Question

Cobalt- 60 , which undergoes beta decay, has a half-life of 5.26 yr. (a) How many beta particles are emitted in \(600 \mathrm{~s}\) by a \(3.75-\mathrm{mg}\) sample of \({ }^{60} \mathrm{Co} ?\) (b) What is the activity of the sample in \(\mathrm{Bq}\) ?

Short answer

The number of Cobalt-60 nuclei in the $3.75 \mathrm{ ~mg}$ sample can be calculated using the formula \(N = \frac{m \cdot N_A}{M}\), giving us \(N \approx 3.77 \times 10^{19}\) nuclei. The decay constant \(\lambda\) is found using the half-life formula, resulting in \(\lambda \approx 1.32 \times 10^{-11} \mathrm{~s}^{-1}\). The number of beta particles emitted in $600 \mathrm{~s}$ can be calculated as \(N_{\text{decay}} = N \cdot (1 - e^{-\lambda t})\), giving us \(N_{\text{decay}} \approx 3.14 \times 10^{9}\) beta particles. The activity of the sample in Bq can be calculated as \(A = \lambda \cdot N\), resulting in \(A \approx 4.98 \times 10^{8} \mathrm{~Bq}\).

Step-by-step solution

Calculate the number of Cobalt-60 nuclei in the sample

To find the total number of Cobalt-60 nuclei in the sample, we will use the formula: \[N = \frac{m \cdot N_A}{M}\] Where N is the number of nuclei, m is the mass of the sample, \(N_A\) is Avogadro's constant \((6.022 \times 10^{23} \ \text{mol}^{-1})\), and M is the molar mass of Cobalt-60 which is 59.9338 u. First, we need to convert the mass of the sample to grams: \[3.75 \ \text{mg} = 3.75 \times 10^{-3} \ \text{g}\] Now we can plug in the values and calculate the number of nuclei: \[N = \frac{(3.75 \times 10^{-3} \ \text{g}) \times (6.022 \times 10^{23} \ \text{mol}^{-1}) }{59.9338 \ \text{g/mol}}\]

Calculate the decay constant

To find the decay constant, we will use the half-life formula: \[\lambda = \frac{\ln{2}}{t_{1/2}}\] Where \(\lambda\) is the decay constant and \(t_{1/2}\) is the half-life of the substance. We are given the half-life of Cobalt-60 as 5.26 yr. Plugging in the values, we get: \[\lambda = \frac{\ln{2}}{5.26 \ \text{yr}}\]

Find the number of beta particles emitted in 600 s

Now that we have the decay constant, we can find the number of beta particles emitted during a specific time interval using the formula: \[N_{\text{decay}} = N \cdot (1 - e^{-\lambda t})\] We are given a time interval of 600 s. Since we have calculated \(\lambda\) in years, we should convert the time to years: \[600 \ \text{s} = \frac{600}{60 \times 60 \times 24 \times 365.25} \ \text{yr}\] Now we can calculate the number of beta particles emitted: \[N_{\text{decay}} = N \cdot (1 - e^{-\lambda t})\]

Calculate the activity in Bq

The activity of the sample is defined as the rate of decay, which we can find using the decay constant and the number of nuclei in the sample: \[A = \lambda \cdot N\] The activity should be expressed in Bq which corresponds to decays per second (s^-1). Since we have found \(\lambda\) in year^-1, we should convert it to s^-1: \[\lambda = \frac{\lambda}{(60 \times 60 \times 24 \times 365.25) \ s^{-1}}\] Now we can calculate the activity: \[A = \lambda \cdot N\]

Key concepts

Beta Decay

Beta decay is a type of radioactive decay where an unstable atomic nucleus transforms into a more stable one by emitting beta particles. This process happens when there are too many protons or neutrons in a nucleus.

• In beta-minus (β-) decay, a neutron is converted to a proton and a beta particle, which is an electron, along with an antineutrino, is emitted.
• In beta-plus (β+) decay, a proton is transformed into a neutron, a positron (beta particle), and a neutrino are emitted.

These transformations help the nucleus achieve a more balanced state, making the atom more stable. Understanding beta decay is essential because it tells us how certain radioactive elements transform and release energy.
Cobalt-60, a radioactive isotope, undergoes beta decay by transforming into an isotope of nickel, which is a more stable state. In practical applications, Cobalt-60 is widely used in radiation therapy to treat cancer, exploiting its beta decay property.

Half-Life

The half-life of a radioactive substance is the time it takes for half of the material to decay and transform into another element or isotope. It's a crucial concept because it helps to predict how long a radioactive substance will remain active.
For example, Cobalt-60 has a half-life of 5.26 years. This means that after 5.26 years, only half of the original amount of Cobalt-60 in a sample will remain, with the rest having decayed into a different substance.
This predictable rate of decay allows scientists to use radioactive isotopes in various fields.

• In medical treatment, the half-life helps figure out the duration for which a radioactive material will be effective.
• In environmental science, it helps in assessing the impact of radioactive waste.

Understanding half-life can also guide safe storage and disposal of radioactive materials.

Decay Constant

The decay constant (\(\lambda\)) is a value that describes the probability per unit time that a nucleus will decay. It is related directly to an element’s half-life and provides insight into the speed of the decay process.
Mathematically, it is expressed as: \(\lambda = \frac{\ln{2}}{t_{1/2}}\) where \(t_{1/2}\) is the half-life. The natural logarithm of 2, \(\ln{2}\), approximately equals 0.693.
A higher decay constant indicates a faster decay process, meaning the substance will lose its radioactivity more quickly.

• In practical terms, knowing the decay constant helps scientists and engineers develop precise models of nuclear decay.
• It aids in calculating the time over which a radioactive substance will maintain its key properties.

This constant is foundational in nuclear physics, providing clarity in scenarios involving radioactive dating and nuclear medicine applications.

Activity in Becquerels

The activity of a radioactive sample is a measure of the number of decays that happen per second within that sample. It's measured in becquerels (Bq), where 1 Bq is equal to one decay per second. This is a crucial unit since it provides a quantitative measure of radioactivity.
To find the activity (\(A\)) of a sample, you can use the formula: \(A = \lambda \cdot N\) where \(\lambda\) is the decay constant and \(N\) is the number of radioactive nuclei present.
A higher activity value means more decays are happening per second, indicating a more radioactive and potentially more dangerous sample.

• In environmental monitoring, it helps in assessing potential exposure risks.
• In medical settings, it assists in planning treatment doses using radioactive materials.

Understanding the activity of a sample helps in evaluating its impact and handling it appropriately with safety in mind.