A substance evaporates at a rate proportional to the exposed...

A substance evaporates at a rate proportional to the exposed surface. If a spherical mothball of radius $\frac{1}{2} \mathrm{~cm}$ has radius $0.4 \mathrm{~cm}$ after 6 months, how long will it take:\} (a) For the radius to be $\frac{1}{4} \mathrm{~cm}$ ? (b) For the volume of the mothball to be half of what it was originally?

A substance evaporates at a rate proportional to the exposed surface. If a spherical mothball of radius $\frac{1}{2} \mathrm{~cm}$ has radius $0.4 \mathrm{~cm}$ after 6 months, how long will it take:\}
(a) For the radius to be $\frac{1}{4} \mathrm{~cm}$ ?
(b) For the volume of the mothball to be half of what it was originally?

Key Concepts

Differential Equations in Decay Processes

This concept involves setting up and solving differential equations that model change over time. In decay processes, one often uses a rate law where the rate of change is proportional to a specific geometric or physical property of the object. Here, the evaporation rate being proportional to the exposed surface area leads to a differential equation that governs how the object's volume or radius changes with time.

Scaling Laws in Spherical Geometry

Understanding how geometric quantities such as volume and surface area scale with the radius of a sphere is crucial. The volume of a sphere scales with the cube of the radius while the surface area scales with its square. This difference in scaling is significant because, in processes like evaporation (or other surface-dependent phenomena), the interplay between surface area and volume determines the overall rate of change in the object’s size.

Proportional Rate Laws

Proportional rate laws describe systems where the rate of a process (such as evaporation) is directly proportional to a particular variable, such as the surface area. This means that as the surface area changes, the rate of evaporation changes in a directly proportional manner. This concept is fundamental in formulating the differential equation that relates the change in volume (or radius) to time.

Linear versus Exponential Change

This concept distinguishes between different types of time-evolution in dynamic systems. Although many decay processes exhibit exponential decay, in this case the cancellation of terms due to the geometry (surface area vs. volume) results in a linear change in the radius over time. Recognizing the type of change—linear in the radius despite the underlying exponential processes in other quantities—is key to solving problems of this nature.

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