27.4: The Frequency of Collisions with a Wall
- Page ID
- 14538
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)In the derivation of an expression for the pressure of a gas, it is useful to consider the frequency with which gas molecules collide with the walls of the container. To derive this expression, consider the expression for the "collision volume".
\[V_{col} = v_x \Delta t\ \cdot A\nonumber \]
in which the product of the velocity \(v_x\) and a time interval \(\Delta t \) is multiplied by \(A\), the area of the wall with which the molecules collide.
All of the molecules within this volume, and with a velocity such that the x-component exceeds \(v_x\) (and is positive) will collide with the wall. That fraction of molecules is given by
\[ N_{col} = \dfrac{N}{V} \dfrac{\langle v_x \rangle \Delta t \cdot A}{2}\nonumber \]
and the frequency of collisions with the wall per unit area per unit time is given by
\[z_w = \dfrac{N}{V} \dfrac{\langle v_x \rangle}{2}\nonumber \]
In order to expand this model into a more useful form, one must consider motion in all three dimensions. Considering that
\[\langle v \rangle = \sqrt{\langle v_x \rangle +\langle v_y \rangle +\langle v_z \rangle}\nonumber \]
and that
\[\langle v_x \rangle = \langle v_y \rangle =\langle v_z \rangle\nonumber \]
it can be shown that
\[ \langle v \rangle = 2 \langle v_x \rangle\nonumber \]
or
\[ \langle v_x \rangle = \dfrac{1}{2} \langle v \rangle\nonumber \]
and so
\[z_w = \dfrac{1}{4} \dfrac{N}{V} \langle v \rangle\nonumber \]
A different approach to determining \(z_w\) is to consider a collision cylinder that will enclose all of the molecules that will strike an area of the wall at an angle \(\theta\) and with a speed \(v\) in the time interval \(dt\). The volume of this collision cylinder is the product of its base area (\(A\)) times its vertical height (\(v\text{cos}\theta dt\)), as shown in figure \(\PageIndex{1}\).
The number of molecules in this cylinder is \(\rho·A·v·\text{cos}\theta dt\), where \(\rho\) is the number density \(\dfrac{N}{V}\). The fraction of molecules that are traveling at a speed between \(v\) and \(v + dv\) is \(F(v)dv\). The fraction of molecules traveling within the solid angle bounded by \(\theta\) and \(\theta + d\theta\) and between \(\phi\) and \(\phi + d\phi\) is \(\dfrac{\text{sin}\theta d\theta d\phi}{4\pi}\). Multiplying these three terms together results in the number of molecules colliding with the area \(A\) from the specified direction during the time interval \(dt\)
\[dN_w = \rho·A·v·\text{cos}\theta \, dt \, · \, F(v)dv \, · \, \dfrac{\text{sin}\theta d\theta d\phi}{4\pi}\nonumber \]
This equation can be rearranged to obtain
\[\dfrac{1}{A}\dfrac{dN_w}{dt} = \dfrac{\rho}{4\pi} vF(v)dv · \text{cos}\theta \, \text{sin}\theta \, d\theta d\phi = dz_w \nonumber \]
Integrating this equation over all possible speeds and directions (on the front side of the wall only), we get
\[z_w = \dfrac{\rho}{4\pi} \int_0^{\infty} vF(v)dv · \int_0^{\pi/2}\text{cos}\theta \, \text{sin}\theta \, d\theta \int_0^{2\pi} d\phi \nonumber \]
The result is that
\[z_w = \dfrac{1}{A}\dfrac{dN_w}{dt} = \dfrac{1}{4} \dfrac{N}{V} \langle v \rangle = \rho\dfrac{\langle v \rangle}{4}\label{27.4.1} \]
Example 27.4.1
Calculate the collision frequency per unit area (\(Z_w\)) for oxygen at 25.0°C and 1.00 bar using equation \(\ref{27.4.1}\):
\[z_w = \dfrac{1}{4} \dfrac{N}{V} \langle v \rangle \nonumber \]
Solution
N molecules = \(N_A\) x \(n\), so that
\[ \dfrac{N}{V} = \dfrac{(N_A) \cdot n}{V} = \dfrac{(N_A) \cdot P}{R \cdot T} \nonumber \]
\[ \dfrac{(6.022 x 10^{23} \, mole^{-1})(1.00 \, bar)}{(0.08319 \, L \cdot bar \cdot mole^{-1} \cdot K^{-1})(298 \, K)} = 2.43 \times 10^{22} \, L^{-1} = 2.43 \times 10^{25} \, m^{-3} \nonumber \]
and
\[ \langle v \rangle = \left({\dfrac{8RT}{\pi M}} \right)^{\dfrac {1}{2}} = \left({\dfrac{8(8.314 J \cdot K^{-1} \cdot mol^{-1})(298K)}{\pi \cdot (0.031999 \, kg)}} \right)^{\dfrac {1}{2}} = 444 \, m\cdot s^{-1} \nonumber \]
Thus
\[z_w = \dfrac{1}{4} (2.43 \times 10^{25} m^{-3})(444 \, m\cdot s^{-1}) \left({\dfrac{1 \, m}{100 \, cm}} \right)^2 = 2.70\times 10^{23} s^{-1} \cdot cm^{-2} \nonumber \]
The factor of N/V is often referred to as the “number density” as it gives the number of molecules per unit volume. At 1 atm pressure and 298 K, the number density for an ideal gas is approximately 2.43 x 1019 molecule/cm3. (This value is easily calculated using the ideal gas law.) By comparison, the average number density for the universe is approximately 1 molecule/cm3.
Exercise 27.4.1
Calculate the collision frequency per unit area (\(Z_w\)) for hydrogen at 25.0°C and 1.00 bar using equation \(\ref{27.4.1}\):
\[z_w = \dfrac{1}{4} \dfrac{N}{V} \langle v \rangle \nonumber \]
- Answer
-
\[ \langle v \rangle = 1770 \, m\cdot s^{-1} \nonumber \] and \[Z_w = 1.08\times 10^{24} s^{-1} \cdot cm^{-2} \nonumber \]
Contributors and Attributions
Patrick E. Fleming (Department of Chemistry and Biochemistry; California State University, East Bay)
- Tom Neils, Grand Rapids Community College