What Is the Average Time Required for H2 to Travel 1.00 M at 298 K and 1 Atm?
Problem i
Consider a collection of gas particles confined to translate in two dimensions (for example, a gas molecule on a surface). Derive the Maxwell speed distribution for such a gas.
Problem 2
Determine $\nu_{k p}, \nu_{\text {ave}},$ and $\nu_{r yard southward}$ for the post-obit species at $298 \mathrm{K}$
a. $\mathrm{Ne}$
b. Kr $\mathbf{c} . \mathrm{CH}_{four}$
d. $\mathrm{C}_{2} \mathrm{H}_{six}$
east. $C_{60}$
Problem 3
Compute $\nu_{yard p}, \nu_{\text {ave}},$ and $\nu_{r m s}$ for $\mathrm{O}_{2}$ at 300 . and 500. 1000. How would your answers change for $\mathrm{H}_{2}$ ?
Problem 4
Compute $\nu_{\text {ave}}$ for $\mathrm{H}_{2} \mathrm{O}, \mathrm{HOD},$ and $\mathrm{D}_{2} \mathrm{O}$ at $298 \mathrm{K} .$ Do you demand to perform the same calculation each time, or can y'all derive an expression that relates the ratio of average speeds for 2 gases to their respective masses?
Problem 5
Compare the average speed and average translational kinetic free energy of $\mathrm{O}_{2}$ with that of $\mathrm{CCl}_{4}$ at $298 \mathrm{M}$.
Problem 6
How far, on boilerplate, does $\mathrm{O}_{2}$ travel in ane second at $298 \mathrm{K}$ and $1 \mathrm{atm} ?$ How does this distance compare to that of Kr under identical weather?
Trouble 7
a. What is the average time required for $\mathrm{H}_{2}$ to travel $1.00 \mathrm{1000}$ at $298 \mathrm{K}$ and 1 atm?
b. How much longer does it have $\mathrm{North}_{2}$ to travel $ane.00 \mathrm{m},$ on boilerplate, relative to $\mathrm{H}_{2}$ under these same conditions?
c. (Challenging) What fraction of $\mathrm{N}_{2}$ particles will crave more than this average time to travel $1.00 \mathrm{m} ?$ Answering this question will crave evaluating a definite integral of the speed distribution, which requires using numerical methods such as Simpson's rule.
Problem eight
As mentioned in Section 33.three, the only differences between the quantities $\nu_{m p}, \nu_{a v eastward},$ and $\nu_{r m s}$ involve constants.
a. Derive the expressions for $\nu_{\text {ave}}$ and $\nu_{r g s}$ relative to $\nu_{m p}$ provided in the text.
b. Your result from part (a) volition involve quantities that are independent of gas-specific quantities such equally mass or temperature. Given this, information technology is possible to construct a "generic" speed distribution curve for speed in reduced units of $\nu / \nu_{m p} .$ Transform the Maxwell distribution into a corresponding expression involving reduced speed.
Problem 9
At what temperature is the $\nu_{r k s}$ of Ar equal to that of $\mathrm{SF}_{six}$ at $298 \mathrm{Grand} ?$ Perform the same adding for $\nu_{thousand p}$.
Problem 10
Determine the temperature at which $\nu_{\text {ave}}$ for Kr is equal to that of $\mathrm{Ne}$ at $298 \mathrm{Thousand}$.
Trouble 11
Compare the average speed and boilerplate translational kinetic energy of $\mathrm{O}_{2}$ with that of $\mathrm{CCl}_{4}$ at $298 \mathrm{Grand}$
$\begin{aligned} f\left(-v_{x 0} \leq v_{x} \leq v_{ten 0}\correct) &=\left(\frac{m}{two \pi k T}\right)^{1 / 2} \int_{v_{x_{o}}}^{v_{x_{o}}} due east^{-m v_{x}^{ii} / two g T} d v_{x} \\ &=\left(\frac{ii m}{\pi k T}\correct)^{i / two} \int_{0}^{v_{x_{s}}} due east^{-m v_{x}^{ii} / 2 k T} d v_{10} \end{aligned}$
The preceding integral can be rewritten using the following substitution: $\xi^{2}=m \nu_{x}^{two} / 2 grand T,$ resulting in $f\left(-\mathrm{v}_{x 0} \leq \mathrm{v}_{x} \leq \mathrm{5}_{x 0}\correct)=$
$2 / \sqrt{\pi}\left(\int_{0}^{\xi_{0}} e^{-\xi^{two}} d \xi\correct),$ which tin be evaluated using the error function defined equally $\operatorname{erf}(z)=2 / \sqrt{\pi}\left(\int_{0}^{z} e^{-ten^{two}} d 10\right)$
The complementary error function is defined as erfc $(z)=$ $1-\operatorname{erf}(z) .$ Finally, a plot of both erf( $z$ ) and erfc $(z)$ as a function of $z$ is shown hither
Using this graph of erf(z), determine the probability that $\left|\mathrm{five}_{ten}\correct| \leq(2 k T / g)^{1 / 2} .$ What is the probability that $\left|\mathrm{v}_{x}\right|>(2 k T / thousand)^{1 / 2} ?$
Trouble 12
The speed of audio is given by $\nu_{\text {sound}}=$ $\sqrt{\gamma k T / m}=\sqrt{\gamma R T / Yard},$ where $\gamma=C_{P} / C_{\nu}.$
a. What is the speed of sound in $\mathrm{Ne}, \mathrm{Kr}$, and $\mathrm{Ar}$ at $m . \mathrm{K} ?$
b. At what temperature will the speed of sound in Kr equal the speed of audio in Ar at $thousand .$ K?
Problem thirteen
For $\mathrm{O}_{ii}$ at 1 atm and $298 \mathrm{Thousand}$, what fraction of molecules has a speed that is greater than $\nu_{r chiliad south} ?$
Problem 14
The escape velocity from Earth's surface is given by $\mathrm{v}_{E}=(ii k R)^{1 / 2}$ where $g$ is the gravitational dispatch $\left(9.807 \mathrm{yard} \mathrm{s}^{-2}\correct)$ and $R$ is the radius of Earth $\left(half-dozen.37 \times 10^{half dozen} \mathrm{m}\right)$
a. At what temperature will $\nu_{\mathrm{mp}}$ for $\mathrm{N}_{2}$ exist equal to the escape velocity?
b. How does the answer for part (a) change if the gas of interest is He?
c. What is the largest molecular mass that is capable of escaping Earth'south surface at $298 \mathrm{K} ?$
Problem fifteen
For $\mathrm{N}_{two}$ at $298 \mathrm{M}$, what fraction of molecules has a speed between 200 . and $300 . \mathrm{thou} / \mathrm{southward} ?$ What is this fraction if the gas temperature is $500 .$ K?
Trouble 16
A molecular beam apparatus employs supersonic jets that allow gas molecules to expand from a gas reservoir held at a specific temperature and force per unit area into a vacuum through a small orifice. Expansion of the gas results for achieving internal temperatures of roughly $10 \mathrm{K} .$ The expansion can be treated as adiabatic, with the modify in gas enthalpy accompanying expansion existence converted to kinetic energy associated with the flow of the gas:
\[\Delta H=C_{P} T_{R}=\frac{1}{2} 1000 \nu^{two}\]
The temperature of the reservoir $\left(T_{R}\right)$ is generally greater than the final temperature of the gas, allowing i to consider the entire enthalpy of the gas to exist converted into translational motion.
a. For a monatomic gas $C_{P}=v / ii R .$ Using this information, demonstrate that the last menses velocity of the molecular axle is related to the initial temperature of the reservoir $\left(T_{R}\right)$ past
\[\nu=\sqrt{\frac{v R T_{R}}{M}}\]
b. Using this expression, what is the flow velocity of a molecular beam of Ar where $T_{R}=298 \mathrm{K} ?$ Notice that this is remarkably similar to the average speed of the gas. Therefore, the molecular beam resulting can exist described as a gas that travels with velocity $\nu$ just with a very low internal energy. In other words, the distribution of molecular speeds effectually the catamenia velocity is significantly reduced in this procedure.
Problem 17
Demonstrate that the Maxwell-Boltzmann speed distribution is normalized.
Problem xviii
(Challenging) Derive the Maxwell-Boltzmann distribution using the Boltzmann distribution introduced in statistical mechanics. Begin by developing the expression for the distribution in translational kinetic energy in 1 dimension then extend it to three dimensions.
Problem 19
Starting with the Maxwell speed distribution, demonstrate that the probability distribution for translational kinetic energy for $\varepsilon_{T} \gg thousand T$ is given by
\[f\left(\varepsilon_{T}\right) d \varepsilon_{T}=2 \pi\left(\frac{i}{\pi k T}\correct)^{iii / 2} e^{-\varepsilon_{T} / k T} \varepsilon_{T}^{one / ii} d \varepsilon_{T}\]
Problem 20
Using the distribution of particle translational kinetic energy provided in Problem $\mathrm{P} 33.nineteen$, derive expressions for the average and nearly probable translational kinetic energies for a collection of gaseous particles.
Problem 21
(Challenging) Using the distribution of particle translational kinetic energy provided in Problem $\mathrm{P} 33.19$ derive an expression for the fraction of molecules that take free energy greater than some energy $\varepsilon^{*} .$ The rate of many chemical reactions is dependent on the thermal free energy available $k T$ versus some threshold energy. Your answer to this question will provide insight into why ane might expect the charge per unit of such chemic reactions to vary with temperature.
Problem 22
As discussed in Chapter 29 the $due north$ th moment of a distribution can be determined as follows: $\left\langle ten^{n}\right\rangle=\int x^{n} f(x) d 10$ where integration is over the entire domain of the distribution. Derive expressions for the $n$ th moment of the gas speed distribution.
Problem 23
Imagine a cubic container with sides $ane \mathrm{cm}$ in length that contains 1 atm of Ar at 298 1000. How many gas-wall collisions are there per second?
Trouble 24
The vapor pressure of various substances tin can be adamant using effusion. In this process, the material of interest is placed in an oven (referred to every bit a Knudsen prison cell) and the mass of material lost through effusion is determined. The mass loss $(\Delta thou)$ is given by $\Delta chiliad=Z_{c} A k \Delta t,$ where $Z_{c}$ is the collisional flux, $A$ is the area of the discontinuity through which effusion occurs, $m$ is the mass of one atom, and $\Delta t$ is the time interval over which the mass loss occurs. This technique is quite useful for determining the vapor pressure level of nonvolatile materials. A $i.00 \mathrm{thousand}$ sample of UF $_{6}$ is placed in a Knudsen cell equipped with a $100 .$ - $\mu$ m-radius hole and heated to $eighteen.2^{\circ} \mathrm{C}$ where the vapor pressure is $100 .$ Torr.
a. The best scale in your lab has an accuracy of $\pm 0.01 \mathrm{g}$. What is the minimum amount of time you must wait until the mass change of the cell tin be determined past your rest?
b. How much UF $_{half-dozen}$ will remain in the Knudsen cell after five.00 minutes of effusion?
Problem 25
Imagine designing an experiment in which the presence of a gas is determined by simply listening to the gas with your ear. The human ear tin can find pressures as low as $two \times 10^{-five} \mathrm{N} \mathrm{grand}^{-two}$. Assuming that the eardrum has an area of roughly $1 \mathrm{mm}^{2},$ what is the minimum collisional rate that can be detected by ear? Assume that the gas of interest is $\mathrm{Due north}_{two}$ at $298 \mathrm{Chiliad}$.
Trouble 26
a. How many molecules strike a $1.00 \mathrm{cm}_{2}$ surface during 1 infinitesimal if the surface is exposed to $\mathrm{O}_{two}$ at 1 atm and $298 \mathrm{K} ?$
b. Ultrahigh vacuum studies typically utilise pressures on the order of $10^{-ten}$ Torr. How many collisions will occur at this pressure level at $298 \mathrm{M} ?$
Problem 27
You are a NASA engineer faced with the task of ensuring that the material on the hull of a spacecraft tin can withstand puncturing by infinite debris. The initial cabin air pressure level in the craft of i atm can drop to 0.7 atm before the prophylactic of the coiffure is jeopardized. The volume of the motel is $100 . \mathrm{m}^{3}$, and the temperature in the cabin is 285 K. Bold information technology takes the space shuttle near 8 hours from entry into orbit until landing, what is the largest circular aperture created by a hull puncture that can be safely tolerated bold that the flow of gas out of the spaceship is effusive? Can the escaping gas from the spaceship be considered as an effusive process? (You can assume that the air is adequately represented by $\mathrm{N}_{2}$.
Problem 28
Many of the concepts developed in this chapter can be practical to agreement the atmosphere. Because atmospheric air is comprised primarily of $\mathrm{N}_{2}$ (roughly $78 \%$ by volume $,$ approximate the atmosphere as consisting only of $\mathrm{North}_{2}$ in answering the post-obit questions:
a. What is the single-particle collisional frequency at body of water level, with $T=298 \mathrm{Yard}$ and $\mathrm{P}=one.0$ atm? The corresponding single-particle collisional frequency is reported as $10^{10} \mathrm{south}^{-one}$ in the CRC Handbook of Chemistry and Physics
$(62 north d \text { ed. }, p . F-171)$
b. At the tropopause $(11 \mathrm{km})$, the collisional frequency decreases to $3.16 \times x^{9} \mathrm{south}^{-one}$, primarily due to a reduction in temperature and barometric pressure (i.e., fewer particles). The temperature at the tropopause is $\sim 220 \mathrm{Grand}$. What is the pressure of $\mathrm{N}_{2}$ at this distance?
c. At the tropopause, what is the mean gratis path for $\mathrm{Northward}_{2} ?$
Trouble 29
a. The stratosphere begins at $11 \mathrm{km}$ above Earth's surface. At this altitude $P=22.half-dozen \mathrm{kPa}$ and $T=-56.5^{\circ} \mathrm{C} .$ What is the mean free path of $\mathrm{N}_{2}$ at this distance assuming $\mathrm{N}_{2}$ is the only component of the stratosphere?
b. The stratosphere extends to $50.0 \mathrm{km}$ where $P=0.085 \mathrm{kPa}$ and $T=18.3^{\circ} \mathrm{C} .$ What is the mean free path of $\mathrm{N}_{two}$ at this altitude?
Problem 30
a. Determine the total collisional frequency for $\mathrm{CO}_{2}$ at one atm and $298 \mathrm{1000}$
b. At what temperature would the collisional frequency exist $10 . \%$ of the value adamant in part (a)?
Problem 31
a. A standard rotary pump is capable of producing a vacuum on the society of $x^{-3}$ Torr. What is the single-particle collisional frequency and mean gratuitous path for $\mathrm{Northward}_{two}$ at this pressure and $298 \mathrm{K} ?$
b. A cryogenic pump can produce a vacuum on the order of $10^{-x}$ Torr. What is the collisional frequency and mean complimentary path for $\mathrm{Northward}_{two}$ at this pressure and $298 \mathrm{M} ?$
Problem 32
Determine the mean free path for Ar at $298 \mathrm{Chiliad}$ at the post-obit pressures:
a. 0.5 atm
b. 0.005 atm
c. $5 \times 10^{-half-dozen}$ atm
Trouble 33
Determine the mean free path at $500 .$ K and 1 atm for the following:
a. $\mathrm{Ne}$
b. $\mathrm{Kr}$
c. $\mathrm{CH}_{4}$
Rather than but calculating the mean free path for each species separately, instead develop an expression for the ratio of mean free paths for two species and utilise the calculated value for one species to determine the other 2.
Problem 34
Consider the following diagram of a molecular beam apparatus:
In the design of the apparatus, it is important to ensure that the molecular beam effusing from the oven does non collide with other particles until the beam is well past the skimmer, a device that selects molecules that are traveling in the appropriate direction, resulting in the creation of a molecular beam. The skimmer is located $10 \mathrm{cm}$ in front of the oven and then that a mean free path of $xx . \mathrm{cm}$ volition ensure that the molecules are well past the skimmer earlier a collision tin occur. If the molecular beam consists of $\mathrm{O}_{2}$ at a temperature of $500 . \mathrm{M}$ what must the pressure level outside the oven be to ensure this mean complimentary path?
Problem 35
A comparison of $\nu_{\text {ave}}, \nu_{m p},$ and $\nu_{r m s}$ for the Maxwell speed distribution reveals that these 3 quantities are not equal. Is the same true for the one-dimensional velocity distributions?
Problem 36
At 30. km above Earth's surface (roughly in the centre of the stratosphere), the pressure is roughly 0.013 atm and the gas density is $3.74 \times x^{23}$ molecules $/ \mathrm{thou}^{3}$ Assuming $\mathrm{N}_{2}$ is representative of the stratosphere, using the collisional diameter information provided in Tabular array 33.i determine
a. the number of collisions a unmarried gas particle undergoes in this region of the stratosphere in 1.0 s.
b. the total number of particle collisions that occur in one.0 s.
c. the mean costless path of a gas particle in this region of the stratosphere.
Source: https://www.numerade.com/books/chapter/kinetic-theory-of-gases-2/