Question

The wavelength of the first line of Lyman series in hydrogen atom is $1216{A^0} $. The wavelength of the first line of Lyman series for $10$ times ionised sodium atom will be:
A) $0.1{A^0}$
B) $1000{A^0}$
C) $100{A^0}$
D) $10{A^0}$

Answer 1

Hint: The quantum is known as a photon as electromagnetic radiation. The phrase quantum means a photon, which is called the smallest primary unit, and a quantity of electromagnetic energy. Use the equation for photon energy involving the atomic number of the atom.

Complete step by step solution:
A photon is the quantum of electromagnetic radiation. The word quantum is the smallest elemental unit of a number, or the smallest distinct sum of anything. Thus, one quantum of electromagnetic radiation is called a photon. The plural of quantum is quanta. Quantum mechanics and quantum theory give rise to the idea of photons and quanta. Quantum mechanics is an atomic and subatomic-level statistical model explaining particles' behaviour. It indicates that matter and energy are quantized or come in the smallest possible size in thin, discrete packets. At light speed, a photon spreads.

A photon represents an electric wave's particle properties instead of the whole wave. In other words, an electromagnetic wave may be represented as consisting of individual particles known as photons. All interpretations of electromagnetic waves are right and reciprocal. In conditions of refraction or interference, light experiences wave properties.
Using the equation for photon energy, we get,
$\dfrac {{hc}} {\lambda} = \dfrac {{- 13.6 \times {Z^2}}} {n} $
$h$ is Planck’s constant
$c$ is speed of light
$\lambda $ is wavelength
$Z$ is atomic number.
$n = 1$ and $Z = 1$
$\lambda = \dfrac{{hc}} {{- 13.6}} $
$ \Rightarrow \lambda = 1216{A^0} $
Now, wavelength for ten times ionised sodium atom, the ${10^ {th}} $ ionised sodium atom is taken from $n = 1$and $Z = 11$.
$\dfrac {{hc}} {\lambda} = - 13.6 \times {11^2} $
$ \Rightarrow \lambda = \dfrac{{hc}} {{- 13.6 \times {{11} ^2}}} $
$ \Rightarrow \lambda = 10{A^0} $
The wavelength of the first line of Lyman series for $10$ times ionised sodium atom will be $10{A^0} $.

Hence the correct option is D.

Note: The energy of the photon is the energy that a photon carries. The energy is directly proportional to the electric frequency of the photon and thus equivalently to the wavelength. The further wavelengths the photon, the more energy it generates. The longer the wavelength of the photon, the lower its energy. Using any power unit, photon energy may be expressed. The electron volt and the joule are among the units used to characterise photon energy.