Problem 68 - Chemistry: The Central Science

Question

Chlorine has two stable nuclides, \({ }^{35} \mathrm{Cl}\) and \({ }^{37} \mathrm{Cl}\). In contrast, \({ }^{36} \mathrm{Cl}\) is a radioactive nuclide that decays by beta emission. (a) What is the product of decay of \({ }^{36} \mathrm{Cl}\) ? (b) Based on the empirical rules about nuclear stability, explain why the nucleus of \({ }^{36} \mathrm{Cl}\) is less stable than either \({ }^{35} \mathrm{Cl}\) or \({ }^{37} \mathrm{Cl}\)

Step-by-Step Solution

Answer

(a) The product of decay of \({ }^{36}\mathrm{Cl}\) after beta emission is \({ }^{36}_{18}\mathrm{Ar}\). (b) \({ }^{36}\mathrm{Cl}\) is less stable than \({ }^{35}\mathrm{Cl}\) or \({ }^{37}\mathrm{Cl}\) due to the odd-even rule (odd numbers of both protons and neutrons) and its neutron-proton ratio, which is not as close to 1 as the other isotopes.

Step 1: Part (a): Determine the product of decay of \({ }^{36}\mathrm{Cl}\) after beta emission

Beta emission occurs when a neutron inside the nucleus is converted into a proton and an electron (beta particle). The electron is emitted from the nucleus, and as a result, a neutron turns into a proton, causing an increase in atomic number while keeping the same atomic mass. In the case of \({ }^{36}\mathrm{Cl}\), the decay by beta emission can be represented as follows: \[{}^{36}\mathrm{Cl} \rightarrow {}^{A}\mathrm{X} + {}^{0}_{-1}\beta\] Where A is the mass number of the product nucleus, X is the chemical symbol of the product nucleus, and β is the emitted beta particle. Now let's consider the available information: 1. The atomic mass number, A, stays constant during the decay process: \(A=36\) 2. The atomic number, Z, increases by 1 unit during the decay process. Since chlorine has an atomic number of 17 (\({ }^{36}_{17}\mathrm{Cl}\)), it means the product of the decay will have an atomic number of \(17+1=18\). The element with atomic number 18 is Argon (Ar). Therefore, the product of decay of \({ }^{36}\mathrm{Cl}\) will be: \({ }^{36}_{18}\mathrm{Ar}\)

Step 2: Part (b): Stability comparison of \({ }^{35}\mathrm{Cl}\), \({ }^{37}\mathrm{Cl}\), and \({ }^{36}\mathrm{Cl}\)

To analyze the stability of these isotopes, we need to consider various empirical rules about nuclear stability. Two important rules for this case are: 1. The Odd-Even Rule: Nuclei with both an even number of protons and neutrons are more stable than nuclei with odd numbers of protons and neutrons. 2. The Neutron-Proton Ratio: In general, stable isotopes have a neutron-proton ratio close to 1. Now let's examine the neutron-proton ratio and the odd-even rule for each of these isotopes. For \({ }^{35}\mathrm{Cl}\): Atomic number (Z) = 17 (odd) Mass number (A) = 35 (odd) Number of neutrons (N) = A - Z = 35 - 17 = 18 (even) Neutron-Proton ratio = \(N/Z \approx 1.06\) For \({ }^{37}\mathrm{Cl}\): Atomic number (Z) = 17 (odd) Mass number (A) = 37 (odd) Number of neutrons (N) = A - Z = 37 - 17 = 20 (even) Neutron-Proton ratio = \(N/Z \approx 1.18\) For \({ }^{36}\mathrm{Cl}\): Atomic number (Z) = 17 (odd) Mass number (A) = 36 (even) Number of neutrons (N) = A - Z = 36 - 17 = 19 (odd) Neutron-Proton ratio = \(N/Z \approx 1.12\) Comparing these isotopes: In both \({ }^{35}\mathrm{Cl}\) and \({ }^{37}\mathrm{Cl}\), the number of protons is odd, and the number of neutrons is even, which makes them more stable according to the odd-even rule. In contrast, \({ }^{36}\mathrm{Cl}\) has odd numbers of both protons and neutrons, making it less stable according to the odd-even rule. Regarding the neutron-proton ratio, both \({ }^{35}\mathrm{Cl}\) and \({ }^{37}\mathrm{Cl}\) have ratios closer to 1 than \({ }^{36}\mathrm{Cl}\), making them more stable. Based on the empirical rules about nuclear stability, the \({ }^{36}\mathrm{Cl}\) nucleus is less stable than either \({ }^{35}\mathrm{Cl}\) or \({ }^{37}\mathrm{Cl}\) due to the odd-even rule and the neutron-proton ratio.

Key Concepts

Radioactive Decay

Radioactive decay is a fascinating process that involves an unstable atomic nucleus losing energy by emitting radiation. This often results in the transformation of the original element into a different element. Many naturally occurring elements have isotopes that are radioactive. These isotopes undergo decay until a stable form is achieved.
Radioactive decay includes several types of radiation emissions: alpha particles, beta particles, and gamma radiation. Each has different properties and consequences for the decay process. Importantly:

• Alpha decay decreases the mass number and atomic number.
• Beta decay often increases the atomic number.
• Gamma decay typically does not alter the mass number or atomic number.

Understanding these differences is key to predicting the behavior of radioactive substances.

Beta Emission

Beta emission is a form of radioactive decay where a neutron in the nucleus transforms into a proton, emitting a beta particle (an electron) in the process. This conversion increases the atomic number of the element by one, though the mass number remains constant. For example, during the beta decay of element \({ }^{36}\mathrm{Cl} \), chlorine converts to \({ }^{36}_{18}\mathrm{Ar} \), argon.
Beta particles are relatively high energy and travel farther than alpha particles but are still less penetrating than gamma rays. They can penetrate the skin, so protection is important during exposure. The overall process in beta decay can be summarized as follows:

• The atomic nucleus emits an electron (beta particle).
• The atomic number increases by one.
• The mass number remains unchanged.

Through beta emission, radioactive elements adjust their neutron-to-proton ratio to achieve stability.

Neutron-Proton Ratio

The neutron-proton ratio is a critical factor that dictates the stability of an atomic nucleus. A stable nucleus typically has a balanced ratio of neutrons to protons, often close to one for lighter elements. This ratio becomes increasingly important for maintaining nuclear stability as elements get heavier.
The concept is crucial because:

• Isotopes with too few or too many neutrons compared to protons can be unstable.
• An isotope with a neutron-proton ratio significantly distant from one is often radioactive.

For example,
the isotope \(^{36}\mathrm{Cl} \) has a neutron-proton ratio of approximately 1.12, suggesting a less stable configuration compared to its more stable counterparts, \(^{35}\mathrm{Cl} \) and \(^{37}\mathrm{Cl} \), which have ratios closer to 1.

Odd-Even Rule

The odd-even rule is a helpful guideline within nuclear chemistry that aids in predicting the stability of an isotope based on its number of protons and neutrons. This rule posits that isotopes with even numbers of protons and neutrons are generally more stable than those with odd numbers.
This guideline stems from the pairing energy released when protons and neutrons form pairs within the nucleus:

• Nuclei with even protons and even neutrons tend to be very stable due to maximal pairing.
• Nuclei with odd protons and odd neutrons tend to be less stable.
• Mixed configurations, with one number even and the other odd, are more stable than both numbers being odd.

Hence, the instability of \({ } ^{36}\mathrm{Cl} \) can be explained by its odd-even composition, making it more inclined to decay towards a more stable state.