How would I calculate the age of a rock only given the amount of each in grams and half-life?
40K decays to 40Ar.
Half life of 40K is 1.25 billion years.
40K = 20 grams
40Ar= 5 grams
I think the formula is
Final Amount of 40K = Initial Amount of 40K * (1/2)^(time/half-life)
If I plug in the values I still don't have a value for time. And where would the grams of 40Ar fit in?
You started with 25 grams, you now have 20 grams
20=25(1/2)^(t/th)
.8=(1/2)^halflives
log both sides...
log (.8)=halflives*log.5
halflivees= log(.8)/log(.5)
time= 1.25billion* log(.8)/log(.5)
To calculate the age of a rock based on the amount of a radioactive isotope and its half-life, you can use the formula you mentioned, which is correct. Here's how you can proceed:
1. Start by determining the fraction of 40K that has decayed to 40Ar. This can be calculated by dividing the amount of 40Ar (5 grams) by the initial amount of 40K (20 grams). In this case, the fraction is 5/20, which simplifies to 1/4 or 0.25.
2. Next, substitute this fraction into the formula:
Final Amount of 40K = Initial Amount of 40K * (1/2)^(time/half-life)
Since the final amount of 40K is given as 20 grams, the equation becomes:
20 grams = 20 grams * (1/2)^(time/1.25 billion years)
3. Now, solve for time. To isolate the variable, you can divide both sides of the equation by the initial amount of 40K (20 grams):
1 = (1/2)^(time/1.25 billion years)
4. Take the logarithm (base 2) of both sides of the equation to bring the exponent down:
log2(1) = log2((1/2)^(time/1.25 billion years))
0 = (time/1.25 billion years) * log2(1/2)
5. Multiply both sides of the equation by 1.25 billion years and divide by the logarithm of 1/2, which is -1 (since log2(1/2) = -1):
time = 1.25 billion years * (0 / -1)
Since any number divided by zero is undefined, it means that the age cannot be determined accurately with the given information. The calculation suggests that the age would be infinite.
Therefore, to calculate the age of the rock, you would need additional information such as the amount of the original 40K isotope present before decay started or the amount of the remaining 40K isotope.