Question

A nuclide has a decay constant of $4.28\times {10}^{-4}{\mathrm{h}}^{-1}$. If the activity of a sample is $3.14\times {10}^5{\mathrm{s}}^{-1},$ how many atoms of the nuclide are present in the sample? (a) $2.64\times {10}^{12};$ (b) $7.34\times {10}^8$ (c) $2.04\times {10}^5$ (d) $4.40\times {10}^{10};$ (e) none of these.

Short answer

The number of atoms of the nuclide present in the sample approximates $2.64\times {10}^{12}$, corresponding to option (a).

Step-by-step solution

Data present in the problem

The problem gives the decay constant $\lambda =4.28\times {10}^{-4}{\mathrm{h}}^{-1}$ and the activity $A=3.14\times {10}^5{\mathrm{s}}^{-1}$. Knowing the formula for activity, the goal is to solve for the number of atoms $N$. It's necessary to convert the decay constant to ${\mathrm{s}}^{-1}$ for the units to match.

- Conversion of units

One hour has $3600$ seconds. Therefore, the decay constant in ${\mathrm{s}}^{-1}$ is $\lambda =4.28\times {10}^{-4}\times 3600$

- Formula for number of atoms

With the newly calculated decay constant in ${\mathrm{s}}^{-1}$, use the formula of total activity, $A=\lambda N$, to solve for the number of atoms $N$. Therefore, $N=\frac{A}{\lambda }$

- Substitution and Solution

Plug the values for activity $A$ and decay constant $\lambda$ into the equation: $N=\frac{3.14\times {10}^5}{4.28\times {10}^{-4}\times 3600}$

Key concepts

Nuclear Chemistry

Nuclear chemistry is a subfield of chemistry focusing on the reactions, structures, and properties of atomic nuclei. It encompasses a range of topics including radioactivity, nuclear transmutation, and the interplay of subatomic particles. Fundamental to nuclear chemistry is understanding how radioactive decay works, which involves an unstable atomic nucleus losing energy by emitting radiation. Through methods like alpha decay, beta decay, and gamma radiation, elements can transform into different isotopes or even entirely different elements.

This field has practical applications in various industries, from medical treatments using radioisotopes to energy production in nuclear reactors. With this knowledge, scientists and engineers can harness nuclear reactions in controlled settings for beneficial purposes.

Decay Constant

The decay constant, represented by the symbol $\lambda$, is a vital parameter in nuclear chemistry that characterizes the rate at which a radioactive substance undergoes decay. It is inversely proportional to the half-life of the radioactive substance, meaning that a larger decay constant indicates a faster rate of decay, and a shorter half-life.

Mathematically, when you know the decay constant of a substance, you can predict the remaining number of undecayed nuclei after a certain time period using the exponential decay formula. This decay constant is intrinsic to each radioactive nuclide and remains constant over time, despite the number of nuclei that may have already decayed.

Activity of a Radioactive Sample

The activity of a radioactive sample is a term that describes the number of decay events occurring within that sample per unit time. Measured in becquerels (Bq) in the International System of Units, where one becquerel corresponds to one decay per second, it serves as an indicator of how radioactive a substance is.

In practical terms, a higher activity means that the sample is emitting a larger number of radioactive particles or photons in a given timeframe. The activity is directly proportional to the number of radioactive atoms present and the decay constant of the nuclide ($A=\lambda N$). By measuring activity, researchers and professionals working with radioactive materials can evaluate the potential radiation exposure and take necessary safety precautions.

Nuclide

The term nuclide is used in nuclear chemistry to refer to a species of atom characterized by its number of protons and neutrons, as well as its nuclear energy state. Nuclides are often referred to by their element name followed by their mass number. For example, Carbon-14, with six protons and eight neutrons, is a nuclide.

There are stable and unstable (radioactive) nuclides. Unstable nuclides will decay over time, changing their number of protons and/or neutrons, and this process can be described by their decay constant. By understanding the specific decay properties of a nuclide, scientists can make precise calculations concerning the material's behavior over time, like establishing dates for archaeological finds through carbon dating or in medical applications using tracers.