Question

A certain radioactive nuclide has a half-life of 3.00 hours. a. Calculate the rate constant in \(s^{-1}\) for this nuclide. b. Calculate the decay rate in decays/s for 1.000 mole of this nuclide.

Short answer

The rate constant for the radioactive decay is \(6.42 × 10^{-5} s^{-1}\), and the decay rate for 1 mole of this radioactive nuclide is approximately \(3.864 × 10^{19} decays/s\).

Step-by-step solution

Calculate the rate constant

First, we need to calculate the rate constant for the radioactive decay. The half-life (t1/2) of a first-order reaction can be related to the rate constant (k) using the following equation: t1/2 = 0.693 / k Given the half-life is 3 hours, we can convert it into seconds by multiplying it by 3600 seconds/hour: t1/2 = 3.00 hours × 3600 s/h = 10,800 s Now, we can solve for the rate constant k: k = 0.693 / t1/2 k = 0.693 / 10,800 s k = 6.42 × 10^(-5) s^(-1) Therefore, the rate constant for the decay is 6.42 × 10^(-5) s^(-1).

Calculate the decay rate

Next, we need to calculate the decay rate for 1 mole of the nuclide. The decay rate (N) can be obtained using the following equation: N = k × N0 Where N0 is the initial amount of the radioactive substance. We know there is 1 mole of the nuclide, therefore we use Avogadro's number to calculate the number of radioactive nuclei: N0 = 1 mole × 6.022 × 10^(23) nuclei/mole ≈ 6.022 × 10^(23) nuclei Now, we can calculate the decay rate N: N = (6.42 × 10^(-5) s^(-1)) × (6.022 × 10^(23) nuclei) N = 3.864 × 10^(19) decays/s Therefore, the decay rate for 1 mole of this radioactive nuclide is approximately 3.864 × 10^(19) decays/s.

Key concepts

Half-life

Half-life is a crucial concept in understanding how radioactive decay works. It refers to the time taken for half of the radioactive atoms in a sample to decay. The half-life is independent of the initial amount of substance, remaining constant over time. For instance, if you start with 100 grams of a substance with a half-life of 3 hours, after 3 hours, you'll have 50 grams left, after 6 hours, 25 grams, and so on.
This concept is described using the formula:

• \( t_{1/2} = \frac{0.693}{k} \)

This formula connects half-life \(t_{1/2}\) to the rate constant \(k\), showing how fast or slow a radioactive substance decays.

Rate Constant

The rate constant \(k\) is a parameter that indicates the speed of a radioactive decay process. It is specific to each radioactive material and is inversely related to the half-life.
The formula \( k = \frac{0.693}{t_{1/2}} \) links the rate constant to the half-life \( t_{1/2} \). A higher \(k\) signifies a quicker decay, while a lower \(k\) indicates slower decay.
By understanding \(k\), scientists can predict how long a radioactive sample will continue to emit radiation, which is vital in fields like medicine and archaeology.

Avogadro's Number

Avogadro's number is a fundamental constant in chemistry and physics. It defines the number of atoms or molecules in one mole of a substance, approximating \(6.022 \times 10^{23}\) entities.
This number allows chemists and physicists to calculate the total amount of particles, such as radioactive nuclei, present in a sample. Understanding Avogadro's number is essential to converting between amounts of substance in moles and the actual count of particles, which is important in the calculation of decay rates.

Decay Rate

Decay rate is the measure of how quickly a radioactive substance undergoes decay.It refers to the number of disintegrations per unit time, usually expressed in decays per second (dps).
The value of decay rate \(N\) can be determined using the equation \( N = k \times N_0 \), where \(N_0\) is the initial quantity of radioactive atoms. Knowing the decay rate helps scientists assess the safety and effectiveness of materials that contain radioactive substances.
For instance, in nuclear medicine, precise decay rate calculations ensure accurate doses for treatments.

First-order Reaction

Radioactive decay commonly follows a first-order reaction pattern. In such reactions, the rate of decay is directly proportional to the amount of the substance present. This means as the quantity of the substance decreases, the rate of decay slows down.
The first-order kinetics can be described using the formula \( A_t = A_0 e^{-kt} \), where \(A_t\) is the amount of substance at time \(t\), \(A_0\) is the initial quantity, and \(k\) is the rate constant.The exponential nature of first-order reactions explains the continuous decrease in the rate of decay over time. Understanding first-order kinetics is imperative in predicting the behavior of decaying substances.