Question

demonstrate understanding

1. contrast the energy that can be released during a nuclear

fission reaction with the energy that can be released during a
nuclear fusion reaction.

1. contrast alpha particles, beta particles, and gamma rays.
2. explain why mass - energy equivalence is not apparent for

chemical reactions.
explain your thinking

1. explain why high temperatures are needed for fusion reac - tions to occur but not for fission reactions to occur.
2. math connection in a chain reaction, two additional

fissions occur for each nucleus that is split. if one nucleus is
split in the first step of the reaction, how many nuclei will have
been split after the fifth step?

Explanation:

Step1: Analyze the pattern

This is a geometric - sequence problem. The number of nuclei split in each step forms a geometric sequence where the first term \(a_1 = 1\) (the initial nucleus), and the common ratio \(r=2\) (since two additional fissions occur for each nucleus split). The formula for the sum of the first \(n\) terms of a geometric sequence is \(S_n=\frac{a_1(r^n - 1)}{r - 1}\).

Step2: Substitute the values into the formula

We want to find the sum \(S_5\) where \(a_1 = 1\), \(r = 2\), and \(n = 5\).
Substitute into the formula \(S_n=\frac{a_1(r^n - 1)}{r - 1}\), we get \(S_5=\frac{1\times(2^5-1)}{2 - 1}\).
First, calculate \(2^5=2\times2\times2\times2\times2 = 32\).
Then \(2^5-1=32 - 1=31\).
Since \(r - 1=2 - 1 = 1\), then \(S_5=\frac{1\times31}{1}=31\).

Answer:

31