A free hydrogen atom after absorbing a photon of wavelength $\lambda_{_a}$ gets excited from the state n = 1 to the state n = 4. Immediately after that the electron jumps to n = m state by emitting a photon of wavelength $\lambda_{_e}.$ Let the change in momentum of atom due to the absorption and the emission are ?p and ?p , respectively. If $\lambda_{a}/\lambda_{e}=\frac{1}{5} $ . Which of the option(s) is/are correct ?
[Use hc = 1242 eV nm; 1 nm = 10 m, h and c are Planck's constant and speed of light, respectively]
- $\lambda_{_e}$ = 418 nm
- The ratio of kinetic energy of the electron in the state n = m to the state n = 1 is $\frac{1}{4}$
- m = 2
- $\Delta P_{a}/\Delta P_{e}=\frac{1}{2}$
The Correct Option is C
Solution and Explanation
$\frac{hc}{\lambda_{e}} = 13.6\left[\frac{1}{m^{2}}-\frac{1}{4^{2}}\right]\quad...\left(ii\right)$
(ii) / (i) , we get
$\frac{\lambda_{a}}{\lambda_{e}} = \frac{\left[\frac{1}{m^{2}}-\frac{1}{16}\right]}{\left[1-\frac{1}{16}\right]} = \frac{1}{5}$
$\Rightarrow\, \frac{1}{m^{2}}-\frac{1}{16} = \frac{15}{16}\times\frac{1}{5}$
$\Rightarrow\, \frac{1}{m^{2}}-\frac{1}{16} = \frac{3}{16}$
$\Rightarrow\, \frac{1}{m^{2}} = \frac{3}{16}+\frac{1}{16}$
$\Rightarrow\quad m = 2$
from (ii)
$\frac{hc}{\lambda_{e}} = 13.6\left[\frac{1}{2^{2}}-\frac{1}{4^{2}}\right] = 13.6\times\frac{3}{16}ev$
$\Rightarrow\quad\lambda_{e} = \frac{12400\times16}{13.6\times3}?$
$\Rightarrow\quad\lambda_{e} \approx 4862 ?$
we have $KE_{n} \,\,\propto \frac{z^{2}}{n^{2}}$
$\Rightarrow\, \frac{KE_{2}}{KE_{1}} = \frac{1}{4}$
$\Delta P_{a} = \frac{h}{\lambda_{a}}$
$\Delta P_{e} = \frac{h}{\lambda _{e}}$
$\Rightarrow\quad \frac{\Delta P_{a}}{\Delta P_{e}} = \frac{\lambda_{e}}{\lambda_{a}}$
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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.