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30.2: A Reaction Cross Section Depends Upon the Impact Parameter

  • Page ID
    14563
  • 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 \(\PageIndex{1}\).

    galncing headon.svg

    Figure \(\PageIndex{1}\) In the top collision, the particles collide head-on, and thus all of the kinetic energy can be used to overcome electron-electron repulsion and to break bonds. In the bottom collision, the particles collide with a glancing blow so that only part of the kinetic energy is contributed to the reaction process. With collisions involving larger, more complex molecules, there is often a specific molecular orientation required for an effective collision. (CC BY-NC; Ümit Kaya)

    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 \(\PageIndex{2}\).

    Collision geometry.svg

    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.