Advertisements
Advertisements
Question
In which of the following transitions will the wavelength be minimum?
Options
n = 5 to n = 4
n = 4 to n = 3
n = 3 to n = 2
n = 2 to n = 1
Solution
n = 2 to n = 1
For the transition in the hydrogen-like atom, the wavelength of the emitted radiation is calculated by
`1/lamda = RZ^2 (1/n^1 - 1/n^2)`
Here, R is the Rydberg constant.
For the transition from n = 5 to n = 4, the wavelength is given by
`1/lamda = RZ^2 (1/4^2 - 1/5^2)`
`lamda = 400/(9RZ^2)`
For the transition from n = 4 to n = 3, the wavelength is given by
`1/lamda = RZ^2 (1/3^2 - 1/4^2)`
`lamda = (144)/ (7RZ^2)`
For the transition from n = 3 to n = 2, the wavelength is given by
`1/lamda = RZ^2 (1/2^2 - 1/3^2 )`
`lamda = (36)/(5RZ^2)`
For the transition from n = 2 to n = 1, the wavelength is given by
`1/lamda = RZ^2 (1/2^2 - 1/3^2)`
`lamda = 2/(RZ^2)`
From the above calculations, it can be observed that the wavelength of the radiation emitted for the transition from n = 2 to n = 1 will be minimum.
APPEARS IN
RELATED QUESTIONS
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.
If Bohr’s quantisation postulate (angular momentum = nh/2π) is a basic law of nature, it should be equally valid for the case of planetary motion also. Why then do we never speak of quantisation of orbits of planets around the sun?
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.
In which of the following systems will the radius of the first orbit (n = 1) be minimum?
Which of the following curves may represent the speed of the electron in a hydrogen atom as a function of trincipal quantum number n?
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
Which of the following products in a hydrogen atom are independent of the principal quantum number n? The symbols have their usual meanings.
(a) vn
(b) Er
(c) En
(d) vr
A group of hydrogen atoms are prepared in n = 4 states. List the wavelength that are emitted as the atoms make transitions and return to n = 2 states.
Find the maximum Coulomb force that can act on the electron due to the nucleus in a hydrogen atom.
Whenever a photon is emitted by hydrogen in Balmer series, it is followed by another photon in Lyman series. What wavelength does this latter photon correspond to?
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?
What is the energy of a hydrogen atom in the first excited state if the potential energy is taken to be zero in the ground 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.
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?
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.
