What is the decay constant for a radioactive isotope which decreases to 34% of its original value in 2.48 hours?

ln(No/N) = kt

ln(100/34) = k(2.48)
I started with 100 arbitrarily, then made the N value 34% of that. Solve for k.

thank you so much

To find the decay constant (λ) for a radioactive isotope, we can use the equation:

N(t) = N₀ * e^(-λt)

Where:
N(t) is the amount of the isotope remaining at time t
N₀ is the initial amount of the isotope
e is the mathematical constant approximated as 2.71828
λ is the decay constant
t is the time elapsed

We are given that the isotope decreases to 34% of its original value, which means N(t) = 0.34 and t = 2.48 hours.

Substituting these values into the equation:

0.34 * N₀ = N₀ * e^(-λ * 2.48)

Dividing both sides by N₀:

0.34 = e^(-λ * 2.48)

Taking the natural logarithm of both sides:

ln(0.34) = -λ * 2.48

Dividing both sides by -2.48:

λ = ln(0.34) / -2.48

Using a calculator, we can evaluate this expression to find the decay constant.

To determine the decay constant for a radioactive isotope, we can use the equation:


N(t) = N₀ * e^(-λt)

Where:
- N(t) is the amount of the radioactive isotope at time t
- N₀ is the initial amount of the radioactive isotope
- e is the base of the natural logarithm (approximately 2.71828)
- λ (lambda) is the decay constant
- t is the time elapsed

In this case, we are given that the radioactive isotope decreases to 34% of its original value. This means N(t) is 0.34N₀.

So, we have:

0.34N₀ = N₀ * e^(-λ * 2.48)

By dividing both sides by N₀, we cancel it out and simplify the equation:

0.34 = e^(-λ * 2.48)

To isolate the decay constant (λ), we need to take the natural logarithm (ln) of both sides:

ln(0.34) = -λ * 2.48

Finally, we solve for λ by dividing both sides by -2.48:

λ = ln(0.34) / -2.48

Using a scientific calculator or any online calculator capable of evaluating natural logarithms, we can find the value of λ by plugging in ln(0.34) and -2.48 into the expression.