I was reading this paper, Bertman, S. B., J. M. Roberts, D. D. Parrish, M. P. Buhr, P. D. Goldan, W. C. Kuster, F. C. Fehsenfeld, S. A. Montzka, and H. Westberg (1995), Evolution of alkyl nitrates with air mass age, J. Geophys. Res., 100(D11), 22805–22813, doi:10.1029/95JD02030 (link attached), and had a hard time trying to derive equation (12) from equation (11). If anyone knows how to do that, help would be highly appreciated.
Thank you
Hi Maria!
I apologize for my complete ignorance about the context of the equation, but if I just look at the differential equation, it looks like there are two unknown concentrations but only one differential equation to fix them. Perhaps you need differential equation for [RH] as well? Are the two compounds perhaps reactants/products of the same reaction, with their rates of change thus simply related? If you had an equation for [RH] as well, then it would be a linear system of two ODEs, and both concentrations could be obtained as an exponent of a 2x2 matrix (which is quite easy to compute by hand).
Hi,
the problem is complete only with the second equation (9) written in the article only in terms of the reaction kinetics [RH] -- (k_A \beta)--> [RONO_2].
In terms of differential equations this means
d[RH]/dt = - k_A \beta [RH].
I now abbreviate:
a:= k_A \beta, b:=k_B
x:=[RH], y:= [RONO_2]
We then have two equations I label (X) and (Y):
(X) dx/dt = -a x
(Y) dy/dt = a x -b y
(X) implies x(t)=x_0 exp(-a t)
such that (Y) becomes
dy/dt = a x_0 exp(-a t) - b y
The ansatz y(t) = c exp(-d t) + e exp(- a t)
yields
dy/dt = - c d exp(-d t) - e a exp( - a t)
= a x -b y = a x_0 exp(-a t) - b c exp(-d t) - b e exp(- a t) for all times t,
so - c d = - b c such that d=b
and -e a = a x_0 - b e so e= a x_0 /(b -a)
finally y(0) = c+ e = y_0 so c = y_0 - e
summary:
y(t) = (y_0 - a x_0/(b-a)) exp(- b t) + a x_0/(b-a) exp(- a t)
dividing by x(t) yields
y(t)/x(t) = (y_0/x_0 - a/(b-a)) exp ((a-b)t) + a/(b-a)
Now substituting back the definitions should give your result up to a factor \beta in several terms (k_A + k_B), which I claim are typos in the article as the time constants should be fixed throughout the expressions,
so every (k_A + k_B) should read (\beta k_A + k_B), both as prefactors and in the exponential function.
Voila.
I encourage you to reach out to the authors, despite that some students are reluctant to do this. The first author has a web page at http://homepages.wmich.edu/~bertman/
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