Since you are not integrating the differential equation, this is a simple nonilinear parameter estimation problem.
T = 270:280;
dNdT = rand(size(T));
a = 42;
deltaE = 3;
R = 8.3145;
objfcn = @(b,T) a.*exp(-deltaE./(R*T)) .* (1 - b);
B = fminsearch(@(b) norm(dNdT - objfcn(b,T)), 10);
It might be possible to solve it as a linear problem, however that is not obvious to me just now.
Experiment to get the result you want.