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-glucono lactone into gluconic acid, for example, dy/dt=−0.6y when t is measured in hours. If there are 100 grams of delta-glucono lactone present when t = 0, how many grams will be left after the first hour?

The amount of delta-glucono lactone present after the first 1 hour = 54.88g

Step-by-step explanation:

dy/dt = -0.6y where y₀ (corresponding to t=0) is 100g

dy/dt = -0.6y

dy/y = -0.6dt

Integrating the left hand side from y₀ to y and the right hand side from 0 to t.

In (y/y₀) = -0.6t

(y/y₀) = e⁻⁰•⁶ᵗ

y = y₀ e⁻⁰•⁶ᵗ

But y₀ = 100g

y = 100e⁻⁰•⁶ᵗ

At t = 1,

y = 100e⁻⁰•⁶ = 54.88g

The amount of delta-glucono lactone present after the first 1 hour = 54.88g

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