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Chapter 2: Problem 21

If the mass number of an isotope is more than twice the atomic number, is the neutron-to-proton ratio less than, greater than, or equal to \(1 ?\)

Short Answer

Expert verified
Short Answer: The neutron-to-proton ratio for an isotope with a mass number more than twice its atomic number is greater than 1.

Step by step solution

01

Express the given condition as an inequality

The problem states that the mass number (A) of an isotope is more than twice the atomic number (Z). Mathematically, we can represent this condition as an inequality: \(A > 2Z\)
02

Understanding mass number and atomic number

Recall that: - The mass number (A) is the sum of the number of protons and neutrons. - The atomic number (Z) is the number of protons. - The number of neutrons (N) can be found by subtracting the number of protons from the mass number, i.e., \(N = A - Z\).
03

Find the neutron-to-proton ratio

We are interested in the neutron-to-proton ratio, which can be represented as \(\frac{N}{Z}\). From step 2, we know that \(N = A - Z\). Now, substitute this expression into the neutron-to-proton ratio: \(\frac{N}{Z} = \frac{A - Z}{Z}\)
04

Use the inequality to determine the neutron-to-proton ratio

We know that \(A > 2Z\). Let's express \(A\) in terms of \(Z\) using this inequality: \(A > 2Z \Rightarrow A - Z > Z\) Now, let's compare this inequality with the expression for the neutron-to-proton ratio: \(\frac{N}{Z} = \frac{A - Z}{Z}\) Since \(A - Z > Z\), it follows that \(\frac{A - Z}{Z} > \frac{Z}{Z}\), which gives us: \(\frac{N}{Z} > 1\) So, the neutron-to-proton ratio is greater than 1.

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