Gas of Dissimilar Particles, 1 and 2

The rudimentary concept of molecular collisions will be relevant to future discussions of reaction kinetics and mechanisms. Here we consider what insights kinetic theory (separate from the field of reaction kinetics) can provide.

First, consider a gas of dissimilar particles, 1 and 2. We want to estimate the number of collisions a single particle of 1 will make with all other 2 particles. Then, we'll consider the total number of collisions per unit of time. Since these properties are per unit of time, they are intuitively frequencies

One of the assumptions that we will invoke is that the density is low (or the distance between is particles is "sufficiently" large; or the particles have very weak "effective" interactions). All of these assumptions can justifiably argued and leave room for further exploration. For now, we proceed with a simplistic model.

Consider Figure 33.13 in Engel and Reid. Here we have a particle moving with some effective speed (which we can determine based on our earlier discussion of measures of probability distributions of speeds; we'll see how to do this shortly). Consider the cylindrical volume swept out by a particle moving with average effective speed $<\nu_{eff}>$.

• length of the cylinder generated by particle moving for a time $\Delta t$: $length \ = \ <\nu_{eff}> \ \Delta t$

• surface area of base of cylinder $A \ = \ \sigma \ = \ \pi \ ( r_1 \ + \ r_2)^2$
  $(r_1 + r_2)$ is the effective radius of the cylinder

• The volume is then $V_{cylinder} \ = \ \ \pi \ ( r_1 \ + \ r_2)^2 \ <\nu_{eff}> \ \Delta t \ \ \ \ \ \ \ \ \ \ \ \ \ = \ \sigma \ <\nu_{eff}> \ \Delta t$.

• The effective speed, $<\nu_{eff}>$ is determined from consideration of the apparent speed between two particles in a collision when both have a certain speed. We can approximate this effective speed as follows:

  
  

  $$< \nu_{eff}> \ = \ \left ( <\nu_1>^2 \ + \ <\nu_2>^2 \right )^{1/2} \ = \ \left [ ... ...\ m_1} \right ) + \left ( \frac{8 \ k_B \ T}{\pi \ m_2} \right ) \right ]^{1/2}$$

  $$= \left [ \frac{8 \ k_B \ T}{\pi} \left ( \frac{1}{m_1} \ + \ \frac{1}{m_2} \right ) \right ]^{1/2}$$

  $$= \left ( \frac{8 \ k_B \ T}{\pi \ \mu} \right )^{1/2}$$

  
  

  where the reduced mass $\mu$ is defined as:

  
  

  $$\frac{1}{\mu} \ = \ \frac{m_1 + m_2}{m_1 \ m_2}$$

  
  

The average effective speed, $<\nu_{eff}>$, is determined by considering the relative velocity of two particles traveling with veolcites $\vec{v}_1$ and $\vec{v}_2$.

$$\vec{v}_{eff} \ = \ \vec{v}_1 \ - \ \vec{v}_2$$

The magnitude of this effective velocity, the effective speed, is :

$$\nu_{eff} \ = \ \sqrt{\vec{v}_{eff} \cdot \ \vec{v}_{eff} }$$

$$\nu_{eff} \ = \sqrt{ ( \vec{v}_1 \ - \ \vec{v}_2 ) \cdot ( \vec{v}_1 \ - \ \vec{v}_2 ) }$$

The square of the effective speed is:

$$\nu_{eff}^2 \ = \ \vec{v}_{eff} \cdot \ \vec{v}_{eff}$$

$$\nu_{eff}^2 \ = ( \vec{v}_1 \ - \ \vec{v}_2 ) \cdot ( \vec{v}_1 \ - \ \vec{v}_2 )$$

The average of the square of the effective speed is:

$$\left < \nu_{eff}^2 \right > \ = \ \left< \vec{v}_{eff} \cdot \ \vec{v}_{eff} \right >$$

$$\left < \nu_{eff}^2 \right > \ = \left < ( \vec{v}_1 \ - \ \vec{v}_2 ) \cdot ( \vec{v}_1 \ - \ \vec{v}_2 ) \right >$$

$$\left < \nu_{eff}^2 \right > \ = \left < \vec{v}_1 \cdot \vec{v}_1 \ - \ 2 \ \vec{v}_1 \cdot \vec{v}_2 \ + \ \vec{v}_2 \cdot \vec{v}_2 \right >$$

$$\left < \nu_{eff}^2 \right > \ = \left < \vec{v}_1 \cdot \vec{v}_... ...\vec{v}_1 \cdot \vec{v}_2 \right > + \left < \vec{v}_2 \cdot \vec{v}_2 \right >$$

Because the velocities of particle 1 and particle 2 are uncorrelated, the average of their dot product (or projection on one another) is zero. Thus, $\left< \vec{v}_1 \ \cdot \ \vec{v}_2 \right > = 0$.

$$\left < \nu_{eff}^2 \right > \ = \left < \vec{v}_1 \cdot \vec{v}_1 \right > \ + \ \left < \vec{v}_2 \cdot \vec{v}_2 \right >$$

Thus,

$$\sqrt{\left < \nu_{eff}^2 \right >} \ = \ \sqrt{ \left < \vec{v}_1 \cdot \vec{v}_1 \right > \ + \ \left < \vec{v}_2 \cdot \vec{v}_2 \right > }$$

$$\sqrt{\left < \nu_{eff}^2 \right >} \ = \ \sqrt{ \left < \nu_1^2 \right > \ + \ \left < \nu_2^2 \right > }$$

From our discussion above concerning the relation between root-mean-square and average speeds, we can write the last expression (which is in terms of root-mean-square speeds really) as an expression in terms of average speeds:

$$\sqrt{\left < \nu_{eff}^2 \right >} \ = \ \sqrt{ \left < \nu_1^2 \right > \ + \ \left < \nu_2^2 \right > }$$

$$\sqrt{\frac{3 \ \pi}{8}} \ \left< \nu_{eff} \right> \ = \ \sqrt{ ... ...eft<\nu_1 \right>^2 + \left(\frac{3 \ \pi}{8} \right) \ \left<\nu_2 \right>^2 }$$

$$\sqrt{\frac{3 \ \pi}{8}} \ \left< \nu_{eff} \right> \ = \ \sqrt{\frac{3 \ \pi}{8}} \sqrt{ \left<\nu_1 \right>^2 + \ \left<\nu_2 \right>^2 }$$

$$\left< \nu_{eff} \right> \ = \ \sqrt{ \left<\nu_1 \right>^2 + \ \left<\nu_2 \right>^2 }$$

With the above definitions, we can now define the number of collisions that a single particle will have with other particles in the cylindrical volume in a unit of time as follows:

$$z_{12} \ = \ \frac{N_2}{V} \ \left ( \frac{V_{cyl}}{\Delta t} \right ) \ = \... ... \left ( \frac{\sigma \ \left< \nu_{eff} \right> \ \Delta t}{\Delta t} \right )$$

$$= \frac{N_2}{V} \ \sigma \ \left ( \frac{8 \ k_B \ T}{\pi \ \mu} \right )^{1/2}$$

Are the units of this property consistent?

For the collision frequency of a particle 1 with another particle 1, $z_{11}$, we have:

$$z_{11} = \frac{N_1}{V} \ \sigma \ \sqrt{2} \ \left ( \frac{8 \ k_B \ T}{\pi \ m_1} \right )^{1/2}$$

The total collisional frequency is thus:

$$Z_{12} = \frac{N_1}{V} \ z_{12}$$

$$Z_{12} = \frac{N_1}{V} \ \frac{N_2}{V} \ \sigma \ \left ( \frac{8 \ k_B \ T}{\pi \ \mu} \right )^{1/2}$$

$$Z_{11} = \frac{1}{2} \ \frac{N_1}{V} \ z_{11}$$

$$Z_{11} = \frac{1}{\sqrt{2}} (\frac{N_1}{V})^2 \ \sigma \ \left ( \frac{8 \ k_B \ T}{\pi \ m_1} \right )^{1/2}$$

Note the units for the total collisional frequency; it is per unit volume.