Question

What is the half-life (in years) of \({ }^{4 \mathrm{~T}} \mathrm{~T}\) i if a \(1.0 \mathrm{ng}\) sample decays initially at a rate of \(4.8 \times 10^{3}\) disintegrations/s?

Short answer

The half-life of tritium \(^{4 ext{T}} ext{T} ext{T}\) is approximately 12.3 years.

Step-by-step solution

Identify the given information

The exercise provides two key pieces of information: the initial mass of the sample, which is 1.0 ng, and the initial decay rate, which is \(4.8 \times 10^3\) disintegrations per second.

Understand the relationship between decay constant and half-life

The decay constant \(\lambda\) is related to the half-life \(t_{1/2}\) by the formula \(t_{1/2} = \frac{\ln(2)}{\lambda}\). We need to calculate \(\lambda\) first to find the half-life.

Calculate the number of nuclei

Determine the initial number of nuclei in the sample (\(N_0\)) using the atomic mass. For \(^{4 ext{T}} ext{T} ext{T}\) (tritium), its molar mass is approximately 3 g/mol. Convert 1.0 ng to grams, calculate moles, and then use Avogadro's number \(6.022 \times 10^{23}\, \text{mol}^{-1}\) to find \(N_0\).

Determine the decay constant \(\lambda\)

Using the initial decay rate \(R_0 = 4.8 \times 10^3\) disintegrations/s and \(R_0 = \lambda N_0\), solve for \(\lambda\) by rearranging the equation to \(\lambda = \frac{R_0}{N_0}\).

Calculate the half-life

Substitute the value of \(\lambda\) obtained into the formula for \(t_{1/2}\), \(t_{1/2} = \frac{\ln(2)}{\lambda}\), to compute the half-life in seconds, and then convert it to years by using the conversion factor (1 year = 3.1536 \times 10^7 seconds).

Key concepts

Decay Constant

When studying radioactive decay, the decay constant, represented by the symbol \( \lambda \), is a crucial concept that helps describe how quickly a radioactive substance disintegrates over time. It is one of the fundamental properties of a radioactive material. In simple terms, the decay constant reflects the probability per unit time that a single nucleus will decay.

For any radioactive material, the relationship between its decay constant and half-life \( t_{1/2} \) is given by the formula:

• \( t_{1/2} = \frac{\ln(2)}{\lambda} \)

The natural logarithm of 2, \( \ln(2) \), approximately equals 0.693. This equation shows an inverse relationship: as the decay constant increases, the half-life decreases, meaning the substance decays faster. By knowing the decay constant, we can predict how long it will take for half the sample to decay, which is key to many applications in fields like medicine, archaeology, and geology.

In practice, the decay constant is determined using the initial decay rate and the initial number of nuclei, according to the formula \( R_0 = \lambda N_0 \), where \( R_0 \) is the initial decay rate and \( N_0 \) represents the initial number of nuclei in the sample.

Disintegration Rate

The disintegration rate, also called the decay rate, tells us how many atomic nuclei of a radioactive sample are undergoing decay in a certain time period. It's essentially a measure of the activity of the sample, indicating how quickly transformations are occurring. This rate is usually measured in disintegrations per second (dps) or becquerels (Bq), with 1 Bq equaling 1 disintegration per second.

In our exercise, the initial disintegration rate \( R_0 \) was provided as \( 4.8 \times 10^3 \) disintegrations per second. This means that initially, the sample experiences 4,800 disintegrations every second. The rate of disintegration can help us determine other important characteristics of the radioactive sample, such as its decay constant.

The decay constant \( \lambda \) can be calculated using the formula:

• \( \lambda = \frac{R_0}{N_0} \)

Here, \( N_0 \) is the initial number of nuclei, and \( R_0 \) is the disintegration rate. By understanding the disintegration rate, scientists can gain insights into how quickly the material is losing its radioactive nuclei and how long the material might pose a radiological hazard or remain useful for its intended purpose.

Number of Nuclei

The concept of the number of nuclei \( N_0 \) in a radioactive sample is central to understanding and calculating both the decay constant and the half-life. It refers to the total number of radioactive atoms present initially in the sample. The calculation starts with knowing the mass of the sample and involves a few conversion steps.

For the tritium, \( ^{3} \text{T} \), involved in the exercise, its molar mass is approximately 3 g/mol. By converting the sample's mass from nanograms to grams, you can find the number of moles. Then, using Avogadro's number, which is \( 6.022 \times 10^{23} \text{mol}^{-1} \), you can calculate the initial number of radioactive nuclei \( N_0 \).

Here's a more detailed breakdown of the steps:

• Convert mass from nanograms to grams.
• Calculate moles using molar mass (moles = mass/molar mass).
• Calculate \( N_0 \) using Avogadro's number (\( N_0 = \text{moles} \times 6.022 \times 10^{23} \)).

Once the number of nuclei is known, this value can be combined with the disintegration rate to find the decay constant, ultimately helping to determine the half-life of the radioactive sample.