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I am struggling to understand the following transition (encountered in a paper on Physical Chemistry).

Let $$D=\frac{\tau_0^{-1}\int_0^\infty G(t)dt}{1-\tau_0^{-1}\int_0^\infty G(t)\int w(R)\exp[-tw(R)] d^3R dt},\qquad (1)$$ where $$G(t)= \exp\left\{-\frac{t}{\tau_0}-n\int (1-e^{-tw(R)})dR-n_A\int (1-e^{-tu(R)})dR \right\},$$ $$w(R)=\frac{1}{\tau_0}\left(\frac{R_0}{R}\right)^6,\qquad u(R)=\frac{1}{\tau_0}\left(\frac{R_A}{R}\right)^6.$$ The authors claim that $$\qquad\qquad\quad\qquad D=\frac{1-f(y)}{1-\frac{c}{c+c_A}f(y)}.\qquad\qquad\qquad\qquad(2)$$ Here $$f(y)=\sqrt{\pi} y\exp(y^2)\left[1-\frac{2}{\sqrt{\pi}}\int_0^y\exp(-t^2)dt\right],$$ $$y=\frac{\sqrt{\pi}}{2}(c+c_A),$$ $$c=\frac{4\pi}{3}R_0^3n,\qquad c_A=\frac{4\pi}{3}R_A^3n_A.$$

Question. How does one pass from (1) to (2)?

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Could you please share the link of the paper? –  Shuhao Cao Jul 24 '13 at 4:40

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