25. First-order chemical reactions In some chemical reactions, the rate at which the amount of a substance changes with time is proportional to the amount present. For the change of δ-gluconolactone into gluconic acid, for example,
dy/dt = -0.6y
when t is measured in hours. If there are 100 grams of δ-gluconolactone present when t=0, how many grams will be left after the first hour?

Verified step by step guidance

1. Recognize that the differential equation given is a first-order linear ordinary differential equation of the form \(\frac{dy}{dt} = ky\), where \(k\) is a constant. Here, \(k = -0.6\), indicating the amount decreases over time.
2. Write the general solution to the differential equation \(\frac{dy}{dt} = -0.6y\). This is a separable equation, and its solution can be expressed as \(y(t) = y_0 e^{kt}\), where \(y_0\) is the initial amount at \(t=0\).
3. Identify the initial condition from the problem: at \(t=0\), \(y = 100\) grams. Substitute this into the general solution to find the specific solution for this problem: \(y(t) = 100 e^{-0.6t}\).
4. To find the amount left after the first hour, substitute \(t=1\) into the specific solution: \(y(1) = 100 e^{-0.6 \times 1}\).
5. Evaluate the expression \(100 e^{-0.6}\) to find the amount of \(\delta\)-gluconolactone remaining after one hour (note: do not calculate the numerical value here, just set up the expression).

Key Concepts

First-Order Differential Equations

A first-order differential equation relates a function and its first derivative. In this problem, the rate of change of the substance's amount is proportional to the amount itself, leading to an equation of the form dy/dt = ky. Solving such equations typically involves separation of variables or recognizing the exponential decay/growth pattern.

Exponential Decay

Exponential decay describes processes where a quantity decreases at a rate proportional to its current value. The solution to dy/dt = -ky is y(t) = y(0)e^(-kt), showing how the amount decreases over time. This model applies to many natural phenomena, including radioactive decay and chemical reactions.

Initial Conditions and Solution Evaluation

Initial conditions specify the value of the function at a starting point, allowing determination of constants in the solution. Here, y(0) = 100 grams sets the initial amount. Using this, we can find the exact amount remaining after a given time by substituting t into the solution formula.