a What do you need to write the equation of decay for Potassium?
B
b Use the exponential decay formula to find t.
C
c Substitute the given time and the initial amount into the exponential decay formula.
D
d If a is the original amount, y will be 18a.
A
a 5.545* 10^(-10)
B
b 1 578 843 530 years
C
c ≈30.48 mg
D
d 3 750 120 003 years
Practice makes perfect
a Let's begin with recalling the exponential decay formula. In this formula y is the present amount of a substance, a is the initial amount, k is the rate of continuous decay, and t is the half-life of this substance.
y=ae^(- kt)
Let a be the initial amount of Potassium-40. We are given that the half-life of this chemical element is about 1.25 billion years. This means that after 1.25* 10^9 years the amount of Potassium-40 will be equal to 0.5a.
0.5a=ae^(- k( 1.25* 10^9))
To find the value of k we can solve the above exponential equation.
The rate of continuous decay of Potassium-40 is approximately 5.545 * 10^(-10). With this we can write the full exponential decay formula.
y=ae^(- 5.545* 10^(-10)t)
b We are given that the initial amount of Potassium-40 in a specimen is 36 milligrams, and we are asked to find the time it will take this amount to decay to 15 milligrams. To do this we can use the formula we found in Part A.
y=ae^(- 5.545* 10^(-10)t)
Let's substitute 36 for a and 15 for y, then solve for t.
It will take about 1 578 843 530 years for the specimen to decay to only 15 milligrams of Potassium-40.
c Now, we are asked to find the amount of Potassium after 300 million years when the initial amount of this chemical element is 36 milligrams. To do this we will again use the formula we found in Part A. Let's substitute 300* 10^6 for t, 36 for a, and then solve for y.
After 300 million years about 30.48 milligrams of Potassium-40 will be left.
d This time we are asked to find how long it will take Potassium-40 to decay to 18 of its original amount,a. To do this we can again use the decay formula and substitute 18a for y, and solve for t. Let's do this!
