Question

Use the exponential decay model a=a0 e^kt to solve. The half-life of a certain substance is 15 years. How long will it take for a sample of this substance to decay to 91% of its original amount?

a) t=15log(0.91)
b) t= 15/log(0.91)
c) t=15log(0.09)
d) t= 15/ log (0.09)

Answer 1

To find the time for a substance to decay to 91% of its original amount, we determine the decay constant from the half-life of 15 years and solve the exponential decay model using natural logarithms.

Step-by-step explanation:

Calculating Time for Radioactive Decay to 91% Using Half-Life

To calculate the amount of time it will take for a substance to decay to 91% of its original amount using its half-life of 15 years, we use the exponential decay model a = a0 ekt. First, we need to find the decay constant (k) using the relation between decay constant and half-life: k = 0.693 / t1/2. Substituting the half-life (15 years) yields k = 0.693 / 15.

Now, to find the time (t) when the sample has decayed to 91% of its original amount (0.91a0), we set up the equation 0.91a0 = a0 e(0.693/15)t. Solving for t, we take the natural logarithm on both sides: ln(0.91) = (0.693/15)t. Therefore, t = (15/0.693)ln(0.91).

The answer is calculated by substituting the natural logarithm of 0.91 and multiplying by the factor (15/0.693). This gives us the time it takes for the substance to decay to 91% of its original amount.