Chromium-48 has a short half-life of 21.6 h. How long will it take 360.00 g of chromium-48 to decay to 11.25g.
Please help me
To determine how long it will take for 360.00 g of chromium-48 to decay to 11.25 g, we need to use the concept of half-life.
The half-life of chromium-48 is given as 21.6 hours. This means that in 21.6 hours, the amount of chromium-48 will decrease by half.
We can use the formula for exponential decay to find the number of half-lives required to reach the desired amount:
N = N₀ * (1/2)^(t/h)
Where:
N is the remaining amount of chromium-48 (11.25 g)
N₀ is the initial amount of chromium-48 (360.00 g)
t is the time
h is the half-life (21.6 hours)
Substituting the given values, we have:
11.25 = 360.00 * (1/2)^(t/21.6)
Now, let's solve for t:
(1/2)^(t/21.6) = 11.25 / 360.00
(1/2)^(t/21.6) = 0.03125
Take the logarithm of both sides (with base 1/2):
(t/21.6) * log(1/2) = log(0.03125)
(t/21.6) = log(0.03125) / log(1/2)
Using a calculator, we find:
(t/21.6) ≈ -4.97
Multiply both sides by 21.6 to solve for t:
t ≈ -4.97 * 21.6
t ≈ -107.47
The negative value for t doesn't make physical sense, so we can take the absolute value:
t ≈ 107.47 hours
Therefore, it will take approximately 107.47 hours for 360.00 g of chromium-48 to decay to 11.25 g.
To solve this problem, we can use the formula for exponential decay:
N(t) = N₀ * e^(-λt),
where:
N(t) is the amount of the substance at time t,
N₀ is the initial amount of the substance,
e is the base of the natural logarithm (approximately 2.718),
λ is the decay constant, equal to ln(2) / half-life,
t is the time passed.
First, let's find the decay constant (λ). The half-life (t₁/₂) is given as 21.6 hours. The decay constant can be calculated as:
λ = ln(2) / t₁/₂.
λ = ln(2) / 21.6.
Calculating the value of λ, we get:
λ ≈ 0.032 / hour.
Now we can use the decay equation to determine the time it takes for the chromium-48 to decay from 360.00g to 11.25g. We need to solve for t, so rearrange the equation:
t = ln(N(t) / N₀) / -λ.
Substituting the given values:
t = ln(11.25 / 360) / -0.032.
Using a scientific calculator, we can evaluate the natural logarithm:
t ≈ ln(0.03125) / -0.032.
t ≈ -3.456 / -0.032.
Finally, we can calculate the time:
t ≈ 108 hours.
Therefore, it will take approximately 108 hours for 360.00g of chromium-48 to decay to 11.25g.