An ancient rock is found to contain 40Ar gas, indicating that 77% of the 40K in the rock has decayed since the rock solidified. Any argon would have boiled out of liquid rock. How long ago did the rock solidify? The half-life of 40K is 1.25 billion years.
A) 2.6 billion years
B) 0.5 billion years
C) 1.8 billion years
D) 0.3 billion years
Answer: A
The amount left after t billion years is
(1/2)^(t/1.25)
So, you want
(1/2)^(t/1.25) = 0.77
t/1.25 log 0.5 = log 0.77
t = 1.25 log(0.77)/log(0.5) = 0.471
Looks like (B) to me.
To determine how long ago the rock solidified, we can use the concept of half-life and the given information about the amount of 40Ar gas and 40K in the rock.
The half-life of 40K is 1.25 billion years, which means that after 1.25 billion years, half of the 40K would have decayed into 40Ar.
Given that 77% of the 40K in the rock has decayed, it means that 100% - 77% = 23% of the 40K remains in the rock.
We can calculate the number of half-lives that need to occur to reach this remaining amount of 40K. We'll start with 100% and divide it by 2 until we reach 23%.
100% ÷ 2 = 50%
50% ÷ 2 = 25%
25% ÷ 2 = 12.5%
12.5% ÷ 2 = 6.25%
6.25% ÷ 2 = 3.125%
3.125% ÷ 2 = 1.5625%
1.5625% ÷ 2 ≈ 0.78125%
0.78125% ÷ 2 ≈ 0.390625%
It takes approximately 7 half-lives to reach 0.390625% (which is very close to 0.423%) of the original 40K amount.
Now, we can multiply the half-life time (1.25 billion years) by the number of half-lives (7) to find the approximate age of the rock:
1.25 billion years * 7 = 8.75 billion years
Therefore, the rock solidified approximately 8.75 billion years ago.
Since none of the provided answer choices are close to this value, it appears there may be an error in the question or the available answer choices.
To determine how long ago the rock solidified, we need to calculate the number of half-lives that have passed based on the percentage of potassium-40 (40K) that has decayed into argon-40 (40Ar).
Given that 77% of the 40K has decayed, this means that 23% of the original 40K remains in the rock.
Next, we need to determine the number of half-lives that corresponds to this remaining amount of 40K. Since the half-life of 40K is 1.25 billion years, we can divide 1.25 billion years by the number of half-lives to get the time for each half-life.
Calculating the number of half-lives can be done using the formula:
Number of Half-lives = ln(Ratio of Remaining 40K / Initial 40K) / ln(0.5)
Let's plug in the values:
Remaining 40K / Initial 40K = 0.23
Number of Half-lives = ln(0.23) / ln(0.5)
Now, we can calculate the number of half-lives:
Number of Half-lives = -2.26475 / -0.693147 = 3.27
Since 3.27 half-lives have passed, we can multiply this by the half-life to get the time elapsed since the rock solidified:
Time Elapsed = 3.27 * 1.25 billion years ≈ 4.09 billion years
Therefore, the rock solidified approximately 4.09 billion years ago.
Among the provided options, the closest one to 4.09 billion years is option A) 2.6 billion years.