if 60% of a radioactive element remains radioactive after 300 million years, then what percent remains radioactive after 500 million years? what is the half-life of this element?
m = M e^-kt
m/M = 0.60 = e^-k(300)
ln 0.60 = -300 k
-.511 = -300 k
k = 0.00170/million years
m/M = e^-0.00170(500)
= 0 .427
so
42.7 %
for half life
0.5 = e^-.00170 t
ln .5 = -.0017 t
solve for t in millions of years
To find the percent of the radioactive element remaining after 500 million years, we can use the concept of exponential decay and the given information.
First, let's calculate the decay constant (k) using the formula for exponential decay:
N(t) = N₀ * e^(-kt)
Where:
- N(t) is the amount of the radioactive element remaining after time t
- N₀ is the initial amount of the radioactive element
- e is the mathematical constant approximately equal to 2.71828
Let's assume the initial amount of the radioactive element is 100 (for simplicity). Therefore, after 300 million years, 60% of it remains:
N(300) = 100 * 0.6 = 60
Now, we can use this information to find the decay constant (k) by rearranging the formula:
k = -ln(N(300) / N₀) / t
Where:
- ln is the natural logarithm function
- t is the time in years
Using the given values:
k = -ln(60 / 100) / 300
Next, we can use the decay constant to find the percent remaining after 500 million years:
N(500) = N₀ * e^(-k * 500)
Substituting the values:
N(500) = 100 * e^(-k * 500)
To find the half-life of the element, we can use the formula:
t₁/₂ = (ln 2) / k
Let's calculate these values step-by-step:
1. Calculate the decay constant (k):
k = -ln(60 / 100) / 300
k ≈ -0.00445
2. Find the percent remaining after 500 million years:
N(500) = 100 * e^(-k * 500)
N(500) ≈ 12.19%
Therefore, approximately 12.19% of the radioactive element remains after 500 million years.
3. Calculate the half-life of the element:
t₁/₂ = (ln 2) / k
t₁/₂ ≈ 156 years
Thus, the half-life of this element is approximately 156 years.
To find out what percent of the radioactive element remains radioactive after 500 million years, we need to understand the concept of half-life.
Half-life is the time it takes for half of a radioactive substance to decay. In this case, we know that after 300 million years, 60% of the element remains radioactive. This implies that after every 300 million years, the amount of radioactive substance is halved.
To calculate the percent remaining after 500 million years, we need to determine how many half-lives have passed. We can do this by dividing the time elapsed (500 million years) by the half-life (300 million years):
Number of half-lives = Time elapsed / Half-life
Number of half-lives = 500 million years / 300 million years
Number of half-lives ≈ 1.67
Since we can't have a fraction of a half-life, we can consider that one full half-life has passed (which is equal to 100% decay). By doing this, we can calculate the remaining percentage:
Percent remaining = 100% / 2
Percent remaining = 50%
Thus, after 500 million years, approximately 50% of the radioactive element remains radioactive.
Now, let's determine the half-life of this element. We already know that after 300 million years, 60% of the element remains radioactive. This means that 40% has decayed, or half of the substance has decayed.
Percent decayed = 100% - Percent remaining
Percent decayed = 100% - 60%
Percent decayed = 40%
Since half of the substance has decayed, we can set up the equation:
40% = 50% / 2
To solve for the half-life, we can divide both sides of the equation by 40%:
40% / 40% = (50% / 2) / 40%
1 = 1.25 / Half-life
Now, we can calculate the half-life:
1.25 / Half-life = 1
Half-life = 1.25
Therefore, the half-life of this radioactive element is 1.25 million years.