In a hydrogen like atom, when an electron jumps from the $M$ - shell to the $L$ - shell, the wavelength of emitted radiation is $\lambda$. If an electron jumps from $N$-shell to the $L$-shell, the wavelength of emitted radiation will be :
Show Hint
- $\frac{27}{20} \lambda $
- $\frac{16}{25} \lambda $
- $\frac{20}{27} \lambda $
- $\frac{25}{16} \lambda $
The Correct Option is C
Approach Solution - 1
$\frac{1}{\lambda} =K \left(\frac{1}{2^{2}} - \frac{1}{3^{2}}\right) = \frac{K\times5}{36} $
for $N\to L$
$ \frac{1}{\lambda'} =K \left(\frac{1}{2^{2}} - \frac{1}{4^{2}}\right) = \frac{K\times3}{16} $
$ \lambda' = \frac{20}{27} \lambda $
Approach Solution -2
- Work with energy levels instead of the wavelength formula directly. Since photon energy and wavelength are inversely related, comparing energy gaps gives the same ratio without writing the Rydberg formula twice.
Step 1: Write the energy of each shell using $E_n \propto -\dfrac{1}{n^2}$.
Drop the common constant for now and just track the shell number: $E_2 \propto -\dfrac14$, $E_3 \propto -\dfrac19$, $E_4 \propto -\dfrac{1}{16}$.
Step 2: Find the energy released for $M \to L$ (shell 3 to shell 2).
$|\Delta E_1| = \left|-\dfrac14 - \left(-\dfrac19\right)\right| = \dfrac14-\dfrac19 = \dfrac{9-4}{36} = \dfrac{5}{36}$
Step 3: Find the energy released for $N \to L$ (shell 4 to shell 2).
$|\Delta E_2| = \left|-\dfrac14 - \left(-\dfrac{1}{16}\right)\right| = \dfrac14-\dfrac{1}{16} = \dfrac{4-1}{16} = \dfrac{3}{16}$
Step 4: Use $\lambda \propto \dfrac{1}{E}$ to find the wavelength ratio.
Since energy and wavelength are inversely proportional, the wavelength ratio is the FLIP of the energy ratio:
$\dfrac{\lambda_N}{\lambda} = \dfrac{|\Delta E_1|}{|\Delta E_2|} = \dfrac{5/36}{3/16} = \dfrac{5}{36}\times\dfrac{16}{3} = \dfrac{80}{108} = \dfrac{20}{27}$
Final Answer: $\lambda_N = \dfrac{20}{27}\lambda$
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Concepts Used:
Atoms
- The smallest unit of matter indivisible by chemical means is known as an atom.
- The fundamental building block of a chemical element.
- The smallest possible unit of an element that still has all the chemical properties of that element.
- An atom is consisting of a nucleus surrounded by one or more shells of electrons.
- Word origin: from the Greek word atomos, which means uncuttable, something that cannot be divided further.
All matter we encounter in everyday life consists of smallest units called atoms – the air we breath consists of a wildly careening crowd of little groups of atoms, my computer’s keyboard of a tangle of atom chains, the metal surface it rests on is a crystal lattice of atoms. All the variety of matter consists of less than hundred species of atoms (in other words: less than a hundred different chemical elements).
Every atom consists of an nucleus surrounded by a cloud of electrons. Nearly all of the atom’s mass is concentrated in its nucleus, while the structure of the electron cloud determines how the atom can bind to other atoms (in other words: its chemical properties). Every chemical element can be defined via a characteristic number of protons in its nucleus. Atoms that have lost some of their usual number of electrons are called ions. Atoms are extremely small (typical diameters are in the region of tenths of a billionth of a metre = 10-10 metres), and to describe their properties and behaviour, one has to resort to quantum theory.