The decay constant of a radioactive nuclide is 9.5x10^-3 s^-1 . At given instant, the activity of a specimen of the nuclide is 80 mCi. The time interval required for the activity to decline to 40 mCi is closest to: ?

To find the time interval required for the activity to decline to 40 mCi, we can use the radioactive decay formula:

A = A₀ * e^(-λt)

where:
A = current activity
A₀ = initial activity
λ = decay constant
t = time interval

We are given that:
A₀ = 80 mCi
A = 40 mCi
λ = 9.5x10^-3 s^-1

Substituting these values into the formula, we get:
40 = 80 * e^(-9.5x10^-3 * t)

Divide both sides of the equation by 80:
0.5 = e^(-9.5x10^-3 * t)

To solve for t, we need to take the natural logarithm of both sides:
ln(0.5) = -9.5x10^-3 * t

Now we solve for t:
t = ln(0.5) / (-9.5x10^-3)

Using a scientific calculator, we can evaluate ln(0.5) and perform the division to find the value of t.

Therefore, the time interval required for the activity to decline to 40 mCi would be closest to the calculated value of t.

To find the time interval required for the activity to decline from 80 mCi to 40 mCi, we can use the formula for radioactive decay:

A(t) = A₀ * e^(-λt)

Where:
A(t) is the activity at time t,
A₀ is the initial activity,
λ is the decay constant, and
t is the time elapsed.

We're given A₀ = 80 mCi and we need to find the value of t when A(t) = 40 mCi.

To do this, we can rearrange the formula as follows:

A(t) = A₀ * e^(-λt)
40 = 80 * e^(-9.5x10^-3 * t)

Next, we divide both sides of the equation by 80:

40/80 = e^(-9.5x10^-3 * t)
0.5 = e^(-9.5x10^-3 * t)

To solve for t, we need to take the natural logarithm of both sides:

ln(0.5) = ln(e^(-9.5x10^-3 * t))
ln(0.5) = -9.5x10^-3 * t

Now, we can solve for t:

t = ln(0.5) / (-9.5x10^-3)

Using a calculator, we find:

t ≈ 72.77 seconds

Therefore, the time interval required for the activity to decline from 80 mCi to 40 mCi is closest to 72.77 seconds.