# 27.4: The Frequency of Collisions with a Wall

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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 $$\ref{27.4.1}$$:

$z_w = \dfrac{1}{4} \dfrac{N}{V} \langle v \rangle \nonumber$

$\langle v \rangle = 1770 \, m\cdot s^{-1} \nonumber$ and $Z_w = 1.08\times 10^{24} s^{-1} \cdot cm^{-2} \nonumber$