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
\[ \langle v \rangle = 2 \langle v_x \rangle\nonumber \]
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 \(dt\)
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 \(\ref{27.4.1}\):
\[z_w = \dfrac{1}{4} \dfrac{N}{V} \langle v \rangle \nonumber \]