Question

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

Short answer

a. The rate constant (k) for this nuclide is approximately \(6.417 \times 10^{-5} \, s^{-1}\). b. The decay rate for 1 mole of this nuclide is approximately \(3.861 \times 10^{19} \, decays/s\).

Step-by-step solution

(Step 1: Convert half-life to seconds)

In order to calculate the rate constant, we need to convert the half-life from hours to seconds: \(3.00 \, h \times \frac{60 \, min}{1 \, h} \times \frac{60 \, s}{1 \, min} = 10,800 \, s\)

(Step 2: Calculate the rate constant)

Now, using the formula for the rate constant, we can find k: \(k = \frac{ln(2)}{half-life} = \frac{ln(2)}{10800 \, s} ≈ 6.417 \times 10^{-5} \, s^{-1}\)

(Step 3: Calculate the number of radioactive nuclides in 1 mole)

We are given 1 mole of this radioactive nuclide, we must convert this into the number of radioactive nuclides using Avogadro's number: \(1.000 \, mole \times \frac{6.022 \times 10^{23} \, nuclides}{1 \, mole} = 6.022 \times 10^{23} \, nuclides\)

(Step 4: Calculate the decay rate)

Now, using the decay rate formula, k from step 2, and the number of radioactive nuclides from step 3, we can find the decay rate in decays/s: decay_rate = k * N decay_rate = (6.417 x 10^-5 s^-1)(6.022 x 10^23 nuclides) decay_rate ≈ 3.861 x 10^19 decays/s So, the decay rate of 1 mole of this radioactive nuclide is approximately \(3.861 \times 10^{19} \, decays/s\).

Key concepts

Half-Life

Understanding the concept of half-life is fundamental when studying radioactive decay. The half-life of a radioactive isotope is the time it takes for half of the radioactive atoms in a sample to decay. Unlike the lifespan of a living organism, the half-life is statistically consistent for a given substance, regardless of the amount present or the conditions surrounding it.

When we say that a nuclide has a half-life of 3.00 hours, like in our exercise, it means that if we start with a particular amount of that nuclide, only half of it will remain after 3.00 hours. After another 3.00 hours, half of the remaining will decay, leaving us with a quarter of the initial amount, and this process continues. Knowing the half-life of an isotope is essential for dating substances, determining the safety of radioactive materials, and in medical applications where isotopes are used for diagnosis or treatment.

Rate Constant Calculation

The rate constant, often denoted as \(k\), is at the heart of the kinetic analysis of radioactive decay. It defines the proportionality between the decay rate and the number of undecayed nuclei. The calculation of the rate constant is linked directly to the half-life of a substance.

To calculate the rate constant from half-life, we use the formula \(k = \frac{ln(2)}{half-life}\). The natural logarithm of 2 in this formula comes from the exponential decay equation. This calculation gives us a clear view of how fast the radioactive decay process occurs; it is specific to each radioisotope and can be used to predict future behavior of radioactive samples.

Decay Rate

The decay rate, expressed in decays per second, is a measure of the activity of a radioactive sample—it tells us how many nuclei are decaying in a given time period. To find the decay rate, we use the rate constant \(k\) we calculated earlier and multiply it by the number of undecayed nuclei present, which we can deduce from Avogadro's number and the mole amount of the substance.

The formula is as follows: \(decay\_rate = k \times N\), where \(N\) represents the number of radioactive nuclides. In our exercise, after calculating the rate constant, we determined that the decay rate of 1 mole of the nuclide is approximately \(3.861 \times 10^{19}\) decays per second, highlighting the immense scale of atomic processes.

Avogadro's Number

Avogadro's number, approximately \(6.022 \times 10^{23}\), is a constant that represents the number of particles found in one mole of a substance. This incredibly large number bridges the gap between the macroscopic scale, familiar in the laboratory, and the microscopic scale of atoms and molecules.

When dealing with substances like radioactive nuclides, Avogadro's number allows us to translate amounts in moles (a concept familiar in chemistry) to an actual number of atoms or molecules. This is particularly useful in radioactivity calculations, as we saw in our exercise, where we used Avogadro's number to determine how many nuclides were present in a 1 mole sample, providing the necessary figures to calculate the decay rate.