A Rock Formation Mystery: How Old Is That Rock?
The Age of a Rock Formation
A rock has 12.5 percent of its original amount of potassium-40 remaining in it. Potassium-40 has a half-life of 1.25 billion years. The question arises, how long ago was the rock formed?
To solve this mystery, we need to take into account the percentage of potassium-40 remaining and its half-life. When a rock has 12.5% of its original potassium-40 left, this indicates that the rock was formed approximately 3.75 billion years ago.
Calculating the Time
Answer: The correct option is (C) 3.75 billion years ago.
Explanation: The half-life of potassium-40 is 1.25 billion years. To determine the time passed since the rock formation, we first calculate the rate constant using the formula:
k = 0.693 / t1/2
k = 0.693 / 1.25 years = 0.554 billion years-1
Next, we apply the rate law for first-order kinetics to find the time passed:
t = 2.303 / k * log(a / (a - x))
Where:
k = rate constant = 0.554 billion years-1
t = time passed by the sample = ?
a = initial amount of the reactant = X g
a - x = amount left after decay process = 0.125X g
By substituting the values into the equation, we get:
t = 3.75 billion years
Therefore, the rock formation occurred approximately 3.75 billion years ago.
How did the percentage of remaining potassium-40 in the rock help determine its age?
The percentage of remaining potassium-40 in the rock, along with the half-life of potassium-40, allowed us to calculate the time elapsed since the rock formation. By understanding the decay process and utilizing the rate constant, we could accurately estimate that the rock was formed around 3.75 billion years ago.