Advertisements
Advertisements
Question
Suppose, in certain conditions only those transitions are allowed to hydrogen atoms in which the principal quantum number n changes by 2. (a) Find the smallest wavelength emitted by hydrogen. (b) List the wavelength emitted by hydrogen in the visible range (380 nm to 780 nm).
Advertisements
Solution
Given:
Possible transitions:
From n1 = 1 to n2 = 3
n1 = 2 to n2 = 4
(a) Here, n1 = 1 and n2 = 3
Energy, `E = 13.6 (1/n_1^2- 1/n_2^2)`
`E = 13.6 (1/1 - 1/9)`
`= 13.6xx8/9 ....... (i)`
Energy (E) is also given by
`E = (hc)/lamda`
Here, h = Planck constant
c = Speed of the light
λ = Wavelength of the radiation
`therefore E = (6.63xx10^-34xx3xx10^2)/lamda` ........(i)
`"Equanting equation (1) and (2) , we have"`
`lamda = (6.63xx10^-34xx3xx10^8xx9)/(13.6xx8)`
= 0.027 × 10-7 = 103 nm
(b) Visible radiation comes in Balmer series.
As 'n' changes by 2, we consider n = 2 to n = 4.
`"Energy" , E_1 = 13.6 (1/n_1^2 - 1/n_2^2)`
= `13.6 xx (1/4 - 1/16)`
=2.55 eV
If `lamda_1` is the wavelength of the radiation, when transition takes place between quantum number n = 2 to n = 4, then
`255 = 1242/lamda_1`
or λ1 = 487 nm
APPEARS IN
RELATED QUESTIONS
A 12.5 eV electron beam is used to bombard gaseous hydrogen at room temperature. What series of wavelengths will be emitted?
Classically, an electron can be in any orbit around the nucleus of an atom. Then what determines the typical atomic size? Why is an atom not, say, a thousand times bigger than its typical size? The question had greatly puzzled Bohr before he arrived at his famous model of the atom that you have learnt in the text. To simulate what he might well have done before his discovery, let us play as follows with the basic constants of nature and see if we can get a quantity with the dimensions of length that is roughly equal to the known size of an atom (~ 10−10 m).
(a) Construct a quantity with the dimensions of length from the fundamental constants e, me, and c. Determine its numerical value.
(b) You will find that the length obtained in (a) is many orders of magnitude smaller than the atomic dimensions. Further, it involves c. But energies of atoms are mostly in non-relativistic domain where c is not expected to play any role. This is what may have suggested Bohr to discard c and look for ‘something else’ to get the right atomic size. Now, the Planck’s constant h had already made its appearance elsewhere. Bohr’s great insight lay in recognising that h, me, and e will yield the right atomic size. Construct a quantity with the dimension of length from h, me, and e and confirm that its numerical value has indeed the correct order of magnitude.
The first excited energy of a He+ ion is the same as the ground state energy of hydrogen. Is it always true that one of the energies of any hydrogen-like ion will be the same as the ground state energy of a hydrogen atom?
When white radiation is passed through a sample of hydrogen gas at room temperature, absorption lines are observed in Lyman series only. Explain.
The minimum orbital angular momentum of the electron in a hydrogen atom is
In which of the following transitions will the wavelength be minimum?
As one considers orbits with higher values of n in a hydrogen atom, the electric potential energy of the atom
The radius of the shortest orbit in a one-electron system is 18 pm. It may be
An electron with kinetic energy 5 eV is incident on a hydrogen atom in its ground state. The collision
Find the radius and energy of a He+ ion in the states (a) n = 1, (b) n = 4 and (c) n = 10.
A hydrogen atom in a state having a binding energy of 0.85 eV makes transition to a state with excitation energy 10.2 e.V (a) Identify the quantum numbers n of the upper and the lower energy states involved in the transition. (b) Find the wavelength of the emitted radiation.
A hydrogen atom in state n = 6 makes two successive transitions and reaches the ground state. In the first transition a photon of 1.13 eV is emitted. (a) Find the energy of the photon emitted in the second transition (b) What is the value of n in the intermediate state?
The average kinetic energy of molecules in a gas at temperature T is 1.5 kT. Find the temperature at which the average kinetic energy of the molecules of hydrogen equals the binding energy of its atoms. Will hydrogen remain in molecular from at this temperature? Take k = 8.62 × 10−5 eV K−1.
Average lifetime of a hydrogen atom excited to n = 2 state is 10−8 s. Find the number of revolutions made by the electron on the average before it jumps to the ground state.
A hydrogen atom in ground state absorbs a photon of ultraviolet radiation of wavelength 50 nm. Assuming that the entire photon energy is taken up by the electron with what kinetic energy will the electron be ejected?
A hydrogen atom moving at speed υ collides with another hydrogen atom kept at rest. Find the minimum value of υ for which one of the atoms may get ionized.
The mass of a hydrogen atom = 1.67 × 10−27 kg.
Consider an excited hydrogen atom in state n moving with a velocity υ(ν<<c). It emits a photon in the direction of its motion and changes its state to a lower state m. Apply momentum and energy conservation principles to calculate the frequency ν of the emitted radiation. Compare this with the frequency ν0 emitted if the atom were at rest.
The Balmer series for the H-atom can be observed ______.
- if we measure the frequencies of light emitted when an excited atom falls to the ground state.
- if we measure the frequencies of light emitted due to transitions between excited states and the first excited state.
- in any transition in a H-atom.
- as a sequence of frequencies with the higher frequencies getting closely packed.
Positronium is just like a H-atom with the proton replaced by the positively charged anti-particle of the electron (called the positron which is as massive as the electron). What would be the ground state energy of positronium?
