The half-life of carbon 14, which is commonly used to date organic materials, is 5700 years. What is the minimum age of sample in which less than 1% of the organic carbon 14 is left?
If 1% is left,
(0.01) = (0.5)^N
where N is the number of half-lives.
N = 6.644
minimum age = 37,900 years
There will be less than 1% left if the sample is older than that, so that is a minimum age.
There is not much about astronomy here
Find the age for a rock for which you determine that 68% of the original uranium-238 remains, while the other 32% has decayed into lead.
To determine the minimum age of a sample in which less than 1% of the organic carbon-14 is left, we can use the equation for exponential decay:
N = N0 * (1/2)^(t / T)
where:
N = remaining amount of carbon-14 after time t
N0 = initial amount of carbon-14
T = half-life of carbon-14
In this case, we want to find the time t when N is less than 1% of N0. This means N/N0 < 0.01.
Let's substitute these values into the equation:
0.01 = (1/2)^(t / 5700)
To solve for t, we can take the logarithm of both sides of the equation:
log(0.01) = log((1/2)^(t / 5700))
Using logarithm properties, we can bring down the exponent:
log(0.01) = (t / 5700) * log(1/2)
To find t, we isolate it on one side of the equation:
t = (5700 * log(0.01)) / log(1/2)
Using a scientific calculator to evaluate the logarithms, we find:
t ≈ (5700 * -2) / -0.301
t ≈ 37,754.15
Therefore, the minimum age of a sample in which less than 1% of the organic carbon-14 is left is approximately 37,754 years.
To determine the minimum age of a sample in which less than 1% of the organic carbon-14 is left, we need to use the concept of half-life.
The half-life of carbon-14 is given as 5700 years. This means that after 5700 years, half of the carbon-14 in a sample will have decayed. After another 5700 years, half of the remaining carbon-14 will have decayed, and so on.
To find the minimum age of a sample with less than 1% of carbon-14 remaining, we need to calculate the number of half-lives it takes for that amount of decay to occur. Since less than 1% is left, we can assume that 99% of the carbon-14 has decayed.
First, we can express 99% as a decimal: 99% = 0.99
Next, we can set up an equation to represent this decay using the formula for exponential decay:
Remaining amount = Initial amount × (1/2)^(number of half-lives)
In this case, we are looking for the number of half-lives needed to reach 0.99 remaining amount:
0.99 = 1 × (1/2)^n
To solve for "n" (the number of half-lives), we can take the logarithm of both sides of the equation:
log(0.99) = log((1/2)^n)
Using logarithm rules, we can bring the exponent "n" down:
log(0.99) = n × log(1/2)
Finally, we can solve for "n" by dividing both sides of the equation by log(1/2):
n = log(0.99) / log(1/2)
Using a calculator, we can find that n ≈ 69.31.
Since "n" represents the number of half-lives, we can round up to 70.
Now, to find the minimum age, we multiply the number of half-lives by the half-life of carbon-14:
Minimum age = 70 half-lives × 5700 years/half-life
Minimum age ≈ 399,000 years
Therefore, the minimum age of a sample in which less than 1% of the organic carbon-14 is left is approximately 399,000 years.