Question

Active oxygen and free radicals are believed to be exacerbating factors in causing cell injury and aging in living tissue. \({ }^{1}\) These molecules also accelerate the deterioration of foods. Researchers are therefore interested in understanding the protective role of natural antioxidants. In the study of one such antioxidant (Hsian-tsao leaf gum), the antioxidation activity of the substance has been found to depend on concentration in the following way: $$ \frac{d A(c)}{d c}=k\left[A^{*}-A(c)\right], \quad A(0)=0 . $$ In this equation, the dependent variable \(A\) is a quantitative measure of antioxidant activity at concentration \(c\). The constant \(A^{*}\) represents a limiting or equilibrium value of this activity, and \(k\) is a positive rate constant. (a) Let \(B(c)=A(c)-A^{*}\) and reformulate the given initial value problem in terms of this new dependent variable, \(B\). (b) Solve the new initial value problem for \(B(c)\) and then determine the quantity of interest, \(A(c)\). Does the activity \(A(c)\) ever exceed the value \(A^{*}\) ? (c) Determine the concentration at which \(95 \%\) of the limiting antioxidation activity is achieved. (Your answer is a function of the rate constant \(k\).)

Short answer

Short Answer: The concentration at which 95% of the limiting antioxidation activity is achieved is given by \(c = \frac{\ln{-0.05}}{-k}\) where k is the rate constant. A(c) never exceeds the limiting antioxidation activity A*.

Step-by-step solution

Find the new differential equation for B(c)

We are given a new dependent variable B(c) = A(c) - A* and asked to rewrite the given initial value problem. Using the chain rule, we can write: $$ \frac{dB}{dc} = \frac{dA}{dc} - \frac{dA^{*}}{dc} $$ Since A* is a constant, its derivative will be zero, so we have: $$ \frac{dB}{dc} = \frac{dA}{dc} $$ Substitute the given relation: $$ \frac{dB}{dc} = k[A^{*}-A(c)] $$ Now, we substitute B(c) = A(c) - A* back into the equation: $$ \frac{dB}{dc} = k[A^{*}-(B(c) + A^{*})] $$ Simplify the equation: $$ \frac{dB}{dc} = -k B(c) $$ And since A(0) = 0, we have B(0) = A(0) - A* = -A*. The initial value problem now becomes: $$ \frac{dB}{dc} = -k B(c), \quad B(0)=-A^{*} $$

Solve the differential equation for B(c)

This is a first-order linear differential equation. We can solve it by integrating both sides with respect to the concentration (c): $$ \int{\frac{1}{B} dB} = -k\int{dc} $$ This gives us: $$ \ln{B} = -kc + C, $$ where C is the constant of integration. To find the constant, we can use the initial condition B(0) = -A*. Substituting the initial values, we get: $$ \ln{-A^{*}} = -k(0) + C $$ Thus, C = ln(-A*). Now, our equation can be written as: $$ \ln{B} = -kc + \ln{-A^{*}} $$ Exponentiating both sides to obtain B(c): $$ B(c) = A^{*}e^{-kc} $$

Obtain A(c) using B(c)

Now we will find A(c) using the relation B(c) = A(c) - A*. Solving for A(c), we get: $$ A(c) = B(c) + A^{*} = A^{*}e^{-kc} + A^{*} $$

Check if A(c) ever exceeds A*

To see if A(c) ever exceeds A*, we can compare the function to A*: $$ A^{*}e^{-kc} + A^{*} > A^{*} $$ Let's subtract A* from both sides of the inequality: $$ A^{*}e^{-kc} > 0 $$ Since A* is positive and e^{-kc} is always positive, then A(c) will never exceed A*.

Find the concentration at which 95% of the limiting antioxidation activity is achieved

Since we want to find the concentration at which 95% of the limiting activity, A* is reached, we set A(c) = 0.95A* and solve for c: $$ 0.95A^{*} = A^{*}e^{-kc} + A^{*} $$ Divide both sides by A*: $$ 0.95 = e^{-kc} + 1 $$ Subtract 1 from both sides: $$ -0.05 = e^{-kc} $$ Take the natural logarithm of both sides: $$ \ln{-0.05} = -kc $$ Divide by -k to get: $$ c = \frac{\ln{-0.05}}{-k} $$ So, the concentration at which 95% of the limiting antioxidation activity is achieved is given by \(c = \frac{\ln{-0.05}}{-k}\).

Key concepts

Initial Value Problem

An initial value problem (IVP) in differential equations is a situation where, at the start of the problem, we know the value of the solution. This often looks like having a differential equation with an accompanying set condition, like knowing the value of a function at a specific point.
In our example, the initial value problem is given by the antioxidant activity relationship\[ \frac{dA}{dc} = k[A^* - A(c)], \ A(0) = 0. \]
Here, the function \(A(c)\) represents the antioxidant activity, and at the concentration \(c = 0\), we know \(A(0)\) equals zero. This knowledge about the value at the starting point helps us solve the differential equation, guiding us throughout the problem.
By introducing \(B(c) = A(c) - A^*\), a transformation occurs, simplifying the equation and enabling us to better solve the problem.

Rate Constants

Rate constants (\(k\)) are crucial in understanding how quickly reactions happen. They provide insight into the rate of change in a process, like how fast antioxidants neutralize free radicals.
In the context of our problem, the rate constant \(k\) dictates the speed at which the antioxidant activity approaches its limiting value \(A^*\). The equation \[ \frac{dB}{dc} = -k B(c) \] shows that as concentration \(c\) changes, the rate of change in activity \(B\) is proportional and inverse to itself, governed by the rate constant \(k\).
Thus, the value of \(k\) directly impacts how quickly the antioxidant activity reaches its equilibrium state \(A^*\). A larger \(k\) means the activity approaches \(A^*\) faster, whereas a smaller \(k\) indicates a slower approach.

Natural Antioxidants

Natural antioxidants play a pivotal role in protecting cells from harmful oxidative damage caused by free radicals. These substances, often found in plants, work to neutralize free radicals and slow down processes like aging and cellular injury.
In our study, Hsian-tsao leaf gum is investigated for its antioxidant properties. It demonstrates a fascinating relationship between its concentration and antioxidation activity. The equation derived for \(A(c)\) allows us to explore how effectively this natural antioxidant performs at different concentrations.
This analysis helps researchers understand at what concentrations these natural substances are most effective. Such knowledge can support developing enhanced dietary supplements and food preservatives, maximizing the beneficial effects of antioxidants against oxidative stress.