Modelling Absorption

The fundamental principle of First-Order Absorption states that the rate of change of the substance in the system is directly proportional to the amount remaining to be absorbed.

This relationship is expressed by the first-order linear differential equation: $\frac{dx}{dt} = k(M - x)$ where $x$ is the amount in the system, $M$ is the total available, and $k$ is the rate constant.

As $x$ approaches $M$, the term $(M - x)$ approaches zero, meaning the rate of absorption slows down as the system nears its capacity.

This model assumes a passive diffusion mechanism where the driving force is the concentration gradient between the source and the system.

Separation of Variables: To solve the differential equation $\frac{dx}{dt} = k(M - x)$, rearrange the terms to group $x$ and $t$: $\int \frac{1}{M - x} dx = \int k dt$

Integration: Integrating both sides yields $-\ln(M - x) = kt + C$. Applying the exponential function results in $M - x = Ae^{-kt}$, where $A$ is a constant determined by initial conditions.

Applying Initial Conditions: If the system starts with zero substance ($x = 0$ at $t = 0$), then $A = M$. This gives the general solution: $x(t) = M(1 - e^{-kt})$

Determining the Rate Constant: If the amount $x$ is known at a specific time $t_1$, the constant $k$ can be found using $k = -\frac{1}{t_1} \ln(1 - \frac{x_1}{M})$.

Check the Asymptote: Always verify that as $t \to \infty$, your solution $x(t)$ approaches the maximum capacity $M$. If it grows infinitely, the model is likely set up incorrectly.

Unit Consistency: Ensure the units of the rate constant $k$ (usually $time^{-1}$) match the units of $t$. If $t$ is in hours, $k$ must be in $hours^{-1}$.

Boundary Conditions: Pay close attention to whether the system starts empty ($x=0$) or with a pre-existing amount ($x=x_0$). This changes the constant of integration $C$.

Rate vs. Amount: Distinguish between the 'rate of absorption' (the derivative $\frac{dx}{dt}$) and the 'amount absorbed' ($x$). Exams often ask for the rate at a specific moment.

Sign Errors: A common mistake is forgetting the negative sign during the integration of $\frac{1}{M-x}$, which results in $-\ln(M-x)$. Forgetting this leads to an incorrect exponential growth model.

Confusing k with Half-life: The rate constant $k$ is not the time it takes to reach half capacity. For first-order absorption, the 'half-time' to reach $M/2$ is calculated as $t = \frac{\ln(2)}{k}$.

Linear Assumption: Students often try to use simple linear proportions for absorption over time. However, because the rate changes as $x$ increases, only calculus-based models provide accurate predictions.