A radioactive sample contains two different types of radioactive nuclei: A, with half-life 5.0 days, and B, with half-life 30.0 days. Initially, the decay rate of the A-type nucleus is 64 times that of the B-type nucleus. When will their decay rates be equal

t = 39.42 days

Explanation:

To do this, we need to use the general expression for decay rate:

R = R₀ exp(-λt)    (1)

According to the problem, decay rate of A initially is 64 times rate of B, so we can say the following:

A₀ = 64B₀   (2)

The value of λ can be determined using:

λ = ln2 / t(1/2)  (3)

Let's calculate first the value of λ for each nuclei:

For A: λ₁ = ln2 / 5 = 0.1286 day⁻¹

For B: λ₂ = ln2 / 30 = 0.0231 day⁻¹

Now, let's write an expression using (1) for each nuclei.

A = A₀ exp(-0.1286t)

A = 64B₀ exp(-0.1286t)   (3)

B = B₀ exp(-0.0231t)    (4)

We want to know the time when A = B , therefore, we can actually equals (3) and (4) and solve for t:

64B₀ exp(-0.1286t) = B₀ exp(-0.0231t)

64 exp(-0.1286t) = exp(-0.0231t)

ln(64 * exp(-0.1286t)) = ln(exp(-0.0231t))

ln64 + ln(exp(-0.1286t))= -0.0231t

4.1589 - 0.1286t = -0.0231t

4.1589 = (0.1286 - 0.0231)t

t = 4.1589 / 0.1055

So, in 39 days, the decay rates of A and B will be the same.

Hope this helps