Skip to main content
Chemistry LibreTexts

27.4: The Frequency of Collisions with a Wall

  • Page ID
    14538
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

    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 \(\PageIndex{1}\).

    collision with wall.svg
    Figure \(\PageIndex{1}\): The collision cylinder for determining the number of collisions of gas molecules with a wall. (CC BY-NC; Ümit Kaya)

    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

    27.4: The Frequency of Collisions with a Wall is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by LibreTexts.