Question

A scientist has a 2.1-gg sample of pure strontium-91, 9138Sr3891Sr. The half-life of strontium-91 is 9.70h9.70h.

A.) Determine its decay constant in units of s−1.

B.) Determine the initial number of nuclei strontium-91 nuclei initially in the sample.

C.) Determine the activity in units of becquerel after 16 hours.

Answer 1

A.) The decay constant of strontium-91 is 0.257 s⁻¹.
B.) The initial number of strontium-91 nuclei in the sample is 2.1 grams.
C.) The activity of the sample after 16 hours is 0.067 becquerel.

Step-by-step explanation:

A. The decay constant (λ) can be calculated using the formula:

λ = ln(2) / T1/2

Given that the half-life (T1/2) of strontium-91 is 9.70 hours, the decay constant can be calculated as follows:

λ = ln(2) / 9.70 hours

Calculating λ gives a value of 0.0714 hours-1. To convert this to seconds, multiply by 3600 (the number of seconds in an hour), giving a decay constant of 0.257 s-1.

B. The initial number of strontium-91 nuclei in the sample can be calculated using the formula:

N0 = N / (1 - e-λt)

Where N is the mass of the sample (in grams), λ is the decay constant (in s-1), and t is the time elapsed (in seconds).

Given that the sample has a mass of 2.1 grams and the decay constant is 0.257 s-1, and solving for N0 after converting the time of 0 seconds to seconds:

N0 = 2.1 g / (1 - e-(0.257 s-1 * 0 s))

Calculating N0 gives a value of 2.1 g.

C. To determine the activity of the sample in becquerel after 16 hours, we need to use the formula:

A = A0 * e-λt

Where A0 is the initial activity (in becquerel), λ is the decay constant (in s-1), and t is the time elapsed (in seconds).

Since we don't have the initial activity (A0), we can use the formula:

A = λ * N

Where A is the activity (in becquerel), λ is the decay constant (in s-1), and N is the number of nuclei.

Given that N is equal to the initial number of nuclei (N0) we calculated earlier, and the decay constant is 0.257 s-1, and converting the time of 16 hours to seconds:

A = (0.257 s-1) * (2.1 g / (1 - e-(0.257 s-1 * 57600 s)))

Calculating A gives a value of 0.067 becquerel.

Answer 2

A.) The decay constant of strontium-91 is 1.98 x 10^-5 s^-1. B.) The initial number of strontium-91 nuclei in the sample is 1.39 x 10^22 nuclei. C.) The activity of the sample after 16 hours can be calculated using the formula A = A₀ * e^-1.65.

Step-by-step explanation:

A.) The decay constant of strontium-91 can be determined using the formula: λ = ln(2) / t½. Given that the half-life of strontium-91 is 9.70 hours, we can calculate the decay constant:

λ = ln(2) / 9.70 = 0.0713 hours-1

Since there are 3600 seconds in an hour, we can convert the units:

λ = 0.0713 / 3600 = 1.98 x 10-5 s-1

B.) The initial number of strontium-91 nuclei in the sample can be calculated using the formula: N₀ = N * eλt. Given that the sample has a mass of 2.1 grams and the molar mass of strontium-91 is 90.90 g/mol, we can determine the number of moles:

moles of strontium-91 = 2.1 / 90.90 = 0.0231 mol

Finally, we can calculate the number of nuclei using Avogadro's number:

N₀ = 0.0231 * 6.022 x 1023 = 1.39 x 1022 nuclei

C.) The activity of the sample after 16 hours can be calculated using the formula: A = A₀ * e-λt. Given that the half-life of strontium-91 is 9.70 hours and the initial activity is unknown, we need to convert the time to the units of the half-life:

t = 16 / 9.70 = 1.65 half-lives

The activity can then be determined as:

A = A₀ * e-1.65