Question

For lighter, stable isotopes, the ratio of the mass number to the atomic number is close to a certain value. What is the value? What happens to the value of the mass number to atomic number ratio as stable isotopes become heavier?

Short answer

For lighter, stable isotopes, the ratio of the mass number (A) to the atomic number (Z) is close to 1 or 2. As stable isotopes become heavier, this ratio increases, moving away from unity, because the number of neutrons in the nucleus increases more than the number of protons to maintain stability.

Step-by-step solution

Determine the ratio for lighter, stable isotopes

For lighter, stable isotopes, such as hydrogen, helium, lithium, and others, the ratio of the mass number (A) to the atomic number (Z) is close to 1. This means that their mass number is approximately equal to their atomic number. Hydrogen (Isotope H-1): \( A = 1, Z = 1, \frac{A}{Z} = \frac{1}{1} = 1 \) Helium (Isotope He-4): \( A = 4, Z = 2, \frac{A}{Z} = \frac{4}{2} = 2 \) Lithium (Isotope Li-6): \( A = 6, Z = 3, \frac{A}{Z} = \frac{6}{3} = 2 \) Although these ratios are not always equal to 1, they are quite close, and the ratio slightly increases as you move to the heavier isotopes. We can now analyze the trend for heavier isotopes.

Analyzing the trend for heavier, stable isotopes

As stable isotopes become heavier, the ratio of the mass number to the atomic number increases. This is because the number of neutrons in the nucleus increases more than the number of protons to maintain stability. Hence, the ratio of mass number to atomic number moves away from unity. For example, let's consider the ratios for the heavier isotopes of carbon and oxygen: Carbon (Isotope C-12): \( A = 12, Z = 6, \frac{A}{Z} = \frac{12}{6} = 2 \) Oxygen (Isotope O-16): \( A = 16, Z = 8, \frac{A}{Z} = \frac{16}{8} = 2 \) Iron (Isotope Fe-56): \( A = 56, Z = 26, \frac{A}{Z} = \frac{56}{26} \approx 2.15 \) Uranium (Isotope U-238): \( A = 238, Z = 92, \frac{A}{Z} = \frac{238}{92} \approx 2.59 \) As you can see from the examples, the ratio of mass number to atomic number increases as the isotopes become heavier. In essence, this means that heavier isotopes have considerably more neutrons than protons to maintain their stability.

Key concepts

Mass Number

The mass number of an atom is the total count of protons and neutrons present in its nucleus. These particles are known as nucleons. The mass number is often abbreviated as \( A \), where the formula is given by:

• Mass Number \( (A) = \text{Number of protons} + \text{Number of neutrons} \)

For example, in a helium atom (He-4), the mass number is 4 because it has 2 protons and 2 neutrons. The importance of the mass number is that it provides insight into an element's isotopes. Isotopes have the same number of protons but differ in the number of neutrons they contain, thus varying the mass number. Identifying the mass number helps understand the overall weight of the nucleus and differentiates isotopes of the same element.

Atomic Number

The atomic number of an element is the number of protons in the nucleus of an atom. It is denoted by the symbol \( Z \). This number is unique to each element and is fundamental in defining the type of element, because the chemical properties are primarily determined by the atomic number. For instance, all carbon atoms have an atomic number of 6, indicating 6 protons in each carbon nucleus.

• The atomic number also determines the element's position on the periodic table.
• As all neutral atoms have equal numbers of protons and electrons, the atomic number also tells us about the number of electrons, affecting chemical behavior.

In different isotopes of the same element, while the atomic number remains the same, the mass number varies. For example, in carbon isotopes like \( ^{12}C \) and \( ^{14}C \), the atomic number remains at 6, distinguishing them both as carbon.

Neutron-Proton Ratio

The neutron-proton ratio, also known as \( n/p \) ratio, is key to understanding nuclear stability. This ratio compares the number of neutrons to the number of protons in the nucleus of an atom. It is a significant factor in determining whether a nucleus is stable or prone to radioactive decay. For light isotopes, the \( n/p \) ratio is close to 1, meaning that the number of neutrons is nearly equal to the number of protons. This balance helps maintain nuclear stability. However, as elements become heavier, the need for additional neutrons increases to counteract the repulsive forces between increased numbers of protons. Thus, for heavier elements, this ratio is greater than 1:

• For elements like oxygen with 8 protons and 8 neutrons, the \( n/p \) ratio is \( \frac{8}{8} = 1 \).
• In heavier elements such as lead, the ratio can increase to greater than 1.5, such as in lead-208, where the ratio is approximately \( \frac{126}{82} = 1.54 \).

Nuclear Stability

The stability of an atomic nucleus depends on its configuration of protons and neutrons. Nuclear stability is a critical concept to understand why certain isotopes are stable while others are not. The nuclear force, which acts between protons and neutrons, is strong enough to overcome the electrostatic repulsion between protons. However, there is a delicate balance:

• Light nuclei are stable when the neutron-proton ratio is around 1.
• For heavier nuclei, more neutrons than protons are required to maintain stability, as they add to the nuclear force without contributing to electrostatic repulsion.

Isotopes that have a balance within this nuclear framework are termed stable isotopes. If there is too much deviation in the neutron-proton ratio, the nucleus becomes unstable, leading to radioactive decay processes such as alpha or beta decay. These processes help the nucleus reach a more stable state by altering the number of protons or neutrons, thus adjusting the neutron-proton ratio accordingly.