Reconcile the Beer Lambert Law with rate of absorption

The Beer-Lambert law gives the proportion of incident light intensity that will be absorbed. Intensity isn't strictly a rate but if the light is monochromatic and the path length is constant it sort of is. $$I=I_0\exp(-\epsilon c l)$$

But the rate of absroption of photons is given by: $$K=B\rho$$ Where B is the Einstein coefficient and $\rho$ is the spectral energy density which is proportional to the intensity of incident radiation.

Surely the second equation should mean that the intensity decays linearly? Would the absorbed intensity double if the concentration was doubled?

To answer your second question first, just use the Beer-lambert law. The first question needs us to go back a few steps and re-define the Einstein B coefficients in terms of the intensity which means that the 'new' B or $B^I=B^{\rho}/c$ and from now on for simplicity we just use B instead of $B^I$.

The intensity I is defined (per unit frequency interval) as flux across unit area per unit time thus has units $\pu{W/m^2.s^{-1}}$ or mass time$^{-2}$ and thus is not a rate. The energy density $\rho$ means density of energy per unit volume and is related to I as $I=c\rho$ where c is the speed of light. (The rate of absorption is unchanged but written as $B^II$ rather than $B^{\rho}\rho$.)

The energy density $\rho$ has units $\pu{J.m^{-3}/s^{-1}}$ which is mass length$^{-1}$ time$^{-1}$. In these units the Einstein B coefficient has dimensions time mass$^{-1}$ or $\pu{s kg^{-1}}$ in SI units.

The Beer Lambert law derives from the differential form $$-dI_{\nu} =I_{\nu}k_{\nu}dx$$ where for clarity x is used to indicate length rather than l. The constant $k_{\nu} = \epsilon [c]$ at frequency $\nu$ and has dimensions of length$^{-1}$.

If the absorption band is wide we take a small element about frequency $\nu$, which is $\delta\nu$, and calculate the rate of energy removal from the beam during absorption from state 1 to state 2 with populations $N_1$ and $N_2$ respectively, with Einstein coefficients for (stimulated) absorption $B_{12}$ and stimulated emission $B_{21}$ $$-dI_{\nu}\delta\nu = h\nu I_{\nu} [N_1B_{12}dx -N_2B_{21}dx ]$$ rearranging gives $$-\frac{1}{I_{\nu}}\frac{dI_{\nu}}{dx}\delta\nu=h\nu[N_1B_{12} -N_2B_{21} ]=k_{\nu}\delta\nu$$

If the light intensity is low, the population of the upper level is only a minute fraction of that of the lower level and can be set to zero. In this case B can be obtained directly as $$B_{12}=\frac{k_{\nu}\delta\nu}{h\nu N_1}$$

Then integration over $\delta\nu$ gives the B value for the absorption band. The value of $N_1$ is effectively the total population.

This shows that the B values are related to the extinction coefficients as expected and that time does not come into the calculation. However, this is only part of the story.

A quantum calculation starting with the Schroedinger equation and using the dipole approximation and first order perturbation theory is necessary to explain the electric dipole interaction. The result of this calculation (see Atkins & Friedman Molecular Quantum Mechanics chapter 6 for gory details) is that the probability of absorption from state a to b is $$P_{ab}= \frac{|V_{ba}|^2}{\hbar^2}\frac{\sin^2((\omega-\omega_{ab})t/2)}{(\omega-\omega_{ab})^2}$$ where $V_{ab}= <a|H|b>$ is the interaction energy, $\omega$ the radiation frequency and $\omega_{ab} = (E_a-E_b)\hbar$. This oscillating behaviour as a function of frequency is not ordinarily observed in thermal samples but has been observed in a cold molecular beam of HCN molecules. Time was made constant by passing the beam at constant speed through the fixed illuminated region (see Dyke et al. J. Chem. Phys. vol 57, 2277, 1972).

The rate of absorption is the time derivative of the probability which is $$R=\frac{|V_{ba}|^2}{2\hbar^2}\frac{\sin((\omega-\omega_{ab})t)}{\omega-\omega_{ab}}$$

and as the the absorption coefficient is directly proportional to the rate of absorption $\gamma(\omega,t)=\beta R$ (where $\beta$ only contains constants) which indicates that near resonance absorption should increase in time. However, this is not what is usually observed for molecules at room temperature as a vapour or in solution. The reason for the generally observed behaviour is that the absorption is continually interrupted by first order processes, for example by collisions.

Suppose that the system is initially in state a and then the radiation field is turned on. The molecule is then in a superposition state involving a and b, for example $\Psi=c_a(t)\psi_a+c_b(t)\psi_b$ where the c are time dependent coefficients that move the system from a to b. A collision will destroy the superposition and return the molecule to state a or b each with a certain probability, then the radiation causes the superposition to grow again.

If we assume that the number of superposition states is n and that a first order process destroys them with lifetime $\tau$ , then $n(t)=n(0)\exp(-t/\tau)$. There is now a competition between the superposition state growing and being destroyed. The average absorption coefficient is obtained by averaging over the probability distribution of it being interrupted thus

$$\gamma(\omega)=\frac{\int_0^{\infty}\gamma(\omega,t)\exp(-t/\tau)dt}{\int_0^{\infty}\exp(-t/\tau)dt}$$ evaluating gives $$\gamma(\omega) \propto L(\omega_{ab}-\omega)$$ where L is the Lorenzian function $$L(\omega_{ab}-\omega)=\frac{1}{\pi }\frac{\tau^{-1}}{\tau^{-2}+(\omega_{ab}-\omega)^2 }$$ and examining the function the half-width at half-height $\Delta \omega = 1/\tau$. The first order process thus explains why the absorption does not increase in time. However, the experimentally line shapes on many molecules do not have a Lorenzian shape and the reasons for this are not that the idea of interrupting the superposition is wrong but that other effects dominate, doppler broadening for example, or in solution there is always inhomogeneous broadening. This is caused by interactions with solvent molecules which shift energy levels up and down and thus the solution effectively contains molecules with a mixture of energy levels. That this is so can be observed at very low temperatures (say 10K) where it is possible to burn a spectral hole in a sample and so remove a small part of the inhomogeneous population. The spectral hole produced is very narrow as expected from the calculation above.