A 10 kg quantity of a radioactive isotope decays to 3 kg after 17 years. Find the decay constant of the isotope.
Well, well, well, looks like we've got a decaying situation here! But don't worry, I'm here to bring some humor to the equation!
To find the decay constant, we can use the equation for exponential decay:
N(t) = N0 * e^(-kt)
Where:
N(t) represents the final amount of the radioactive isotope after time t.
N0 represents the initial amount of the radioactive isotope.
e is Euler's number, roughly equal to 2.71828.
k is the decay constant.
In our case, we have N(t) = 3 kg and N0 = 10 kg. We also know that the time is 17 years. So, plugging in the numbers, we get:
3 kg = 10 kg * e^(-17k)
Now, let's solve this equation and find the decay constant!
Oh, whoops! It seems I misplaced my calculator in my oversized pocket. Give me a sec to fetch it. *rummages through pockets*
Ah, here it is! Now, let's crunch these numbers!
-ln(3/10) = 17k
Oops! I almost got tangled up with those logarithms! But fear not, I'll solve it gracefully.
Taking the natural logarithm on both sides:
ln(3/10) = -17k
Wait, I thought it's impolite to take the log without asking. Let me apologize to the logarithm for this audacious act.
Okay, logarithm, may I take your value, please?
-ln(3/10) ≈ 1.203972804
Now, let's not leave poor little k all alone. Dividing both sides by -17:
k ≈ -1.203972804 / 17 ≈ -0.070871938
Well, it seems that our decay constant is approximately -0.0709. But don't let that negative sign bring you down – it's just indicating the rate of decay!
To find the decay constant of the isotope, we can use the formula:
N = N₀ * e^(-λt)
where:
N is the final quantity of the radioactive isotope (3 kg),
N₀ is the initial quantity of the radioactive isotope (10 kg),
e is the base of the natural logarithm (approximately 2.71828),
λ is the decay constant we want to find,
t is the time period (17 years).
We can rearrange the formula to solve for λ:
λ = - ln(N/N₀) / t
Substituting the values, we have:
λ = - ln(3/10) / 17
Calculating this expression, we get:
λ ≈ -0.1865 / 17
So, the decay constant of the isotope is approximately -0.01097 per year.
To find the decay constant (symbolized as λ) of the isotope, you need to use the formula for exponential decay:
N(t) = N₀ * e^(-λt)
Where:
- N(t) is the quantity of the isotope at time t
- N₀ is the initial quantity of the isotope
- e is the mathematical constant, approximately equal to 2.71828
- λ is the decay constant
- t is the time elapsed
In your case, the initial quantity (N₀) is 10 kg, and the final quantity after 17 years (N(t)) is 3 kg. Therefore, we can set up the following equation:
3 = 10 * e^(-λ * 17)
To find the value of λ, we need to solve this equation. Here's one way to do it step by step:
1. Divide both sides of the equation by 10:
3/10 = e^(-λ * 17)
2. Take the natural logarithm (ln) of both sides:
ln(3/10) = ln(e^(-λ * 17))
3. Apply the logarithmic property that ln(a^b) = b * ln(a):
ln(3/10) = -λ * 17 * ln(e)
4. Simplify ln(e) since ln(e) = 1:
ln(3/10) = -17λ
5. Divide both sides of the equation by -17:
ln(3/10) / -17 = λ
Now, use a calculator to find the value of ln(3/10) / -17, and that will give you the decay constant (λ) for the radioactive isotope.