the half life of isotope X is 2.0 years how many years would it take for 4.0 mg sample of X to decay and have .50 mg of it remain?
To determine the time required for the decay of a sample, we can use the concept of the half-life.
The half-life is the time it takes for half of a sample to decay. In this case, the half-life of isotope X is given as 2.0 years.
We want to know the time it takes for a 4.0 mg sample to decay to 0.50 mg. Let's calculate the number of half-lives it takes first:
Initial mass of sample = 4.0 mg
Final mass of sample = 0.50 mg
Number of half-lives = (log(final mass / initial mass)) / (log(0.5))
Number of half-lives = (log(0.50 / 4.0)) / (log(0.5))
Number of half-lives = -2 / -0.301 = 6.64
Since we can't have 0.64 of a half-life, we will consider it as 7 half-lives.
Now, we know the number of half-lives required. To find the time, we need to multiply it by the half-life:
Time for decay = Number of half-lives × Half-life
Time for decay = 7 × 2.0 years
Time for decay = 14 years
Therefore, it would take approximately 14 years for a 4.0 mg sample of isotope X to decay and have 0.50 mg remaining.
To answer this question, we can use the formula for exponential decay:
N = N₀ * (1/2)^(t / T)
where:
N is the final amount remaining (0.50 mg in this case),
N₀ is the initial amount (4.0 mg in this case),
t is the time that has passed, and
T is the half-life of the isotope (2.0 years in this case).
We need to find the time, t. Let's rearrange the formula to solve for t:
t = T * log₂(N / N₀)
Now we can substitute the given values and calculate the result:
t = 2.0 years * log₂(0.50 mg / 4.0 mg)
To do this calculation, we'll use the base-change formula to convert the logarithmic expression:
t = 2.0 years * log(0.50 mg / 4.0 mg) / log(2)
Calculating this expression, we find:
t ≈ 2.0 years * (-1) / (-0.301) ≈ 2.0 years * 3.32 ≈ 6.64 years
Therefore, it would take approximately 6.64 years for the 4.0 mg sample of isotope X to decay to 0.50 mg.
.5mg is 1/8 of 4 mg
1/8 = (1/2)^3
so, that's 3 half-lives