Substance A decomposes at a rate proportional to the amount...

Substance $A$ decomposes at a rate proportional to the amount of A present. a) Write an equation relating $A$ to the amount left of an initial amount $A_{0}$ after time $t$ b) It is found that 8 g of $A$ will reduce to $4 \mathrm{g}$ in $3 \mathrm{hr}$. After how long will there be only 1 g left?

Substance $A$ decomposes at a rate proportional to the amount of A present.
a) Write an equation relating $A$ to the amount left of an initial amount $A_{0}$ after time $t$
b) It is found that 8 g of $A$ will reduce to $4 \mathrm{g}$ in $3 \mathrm{hr}$. After how long will there be only 1 g left?

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Key Concepts

Exponential Decay

Exponential decay refers to a process where the rate of decrease of a quantity is proportional to the current amount. This leads to an equation of the form A = A?e^(–kt), which is commonly used to model natural processes such as radioactive decay or chemical reactions.

First-Order Differential Equations

The relationship between the rate of change of a substance and the amount present is formulated as a first-order differential equation, dA/dt = kA (with a negative sign for decay). Solving this type of differential equation produces an exponential solution that accurately represents the decay process.

Rate Constant

The rate constant, denoted by k, is a parameter that determines the speed of the decay process. It is typically found using experimental data by setting up the relationship between the initial amount and the remaining quantity after a known time, and plays a crucial role in predicting how long it takes for the substance to reach a certain level.

Logarithmic Functions

Logarithmic functions are instrumental in solving equations that involve an exponential term, such as A = A?e^(–kt). They allow us to linearize the exponential relationship and isolate variables like time, which is essential in determining the duration required for the substance to decay to a specified amount.

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