Question

If \(13.6 \mathrm{eV}\) energy is required to ionise the hydrogen atom the energy required to remove the electron form \(n=2\) state is (A) Zero (B) \(10.2 \mathrm{eV}\) (C) \(6.8 \mathrm{eV}\) (D) \(3.4 \mathrm{eV}\)

Short answer

The energy required to remove the electron from the n=2 state is \(10.2 \mathrm{eV}\).

Step-by-step solution

Calculate energy of n=2 state for hydrogen atom

We use the energy level formula for hydrogen atom: \[E_n = -\dfrac{13.6 \, \text{eV}}{n^2}\] Now substituting n=2 \[E_2 = -\dfrac{13.6 \, \text{eV}}{(2)^2} = -\dfrac{13.6 \, \text{eV}}{4} = -3.4\, \text{eV}\]

Calculate energy difference between n=1 and n=2 states

Since, the energy required to ionize the hydrogen atom (remove electron from n=1 state) is 13.6 eV, the energy of the ground state (n=1) is: \[E_1 = -13.6\, \text{eV}\] Now to find the energy required to remove the electron from the n=2 state, we calculate the energy difference between n=1 and n=2 states: \[\Delta E = E_2 - E_1 = (-3.4\, \text{eV}) - (-13.6\, \text{eV}) = 10.2\, \text{eV}\] So, the energy required to remove the electron from the n=2 state is 10.2 eV. The correct answer is (B) \(10.2 \mathrm{eV}\).

Key concepts

Energy Levels

Energy levels refer to the specific energies that electrons can occupy within an atom. In the hydrogen atom, these energy levels are quantized, meaning they can only take specific values. The lowest energy level, known as the ground state, is when the electron is closest to the nucleus (n=1). As the principal quantum number (n) increases, so does the size of the orbit, and the energy level becomes less negative.

• Ground state energy level: corresponds to n=1.
• Excited states: higher energy levels where n > 1.

For hydrogen, the energy levels can be calculated using the formula: \[E_n = -\dfrac{13.6 \, \text{eV}}{n^2}\]Where \(E_n\) is the energy of an electron at a certain level \(n\). This negative value indicates that the electron is bound to the nucleus.

Ionization Energy

Ionization energy is the energy required to remove an electron completely from an atom. For the hydrogen atom, the ionization energy is typically considered the energy needed to remove an electron from its ground state (n=1) which is 13.6 eV.

• Higher ionization energies correspond to more tightly bound electrons.
• The less negative the energy level, the less energy is required to remove the electron.

When considering higher energy levels like n=2, the electron is less tightly bound. Thus, the ionization energy is lower because the electron starts from a higher energy state with respect to becoming completely free.

Bohr Model

The Bohr model is a fundamental theory used to describe the hydrogen atom. Introduced by Niels Bohr in 1913, this model proposed that electrons revolve around the nucleus in fixed orbits called shells with quantized energies. These orbits or shells correspond to the energy levels.

• Electrons transition between energy levels by absorbing or emitting photons of specific energies.
• The model explains spectral lines emitted by hydrogen.

In this model, the allowed orbits are those for which the electron's angular momentum is an integer multiple of \(\hbar = \dfrac{h}{2\pi}\), where \(h\) is Planck's constant. This quantized nature of electron orbits supports the concept of definite energy levels in atoms.

Quantum States

Quantum states are descriptions of all possible states of a system according to quantum mechanics. Each quantum state of an electron in a hydrogen atom is uniquely described by a set of quantum numbers, with the principal quantum number \(n\) being one of them.

• The principal quantum number \(n\) determines the energy level.
• Other quantum numbers characterize shape and orientation of the orbit.

The electron's quantum state not only defines its energy but also its probability distribution around the nucleus. Quantum states help predict physical properties like energy transitions, making them critical in understanding phenomena such as atomic spectra.