Exponential growth and decay
find the half life of a radioactive substance if 220 grams of the substance decays to 200 grams in 4 years?
Let the half life be T years
2^(-4/T) = 200/220 = 0.9091
(-4/T) log 2 = log 0.9091
-4/T = -0.1375
T = 29.1 years
To find the half-life of a radioactive substance, you can use the formula:
N(t) = N₀ * (1/2)^(t/h)
Where:
N(t) is the amount of the substance after time t
N₀ is the initial amount of the substance
t is the given time
h is the half-life of the substance
In this case, N₀ = 220 grams, N(t) = 200 grams, and t = 4 years. We can rearrange the formula to solve for the half-life:
200 = 220 * (1/2)^(4/h)
Now, let's solve for h:
(1/2)^(4/h) = 200/220
1/2^(4/h) = 10/11
2^(4/h) = 11/10
(2^(2/h))^2 = 11/10
Taking the square root of both sides:
2^(2/h) = √(11/10)
2^(2/h) = √(11)/√(10)
2^(2/h) = √(11)/√(10)
Now, we can isolate h by taking the logarithm of both sides:
2/h = log(√(11)/√(10)) / log(2)
Simplifying the right side:
2/h = log(√(11)/√(10)) / log(2)
2/h = log(√(11)) - log(√(10)) / log(2)
2/h = (1/2) * log(11) - (1/2) * log(10) / log(2)
2/h = (1/2) * (log(11) - log(10)) / log(2)
2/h = (1/2) * log(11/10) / log(2)
2/h = log(11/10) / (2 * log(2))
2/h = 0.014628 / (2 * 0.30103)
2/h = 0.014628 / 0.60206
2/h = 0.024267
Finally, solving for h:
2/h = 0.024267
h/2 = 1/0.024267
h = 2 / (1/0.024267)
h ≈ 82.25
Therefore, the half-life of the radioactive substance is approximately 82.25 years.
To find the half-life of a radioactive substance, we need to use the formula for exponential decay. The formula is:
A = A₀ * e^(-kt)
Where:
A₀ is the initial amount of the substance (220 grams in this case).
A is the amount of the substance after time t (200 grams in this case).
k is the decay constant.
t is the time in years (4 years in this case).
e is a mathematical constant approximately equal to 2.71828.
We can rearrange the formula to solve for the decay constant:
k = -ln(A/A₀) / t
Where ln represents the natural logarithm.
Now, let's substitute the values given in the problem:
k = -ln(200/220) / 4
Using a calculator, we can find the value of ln(200/220):
k ≈ -0.096
Now, we have the value of the decay constant. The half-life (t₁/₂) is defined as the time it takes for the substance to decay by half its initial amount. In exponential decay, we can find the half-life using the formula:
t₁/₂ = ln(2) / k
Let's calculate it:
t₁/₂ = ln(2) / -0.096
Using a calculator, we can find the value of ln(2):
t₁/₂ ≈ 7.22 years
Therefore, the half-life of the radioactive substance is approximately 7.22 years.