Chapter 37 - The Foundations of Modern Physics - Exercises and Problems - Page 1123: 38

See the detailed answer below.

$$\color{blue}{\bf [a]}$$ To create the graph of charge-to-mass ratio versus proton number Z for nuclei with \( Z = 5, 10, 15, 20, ..., 90 \), we will use the formula \( q/m = Z/A \), where \( Z \) is the proton number and \( A \) is the average atomic mass. We'll use the average atomic masses from the periodic table to calculate the charge-to-mass ratio for each nucleus. Here are the proton numbers \( Z \) and corresponding average atomic masses \( A \) for the nuclei: \begin{array}{|c|c|c|} \hline Z & A &Z/A\\ \hline 1 & 1& 1\\ \hline 5 & 10.8 &\frac{5}{10.8 } \\ \hline 10 & 20.2&\frac{10 }{20.2} \\ \hline 15 & 31& \frac{15 }{31}\\ \hline 20 & 40.1&\frac{20 }{40.1} \\ \hline 25 &54.9 &\frac{25 }{54.9 } \\ \hline 30& 65.4&\frac{30}{65.4} \\ \hline 35& 79.9 & \frac{35}{79.9 }\\ \hline 40& 91.2 &\frac{40}{91.2 } \\ \hline 45& 102.9 & \frac{45}{102.9 }\\ \hline 50& 121.8 &\frac{50}{121.8 } \\ \hline 55& 132.9 &\frac{55}{132.9 } \\ \hline 60& 144.2&\frac{60}{144.2} \\ \hline 65& 158.9&\frac{65}{158.9} \\ \hline 70& 173&\frac{70}{173} \\ \hline 75&186.2 &\frac{75}{186.2 } \\ \hline 80& 200.6&\frac{80}{200.6} \\ \hline 85& 210 & \frac{85}{210 }\\ \hline 90& 232 &\frac{90}{232 } \\ \hline \end{array} Now we need to plot these dots in the graph of $Z/A$ as a function of $Z$, as we see in the figure below. $$\color{blue}{\bf [b]}$$ From the graph below, we can see that there is no significant trend in the charge-to-mass ratio as a function of proton number $Z$. The values of $Z/A$ appear to be between 0.4 and 0.5 for $1\lt Z\lt85$ while it drops to less than 0.4 at $z=90$ However, we might notice a slight decrease in $Z/A$ as $ Z $ increases towards larger values. $$\color{blue}{\bf [c]}$$ It is obvious that increasing $Z$ adds more positive charge to the nucleus, increasing $A$, (mass number), by adding more neutrons tends to increase the mass more than the charge, resulting in a slight decrease in the charge-to-mass ratio for heavier nuclei. Hence, $Z/ A$ decreases because the number of neutrons in the nuclei increases more rapidly than $Z$.