Question

Radium-226, which undergoes alpha decay, has a half-life of 1600 yr. (a) How many alpha particles are emitted in 5.0 min by a 10.0 -mg sample of \(^{226} \mathrm{Ra}\) ? (b) What is the activity of the sample in mCi?

Short answer

(a) The number of alpha particles emitted in 5 minutes by a 10 mg sample of Radium-226 is found by calculating the number of decayed nuclei (\(N_{decay}\)) using the decay formula: \[ N_{decay} = N \times (1 - e^{-λ \times (9.512 \times 10^{-6}\, \text{years})}) \] (b) The activity of the sample in millicuries (mCi) can be calculated by converting the activity in Becquerels (Bq) to millicuries using the conversion factor: \[ \text{Activity} (\text{mCi}) = \frac{\text{Activity} (\text{Bq})}{3.7 \times 10^{10}\, \frac{\text{Bq}}{\text{Ci}}} × 10^{3}\, \frac{\text{mCi}}{\text{Ci}} \]

Step-by-step solution

Find the Decay Constant

Using the half-life formula, we can find the decay constant (λ) for Radium-226. Half-life formula: \[ T_{1/2} = \frac{0.693}{λ} \] Given half-life of Radium-226: \[ T_{1/2} = 1600\, \text{years} \] Calculate the decay constant: \[ λ = \frac{0.693}{T_{1/2}} \]

Convert Time to the Same Unit

To use the decay constant in our calculations, we need to convert the given time of 5 minutes into the same unit as the half-life (years). \[ 5.0\, \text{min} × \frac{1\, \text{hour}}{60\, \text{min}} × \frac{1\, \text{day}}{24\, \text{hours}} × \frac{1\, \text{year}}{365.25\, \text{days}} = 9.512 \times 10^{-6}\, \text{years} \]

Find the Number of Nuclei in the Sample

Now we need to find the number of Radium-226 nuclei in the 10 mg sample. Given sample mass: \[ m = 10\, \text{mg} \] Mass of one Radium-226 nucleus: \[ m_{Ra} = \frac{226\, \text{g/mol}}{(6.022 \times 10^{23}\, \text{nuclei/mol})} \] Now, calculate the number of Radium-226 nuclei (\(N\)) in the sample: \[ N = \frac{m}{m_{Ra}} \]

Find the Number of Nuclei That Decay in 5 Minutes

Using the decay formula, we can find the number of nuclei that decay in 5 minutes: Decay Formula: \[ N_{decay} = N \times (1 - e^{-λ \times t}) \] Plug in the values from previous steps: \[ N_{decay} = N \times(1 - e^{-λ \times (9.512 \times 10^{-6}\, \text{years})}) \]

Convert the Number of Decayed Nuclei to Alpha Particles

Since one alpha particle is emitted per decay, the number of alpha particles is equal to the number of decayed nuclei: \[ \text{Alpha particles} = N_{decay} \]

Calculate the Activity in Becquerels

Now, we can find the activity of the sample by dividing the number of alpha particles by the elapsed time of 5 minutes. First, we convert 5 minutes to seconds: \[ 5.0\, \text{min} × 60\, \frac{\text{s}}{\text{min}} = 300\, \text{s} \] Next, calculate the activity in Becquerels (Bq): \[ \text{Activity} = \frac{\text{Alpha particles}}{300\, \text{s}} \]

Convert the Activity to millicuries (mCi)

Finally, we convert the activity from Becquerels to millicuries: Conversion factor: \[ 1\, \text{Curie (Ci)} = 3.7 \times 10^{10}\, \text{Bq} \] Convert to millicuries: \[ \text{Activity} (\text{mCi}) = \frac{\text{Activity} (\text{Bq})}{3.7 \times 10^{10}\, \frac{\text{Bq}}{\text{Ci}}} × 10^{3}\, \frac{\text{mCi}}{\text{Ci}} \] By following the above steps, you will find the number of alpha particles emitted in 5 minutes by the 10 mg sample of Radium-226 and its activity in millicuries.

Key concepts

Radium-226

Radium-226 is a naturally occurring radioactive isotope of radium. It undergoes alpha decay, which is a type of radioactive decay where the nucleus emits an alpha particle. An alpha particle is essentially a helium nucleus, comprising two protons and two neutrons.

Radium-226 is significant in the study of radioactivity because it has a relatively long half-life of 1600 years. This means it remains radioactive for extended periods, making it a point of interest in both scientific research and practical applications like medical radiotherapy and radiographic testing.

• Originates through the decay of uranium and thorium.
• Produces radon gas during decay, which is itself radioactive.
• Important historical use: once used in luminous paints for watches and instrument dials.

Half-life Calculation

The concept of half-life is fundamental in nuclear physics and chemistry because it describes how quickly a radioactive substance decays. The half-life of a substance is the time it takes for half of the atoms in a sample to decay.

For Radium-226, its half-life is 1600 years. This value indicates that in 1600 years, half of any given sample of Radium-226 will have transformed into another element through alpha decay. This transformation is not only essential for understanding radioactive decay but also for calculations involving the decay constant and other nuclear properties.

• Used to determine the decay constant, λ, using the formula: \( T_{1/2} = \frac{0.693}{λ} \)
• Crucial for figuring out the age of ancient materials via radiometric dating techniques.
• Helps in predicting the long-term behavior of radioisotopes.

Radioactivity Measurement

Measuring radioactivity is key to understanding the rate of decay in radioactive materials. The activity of a radioactive sample, which describes the number of disintegrations per unit time, is often measured in units such as Becquerels (Bq) or Curies (Ci).

In practice, activity measurements allow us to determine how many radioactive decays occur in a given time frame, like the number of alpha particles emitted by Radium-226 over a 5-minute period. The calculation involves the decay constant and the number of undecayed nuclei at a given time.

• Activity \( A \) can be expressed as: \( A = λN \) where \( N \) is the number of radioactive atoms.
• Different units: 1 Bq = 1 disintegration per second, while 1 Ci = \( 3.7 \times 10^{10} \) Bq.
• Importance in safety: Monitoring the activity helps ensure any radioactive exposure is within safe limits.

Decay Constant

The decay constant, denoted as \( λ \), is an important parameter in the study of radioactive decay. It represents the probability per unit time that a given nucleus will decay.

To find the decay constant for Radium-226, we use its known half-life within the formula: \( λ = \frac{0.693}{T_{1/2}} \). With a half-life of 1600 years, this calculation provides the specific decay rate for Radium-226.

• Defines the rate at which a sample of radioactive atoms decays.
• Inversely related to the stability of a nuclide: larger \( λ \) means less stable.
• Essential in deriving various other properties of radioisotopes and predicting future decay events.

Understanding the decay constant is vital for applications ranging from medical diagnostics to nuclear energy calculations.