STATEMENT-1 : An electron beam from a discharge tube is incident on a sample of atomic H-gas, it is observed that the number of spectral lines emitted from gas increases with increase in potential difference of discharge tube. STATEMENT-2 : de Broglie wavelength of electron is inversely proportional to the square root of voltage applied across discharge tube : (1) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1. (2) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1. (3) Statement-1 is True, Statement-2 is False. (4) Statement-1 is False, Statement-2 is True.

When we talk about electron excitation, we are discussing how electrons absorb energy and move from a lower energy orbit to a higher energy orbit. This process is crucial in understanding atomic spectra. Consider an electron in a hydrogen atom that normally resides in the ground state. When an electron beam from a discharge tube collides with this atom, the energy transferred can excite the electron to a higher energy level. The excited electron doesn’t stay in an elevated state for long, it eventually returns to a lower energy level. As it does this, it releases energy in the form of light.

Each time the electron makes a transition back to a lower energy level, it emits a photon (a particle of light), which creates a spectral line. The important point here is that the number of possible transitions increases as the electron gets more energy and can reach higher energy states. This explains why increasing the potential difference in a discharge tube results in more spectral lines: more energy means higher possible energy levels and more lines in the emission spectrum.

A potential difference (or voltage) applied across a discharge tube plays a key role in electron excitation. The potential difference is essentially the energy supplied to the electrons. When we apply a higher voltage, we give the electrons more kinetic energy.

Think of it like pushing a child on a swing. The harder you push, the higher the child goes. Similarly, the higher the voltage, the more energy the electrons have to move to higher energy levels. In the context of the atomic spectra, this increased potential difference means that the electrons can excite to higher energy levels before emitting light as they return to lower levels. Hence, by increasing the voltage, you increase the number of possible energy transitions, leading to a greater number of spectral lines.

The de Broglie wavelength is a concept that describes the wave-like behavior of particles. Specifically for electrons, this wavelength is inversely proportional to the square root of the energy (or voltage) applied. The de Broglie wavelength \( \lambda \) is given by the formula \[ \lambda = \frac{h}\sqrt{2meV} \] where \( h \) is Planck's constant, \( m \) is the mass of the electron, and \( V \) is the voltage applied.

This formula shows that as the voltage increases, the wavelength decreases. A lower wavelength corresponds to higher energy electrons. Although this relationship shows how the electron’s wavelength changes with voltage, it doesn’t directly explain the increase in the number of spectral lines. That increase, as explained earlier, is due to electrons reaching higher energy levels, not directly their wavelength properties.