon at $0^{\circ} \mathrm{C}$ is $K=4.8 \times 10^{-5} \mathrm{Pa}^{-1}$. What pressures of molecular nitrogen are required to cover $10 \%, 50 \%$, and $90 \%$ of the surface at $0^{\circ} \mathrm{C} ?$

Madi Sousa

Numerade Educator

Problem 9

According to Problem 17.43, the rate with which oxygen molecules strike a surface at 1 bar and $25^{\circ} \mathrm{C}$ is $2.69 \times$ $10^{23} \mathrm{cm}^{-2} \mathrm{s}^{-1} .$ If the oxygen molecules are striking a platinum surface, what are the frequencies of collisions per atom on the $(100),(110), \text { and }(111) \text { planes? (See Problem } 23.32 .)$

Madi Sousa

Numerade Educator

Problem 10

One gram of activated charcoal has a surface area of $1000 \mathrm{m}^{2} .$ If complete surface coverage is assumed, as a limiting case, how much ammonia, at $25^{\circ} \mathrm{C}$ and 1 bar, could be adsorbed on the surface of 45 g of activated charcoal? The diameter of the $\mathrm{NH}_{3}$ molecule is $3 \times 10^{-10} \mathrm{m},$ and it is assumed that the molecules just touch each other in a plane so that four adjacent spheres have their centers at the corners of a square.

Madi Sousa

Numerade Educator

Problem 11

Calculate the surface area of a catalyst that adsorbs $10^{3} \mathrm{cm}^{3}$ of nitrogen (calculated at 1.013 bar and $0^{\circ} \mathrm{C}$ ) per gram in order to form a monolayer. The adsorption is measured at $-195^{\circ} \mathrm{C},$ and the effective area occupied by a nitrogen molecule on the surface is $16.2 \times 10^{-20} \mathrm{m}^{2}$ at this temperature.

Madi Sousa

Numerade Educator

Problem 12

The de Broglie wavelength of an electron that has been accelerated through a potential difference of $\phi$ is given by
\[
\lambda=\left(\frac{h^{2}}{2 m e \phi}\right)^{1 / 2}
\]
Derive this equation and verify that it is correct to write equation 24.12

Madi Sousa

Numerade Educator

Problem 13

In LEED experiments, acceleration voltages of 10 to $200 \mathrm{V}$ are generally used. $(a)$ Calculate the energies of electrons accelerated by these voltages in $\mathrm{kJ} \mathrm{mol}^{-1}$. ( $b$ ) Calculate the wavelengths of electrons accelerated by these voltages in $\mathrm{nm}$

Madi Sousa

Numerade Educator

Problem 14

What volume of oxygen, measured at $25^{\circ} \mathrm{C}$ and 1 bar, is required to form an oxide film on $1 \mathrm{m}^{2}$ of a metal with atoms in a square array $0.1 \mathrm{nm}$ apart? Assume that one oxygen atom combines with each metal atom.

Madi Sousa

Numerade Educator

Problem 15

The following table gives data on the adsorption of benzene by graphitized carbon black (P-33) at two surface coverages and two temperatures:
\[
\begin{array}{ccc}
v\left(\mathrm{cm}^{3} \mathrm{g}^{-1} \text {at } 25^{\circ} \mathrm{C}, 1 \mathrm{bar}\right) & 0.2 & 0.4 \\
t / \mathrm{C}^{\circ} & \multicolumn{2}{c} {\text {Pressure}(\mathrm{Pa})} \\
\hline 0 & 13 & 27 \\
35.0 & 80 & 170
\end{array}
\]
Calculate the enthalpy of adsorption $\Delta_{\text {ads }} H$ at each coverage.
(S. Ross and J. P. Oliver, On Physical Adsorption, p. 238. New York: Wiley-Interscience, 1964.)

Madi Sousa

Numerade Educator

Problem 16

The adsorption of ammonia on charcoal is studied at 30 and $80^{\circ} \mathrm{C}$. It is found that the pressure required to adsorb a certain amount of $\mathrm{NH}_{3}$ per gram of charcoal is $14.1 \mathrm{kPa}$ at $30^{\circ} \mathrm{C}$ and $74.6 \mathrm{kPa}$ at $80^{\circ} \mathrm{C}$. Calculate the enthalpy of adsorption.

Madi Sousa

Numerade Educator

Problem 17

Hydrogen is dissociatively adsorbed on a metal, and the pressure required to obtain half of the saturation coverage of the surface is 10 Pa. $(a)$ What pressure will be required to reach $\Theta=0.75 ?(b)$ What pressure would have been required if the adsorption were not dissociative?

Madi Sousa

Numerade Educator

Problem 18

A fresh metal surface with $10^{15}$ atoms per square centimeter is prepared. This surface is exposed to oxygen at $10^{-2}$ Pa. If every oxygen molecule that strikes the surface reacts so that there is one oxygen atom per metal atom in the surface, how long will it take for half of the surface to become oxidized at $25^{\circ} \mathrm{C} ?$

Madi Sousa

Numerade Educator

Problem 19

Show that the SI units of the diffusion coefficient for surface diffusion are $\mathrm{m}^{2} \mathrm{s}^{-1},$ the same as for three-dimensional diffusion.

Madi Sousa

Numerade Educator

Problem 20

The volume of nitrogen gas $v_{\mathrm{m}}$ (measured at 1.013 bar and $0^{\circ} \mathrm{C}$ ) required to form a complete monolayer on a sample silica gel is $129 \mathrm{cm}^{3} \mathrm{g}^{-1}$ of gel. Calculate the surface area per gram of the gel if each nitrogen molecule occupies $16.2 \times$ $10^{-20} \mathrm{m}^{2}$

Madi Sousa

Numerade Educator

Problem 21

The diameter of the hydrogen molecule is about 0.27 $\mathrm{nm} .$ If an adsorbent has a surface of $850 \mathrm{m}^{2} \mathrm{cm}^{-3},$ what volume of $\mathrm{H}_{2}\left(\text { measured at } 25^{\circ} \mathrm{C} \text { and } 1 \text { bar }\right)$ could be adsorbed by $100 \mathrm{mL}$ of the adsorbent? It may be assumed that the adsorbed molecules just touch in a plane and are arranged so that four adjacent spheres have their centers at the corners of a square.

Madi Sousa

Numerade Educator