Problem 12 - A Complete Resource Book in Chemistry for JEE Main

Question

The atomic number (A) and mass number (M) of the nuclide formed when three \(\alpha\) and \(2 \beta\) particles are emitted from \({ }_{92} \mathrm{U}^{238}\), is (a) \(\mathrm{A}=88, \mathrm{M}=235\) (b) \(\mathrm{A}=88, \mathrm{M}=226\) (c) \(\mathrm{A}=87, \mathrm{M}=233\) (d) \(A=86, M=226\)

Step-by-Step Solution

Answer

The resulting numbers are \( A = 88 \), \( M = 226 \), option (b).

Step 1: Understand Alpha and Beta Emission

An \( \alpha \) particle is composed of 2 protons and 2 neutrons, reducing the atomic number by 2 and the mass number by 4. A \( \beta \) particle emission increases the atomic number by 1, as it converts a neutron to a proton, but does not change the mass number.

Step 2: Calculate the Changes from Alpha Emission

For \(3\) \( \alpha \) particles, the changes are: \(3 \times 2 = 6\) protons and \(3 \times 4 = 12\) in the mass number. Therefore, the atomic number decreases by 6 and the mass number decreases by 12.

Step 3: Calculate the Changes from Beta Emission

For \(2\) \( \beta \) particles, the overall change in atomic number is \(2 \times 1 = 2\), because each \( \beta \) emission increases the atomic number by 1. The mass number remains unchanged as beta emissions do not affect it.

Step 4: Calculate the Resulting Atomic and Mass Numbers

Start with uranium: \( A = 92 \), \( M = 238 \). Subtract the impact of \(3 \alpha\) emissions, then add the impact of \(2 \beta\) emissions:\[\text{New Atomic Number} = 92 - 6 + 2 = 88\]\[\text{New Mass Number} = 238 - 12 = 226\]

Step 5: Verify Final Answer with Choices

The resulting atomic number is \(88\) and the mass number is \(226\), which corresponds to option (b).

Key Concepts

Alpha Emission

When an atom undergoes alpha emission, it releases an alpha particle from its nucleus. This alpha particle is composed of two protons and two neutrons, which means it's identical to a helium nucleus. As a result, the emission of an alpha particle reduces the atomic number of the original atom by 2, because two protons are lost, and also decreases the mass number by 4, since the total number of nucleons (protons + neutrons) is reduced by four.

• Alpha particles are relatively heavy and have a positive charge due to the two protons they carry.
• Because they are charge-heavy, they do not penetrate materials easily and can often be stopped by a sheet of paper or human skin.

Understanding these changes is crucial when trying to predict the resulting elements and isotopes formed after alpha emissions.

Beta Emission

Beta emission involves the transformation of a neutron into a proton, releasing a beta particle, which is essentially a high-speed electron. This process increases the atomic number of the element by 1, as the gain of a proton changes the element's identity. Unlike alpha emission, beta emission does not change the mass number of the atom, since the total count of nucleons remains the same.

• Beta particles are much lighter than alpha particles and carry a negative charge.
• They have greater penetration power and can pass through paper but are usually stopped by materials such as plastic or metal sheets.

This process is vital for understanding how an unstable nucleus can seek stability through nuclear reactions.

Atomic Number

The atomic number of an element is a fundamental property that defines the element. It represents the number of protons in the nucleus of an atom, which corresponds to the element's identity on the periodic table. Each element has a unique atomic number, and it determined the sequence of elements in the periodic table.

• A change in the atomic number alters the identity of the element.
• Alpha emissions decrease the atomic number, while beta emissions increase it.

This property is crucial when analyzing nuclear reactions, where emissions might change both the element and its isotopes.

Mass Number

The mass number of an atom is the total count of protons and neutrons in its nucleus. It fundamentally represents the mass of the atom since electrons contribute negligibly to an atom's mass. The mass number is denoted by the letter "M" in nuclear equations and gives us insight into the isotope of a given element.

• The mass number changes with alpha emissions (since they remove two protons and two neutrons), but it remains unchanged during beta emissions.
• It is always a whole number due to the discrete nature of protons and neutrons.

Knowing how to calculate mass numbers is essential when studying nuclear stability and isotopic variations following nuclear reactions.