# 30.2: A Reaction Cross Section Depends Upon the Impact Parameter

In the previous section, it was assumed that all collisions with sufficient energy would lead to a reaction between the Q and B particles. This is an unrealistic assumption because not all collisions occur with a proper alignment of the particles as shown in Figure $$\PageIndex{1}$$.

Thus, the energy-dependent reaction cross-section,$$\sigma_r(E_r)$$, introduced previously is inaccurate and must be modified to take into account the inefficient collisions. One modification is to employ the line-of-centers (loc) model for $$\sigma_r(E_r)$$. This model incorporates the angle of collision relative to the line drawn between the centers of the two colliding particles, as shown in Figure $$\PageIndex{2}$$. Figure $$\PageIndex{2}$$: This figure shows the geometry of the collision between two particles. The particle velocities are $$\vec{v}_Q$$ and $$\vec {v}_B$$. The distance between the paths traveled by the two particle centers is $$b$$. (CC BY-NC; Ümit Kaya)

In this model, and effective collision occurs when $$E_{loc} > E_0$$ where $$E_{loc}$$ takes into account the fact that all particle collisions are not head-on collisions. If we define vr as the relative velocity of approach of particles Q and B, then vr = $$\vec{v}_Q-\vec{v}_B$$. The relative kinetic energy, $$E_r$$, is then $$\dfrac{1}{2}\mu v_r^2$$. From Figure $$\PageIndex{2}$$ we can see that the fraction of $$E_r$$ that can be applied to the collision, ($$E_{loc})$$, is dependent upon $$b$$, the impact parameter, which is the perpendicular distance between the extrapolated paths traveled by the centers of the particles before the collision occurs. If $$b$$ is 0, then $$E_{loc}$$ = $$E_r$$, but for any other value of $$b$$, $$E_{loc}$$ < $$E_r$$. If $$b$$ is greater than the sum of the radii of Q and B, the particles will not collide, and $$E_{loc}$$ = 0. The calculation for determining the exact relationship between the $$\sigma_r(E_r)$$ and $$E_r$$ for this line of center model rather complicated, but the result is that $$\sigma_r(E_r)$$ is equal to 0 if $$E_r < E_0$$ and is equal to $$\sigma_{QB} \left( 1 - \dfrac{E_0}{E_r} \right)$$ if $$E_r \geq E_0$$.

When compared to the simple hard-sphere collision theory, we see that

$\sigma_r(E_r) = \sigma_{QB} \, \text {if} \, E_r \geq E_0 \, (\text{hard-sphere theory}) \label{30.2.1}$

$\sigma_r(E_r) = \sigma_{QB} \left( 1 - \dfrac{E_0}{E_r} \right) \, \text {if} \, E_r \geq E_0 \, (\text{line of centers theory}) \label{30.2.2}$

If we substitute Equation $$\ref{30.2.2}$$ into Equation $$30.1.4$$ we get

$k = \left(\dfrac{2}{k_BT} \right)^{3/2} \left(\dfrac{1}{\mu\pi}\right)^{1/2} \int_{E_0}^{\infty} dE_r E_r e^{-E_r/k_BT} \sigma_{QB} \left( 1 - \dfrac{E_0}{E_r} \right) \nonumber$

$= \left(\dfrac{8k_BT}{\mu\pi}\right)^{1/2} \sigma_{QB} e^{-E_r/k_BT} \nonumber$

$= \langle v_r \rangle\sigma_{QB} e^{-E_r/k_BT} \nonumber$

When compared to the simple hard-sphere collision theory, we see that

$k = \langle v_r \rangle\sigma_{QB} e^{-E_r/k_BT} \left(1 + \dfrac{E_0}{k_BT}\right) (\text{hard-sphere theory})\nonumber$

$k = \langle v_r \rangle\sigma_{QB} e^{-E_r/k_BT} (\text{line of centers theory})\nonumber$

The line of centers theory expresses $$k$$ in the same terms as the Arrhenius equation, yet experimental values of $$k$$ still differ from those predicted by the line of centers model. The errors come about because $$\sigma_r(E_r)$$ is not accurately described by $$\sigma_{QB} \left( 1 - \dfrac{E_0}{E_r} \right)$$. More work needs to be done to improve the model for describing $$A$$, the Arrhenius factor.