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 to the atomic number (\(R = \frac{A}{Z}\)) is close to 2. As stable isotopes become heavier, the ratio increases at a slower pace due to the higher number of neutrons required to maintain the stability of the nucleus, balancing the repulsive forces between protons.

Step-by-step solution

Define Terms

The first thing we need to do is define our terms. The mass number, represented as \(A\), is the total number of protons and neutrons in the nucleus of an isotope. The atomic number, represented as \(Z\), is the number of protons in the nucleus of an isotope. The ratio we are interested in is given by: \[R = \frac{A}{Z}\]

Determine the Ratio for Lighter Stable Isotopes

The most common lighter, stable isotopes include hydrogen (\(^1_1H\)), helium (\(^4_2He\)), lithium (\(^7_3Li\)), and beryllium (\(^9_4Be\)). Let's calculate the ratio, R, for each of these isotopes: For hydrogen: \[R_H = \frac{1}{1} = 1\] For helium: \[R_{He} = \frac{4}{2} = 2\] For lithium: \[R_{Li} = \frac{7}{3} \approx 2.33\] For beryllium: \[R_{Be} = \frac{9}{4} \approx 2.25\] It appears that the ratio for lighter, stable isotopes is close to 2.

Discuss the Trend as Isotopes Become Heavier

As the isotopes become heavier, the ratio R is expected to deviate from the previously determined value of 2. This is mainly due to the increase in the number of neutrons relative to protons in the nucleus. The presence of more neutrons provides a balance to the repulsive forces between protons, thereby making heavier nuclei more stable. As we progress through the periodic table, the necessity to maintain greater neutron counts to maintain stability increases. Therefore, the ratio R also increases, but at a slower pace. This reflects the need for a higher number of neutrons relative to protons in the nucleus of isotopes with a larger atomic number.