Question

How many \(\alpha\) particles and how many \(\beta\) particles are emitted in the 11 -step decay of \(^{235} \mathrm{U}\) into \({ }^{207} \mathrm{~Pb}\) ?

Short answer

7 \(\alpha\) particles and 4 \(\beta\) particles are emitted.

Step-by-step solution

Understanding the Problem

We are given that the isotope \(^{235} \mathrm{U}\) transforms into \(^{207} \mathrm{Pb}\) through an 11-step decay process. Our task is to find out how many \(\alpha\) and \(\beta\) particles are emitted during this process.

Decay by Alpha Particle

An \(\alpha\) particle consists of 2 protons and 2 neutrons, effectively lowering the atomic mass by 4 and the atomic number by 2. When \(^{235} \mathrm{U}\) decays to \(^{207}\mathrm{Pb}\), the atomic mass decreases by: \[235 - 207 = 28\]Since each \(\alpha\) particle reduces the mass by 4, the number of \(\alpha\) particles emitted is: \[\frac{28}{4} = 7\]

Calculating the Change in Atomic Number

For each \(\alpha\) particle emitted, the atomic number decreases by 2. Therefore, with 7 \(\alpha\) particles:\[7 \times 2 = 14\]Thus, the atomic number decrease due to \(\alpha\) emission is 14.

Finding the Number of Beta Particles

\(\beta\) particles convert a neutron to a proton, increasing the atomic number by 1 each time. The initial atomic number for \(^{235} \mathrm{U}\) is 92, and final atomic number for \(^{207} \mathrm{Pb}\) is 82.The total decrease in atomic number should be:\[92 - 82 = 10\]Given 7 \(\alpha\) particles decrease the atomic number by 14, to achieve a net decrease of 10, 4 \(\beta\) particles must increase it by 4:\[-14 + x = -10\] -> solving for \(x\),\[x = 4 \]

Key concepts

Alpha Decay

Alpha decay is one of the most common types of radioactive decay. In alpha decay, an unstable nucleus releases an alpha particle, which is composed of 2 protons and 2 neutrons. This release causes the nucleus to lose four units of atomic mass and two units of atomic number.
For instance, when uranium-235 ( \(^{235} U \)) undergoes alpha decay, it loses these components in each decay step.

• This reduces its atomic mass from 235 by 4 units per alpha particle emitted.
• As a result, the atomic number decreases, changing the element's identity.

This process repeats until the uranium transforms into a different element entirely, like lead-207 ( \(^{207} Pb \)). Calculating the total change in mass and atomic number helps determine the number of alpha particles emitted. In the exercise example, seven alpha particles were released, lowering the mass by 28 units and the atomic number by 14.

Beta Decay

Unlike alpha decay, beta decay involves the transformation of a neutron into a proton, with the emission of a beta particle (an electron or positron). This increases the atomic number by one, as a proton replaces a neutron, while the atomic mass remains nearly unchanged.
Beta decay is notable because:

• The atomic number increases, resulting in a new element with the same atomic mass.
• The process is crucial for achieving the atomic composition observed in daughter nuclei.

In our exercise, while uranium-235 transforms into lead-207, it is also necessary to account for these beta changes. After calculating the effect of alpha particles, beta decays adjusted the atomic number. Four beta particles increased the atomic number enough to achieve the net decrease needed.

Radioactive Particles

Radioactive particles, such as alpha and beta particles, are emitted during nuclear decay, altering the nucleus as different elements form. Understanding these particles is crucial when studying the decay of radioactive isotopes like uranium-235.
Alpha particles

• Are quite heavy and positively charged.
• They don't penetrate materials very well, making them less harmful externally but dangerous if ingested or inhaled.

Beta particles

• Are much lighter and can penetrate deeper, passing through the skin but generally not through bones.
• They pose a more significant external radiation risk than alpha particles.

When examining nuclear decay processes, like the transformation of uranium to lead, these particles indicate the type and number of changes occurring in the nucleus. Their emission helps scientists track and predict nuclear reactions, vital for applications in medicine and energy.