Problem 20 - Master Resource Book in JEE Main Physics

Question

Electron-volt is the unit of energy \(\left(1 \mathrm{eV}=1.6 \times 10^{-19} \mathrm{~J}\right) .\) In \(\mathrm{H}\)-atom, the binding energy of electron in first orbit is \(13.6 \mathrm{eV}\). The same in joule (J) is (a) \(10 \times 10^{-19} \mathrm{~J}\) (b) \(21.76 \times 10^{-19} \mathrm{~J}\) (c) \(13.6 \times 10^{-19} \mathrm{~J}\) (d) None of these

Step-by-Step Solution

Answer

The same in joule is (b) \(21.76 \times 10^{-19} \mathrm{~J}\).

Step 1: Identify the Given Values

The binding energy of an electron in the first orbit of a hydrogen atom is given as \( 13.6 \) eV. The conversion factor given is \( 1 \text{ eV} = 1.6 \times 10^{-19} \text{ J} \).

Step 2: Use the Conversion Factor to Change Units

To convert the binding energy from electron volts to joules, we multiply the energy in eV by the conversion factor: \[ E(J) = 13.6 \text{ eV} \times 1.6 \times 10^{-19} \text{ J/eV} \]

Step 3: Perform the Multiplication

Calculate the multiplication: \[ E(J) = 13.6 \times 1.6 \times 10^{-19} = 21.76 \times 10^{-19} \text{ J} \]

Step 4: Identify the Correct Answer

Compare the calculated result with the given options to identify the correct choice. The energy \( 21.76 \times 10^{-19} \text{ J} \) matches option (b).

Key Concepts

Binding Energy

Binding energy refers to the energy that holds a system together. In the case of a hydrogen atom, this is the energy needed to remove an electron from its orbit around the nucleus. It is a measure of the strength of the bond between the electron and the atom. The larger the binding energy, the more energy is required to remove the electron. For a hydrogen atom, this energy is specifically measured when considering the electron in its first orbit. - In simpler terms, think of binding energy as the 'glue' that keeps the electron stuck in orbit around the nucleus. - In the hydrogen atom, the binding energy for the first orbit is 13.6 electron-volts (eV). Recognizing the amount of energy involved in atomic bonds helps us understand reactions in physics and chemistry. This critical value of 13.6 eV explains why the hydrogen atom remains stable under normal conditions. It highlights why specific reactions or interactions are needed to alter the hydrogen atom's structure.

Unit Conversion

Unit conversion is the process of changing a measurement from one unit to another. Here, we are concerned with converting energy from electron-volts (eV) to joules (J). Understanding unit conversions can bridge gaps between different scientific conventions, making it essential for interpreting measurements in various fields.- An electron-volt is defined as the energy gained by an electron when it moves through an electric potential difference of one volt. This unit is particularly common in atomic and particle physics.- One electron-volt (1 eV) is equivalent to \(1.6 \times 10^{-19}\) joules.To convert, we multiply the given energy in eV by the conversion factor \(1.6 \times 10^{-19}\) J/eV. For example, converting the 13.6 eV binding energy of a hydrogen atom produces:\[13.6 \text{ eV} \times 1.6 \times 10^{-19} \text{ J/eV} = 21.76 \times 10^{-19} \text{ J}\]This conversion helps express atomic level energy levels widely used in physics into the more universally understood unit of joules.

Hydrogen Atom

The hydrogen atom is the simplest and most fundamental type of atom, consisting of only one proton and one electron. Its simplicity makes it an ideal starting point for understanding atomic structure and behavior. Unlike other atoms, it lacks neutrons. - The electron revolves around the nucleus in defined orbits, and the energy levels of these orbits determine how the atom interacts with other particles. - The energy associated with an electron’s orbit is quantified by the binding energy; for the hydrogen atom's first orbit, this energy is 13.6 electron-volts. Understanding hydrogen's makeup helps us explore other atomic systems, as it sets the foundation for complex quantum mechanical systems. Moreover, hydrogen's ubiquitous presence in the universe - as a major component of stars and galaxies - makes its study fundamental to the field of astronomy as well.