What is the half-life of a 100.0 g sample of nitrogen-16 that decays to 12.5 g of nitrogen-16 in 21.6 s? Please help me!!
ln(No/N) = kt
No = 100g
N = 12.5g
k = solve for this
t = 21.6 sec
Then Substitute k into
k = 0.693/t1/2 and solve for t1/2
Yes
To find the half-life of a substance, we can use the formula:
t(1/2) = (ln(2))/k
Where t(1/2) is the half-life, ln is the natural logarithm, and k is the decay constant.
In this case, we have the initial mass of Nitrogen-16 (100.0 g), the final mass of Nitrogen-16 (12.5 g), and the time it takes for the decay to occur (21.6 s).
To find the decay constant (k), we can use the formula:
k = (ln(Nt/N0)) / (-t)
Where Nt is the final mass, N0 is the initial mass, and t is the time.
Let's calculate k first:
k = (ln(12.5/100.0)) / (-21.6)
k ≈ -0.033 s^-1
Now, we can find the half-life using the formula:
t(1/2) = (ln(2))/(-0.033)
Calculating this, we get:
t(1/2) ≈ 21.0 s
Therefore, the half-life of the Nitrogen-16 sample is approximately 21.0 s.
To find the half-life of a substance, we need to use the formula for exponential decay:
N(t) = N0 * (1/2)^(t / half-life)
Where:
- N(t) is the amount of substance at time t
- N0 is the initial amount of substance
- t is the time that has passed since the initial amount
- half-life is the time it takes for half of the substance to decay
In this case, we have:
- N(t) = 12.5 g
- N0 = 100.0 g
- t = 21.6 s
We want to find the half-life. To do this, we can rearrange the formula:
(1/2)^(t / half-life) = N(t) / N0
Substituting the given values:
(1/2)^(21.6 / half-life) = 12.5 / 100.0
To find the half-life, we need to solve this equation. We can start by taking the logarithm (base 2) of both sides:
log2[(1/2)^(21.6 / half-life)] = log2(12.5 / 100.0)
The logarithm of an exponent is equal to the exponent multiplied by the logarithm of the base. So, we can simplify the equation to:
(21.6 / half-life) * log2(1/2) = log2(12.5 / 100.0)
The logarithm of 1/2 to the base 2 is -1, so the equation becomes:
(21.6 / half-life) * -1 = log2(12.5 / 100.0)
Simplifying further:
-21.6 / half-life = log2(12.5 / 100.0)
Now, we can isolate the half-life:
half-life = -21.6 / log2(12.5 / 100.0)
Using a calculator to evaluate the expression on the right-hand side will give us the value of the half-life.