A 10 kg quantity of a radioactive isotope decays to 3 kg after 17 years. Find the decay constant of the isotope.
Solve this for k:
3 = 10 e^(-kt)
where t = 17 years
ln (3/10) = -17k
k = 0.0798 y^-1
Well, this is quite a radioactive question! To find the decay constant of the isotope, we can use the formula:
N(t) = N(0) * e^(-λt)
Where N(t) is the amount of the isotope at time t, N(0) is the initial amount, e is Euler's number (approximately 2.71828), λ is the decay constant, and t is the time.
In this case, we know that the initial amount N(0) is 10 kg, and the amount after 17 years N(t) is 3 kg. Let's plug these values into the equation:
3 = 10 * e^(-17λ)
To solve for λ, we need to do a little math here. Divide both sides by 10:
0.3 = e^(-17λ)
Now, let's take the natural logarithm (ln) of both sides:
ln(0.3) = -17λ
Finally, divide both sides by -17 to solve for λ:
λ = ln(0.3) / -17
And there you have it! The decay constant of the radioactive isotope is approximately equal to ln(0.3) divided by -17. Just don't let it go to your head, or you may start to feel a bit "radioactive" yourself!
To find the decay constant of the isotope, we can use the formula for exponential decay:
N(t) = N₀ * e^(-λt)
Where:
N(t) = the final quantity of the isotope (3 kg)
N₀ = the initial quantity of the isotope (10 kg)
λ = the decay constant
t = the time (17 years)
Now, we can rearrange the formula to solve for λ:
e^(-λt) = N(t)/N₀
Substituting the given values:
e^(-λ*17) = 3/10
Next, take the natural logarithm (ln) of both sides to eliminate the exponential function:
ln(e^(-λ*17)) = ln(3/10)
Simplifying the left side:
-17λ = ln(3/10)
Finally, divide both sides by -17 to solve for λ:
λ = ln(3/10) / -17
Using a calculator to evaluate this expression, the value of the decay constant (λ) is approximately -0.0622 per year.
To find the decay constant of the radioactive isotope, we can use the formula for exponential decay:
N(t) = N0 * e^(-λt)
where:
N(t) is the amount of the isotope at time t
N0 is the initial amount of the isotope (10 kg in this case)
e is the mathematical constant approximately equal to 2.71828
λ is the decay constant (what we want to find)
t is the time elapsed (17 years in this case)
Since we know that the isotope decays from 10 kg to 3 kg after 17 years, we can set up the following equation:
3 kg = 10 kg * e^(-λ * 17 years)
Now, we need to isolate the decay constant (λ). Divide both sides of the equation by 10 kg:
3 kg / 10 kg = e^(-λ * 17 years)
0.3 = e^(-λ * 17 years)
To isolate λ, we can take the natural logarithm of both sides of the equation:
ln(0.3) = ln(e^(-λ * 17 years))
The natural logarithm of e^(-λ * 17 years) simplifies to (-λ * 17 years) by the properties of logarithms.
ln(0.3) = -λ * 17 years
Now, divide both sides of the equation by (-17 years):
ln(0.3) / (-17 years) = λ
Using a calculator, we can evaluate the left side of the equation:
ln(0.3) / (-17 years) ≈ 0.1015 years^(-1)
Therefore, the decay constant of the isotope is approximately 0.1015 years^(-1).