Advertisements
Advertisements
Question
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.
Advertisements
Solution
Given:
Mass of the hydrogen atom, M = 1.67 × 10−27 kg
Let v be the velocity with which hydrogen atom is moving before collision.
Let v1 and v2 be the velocities of hydrogen atoms after the collision.
Energy used for the ionisation of one atom of hydrogen, ΔE = 13.6 eV = 13.6×(1.6×10−19)J
Applying the conservation of momentum, we get
mv = mv1 + mυ2 ...(1)
Applying the conservation of mechanical energy, we get
`1/2 mv^2=1/2mv_1^2+ 1/2mv_2^2+DeltaE//m` .......(2)
Using equation (1), we get
`V^2 = (V_1 + V_2)^2`
`V^2 = v_1^2 +v_2^2 + 2v_1v_2` ...(3)
In equation(2), on multiplying both the sides by 2 and dividing both the sides by m.
`v^2 = v_1^2 + v_1^2+ 2Delta E//m .......(4)`
On comparing (4) and (3), we get
`[therefore 2v_1v_2 = (2DeltaE)/m]`
`(U_1 - U_2)^2 = (U_1 + U_2)^2 -4U_1U_2`
`(v_1 - v_2)^2 = v^2 - (4DeltaE)/m`
For minimum value of υ,
`V_1 = V_2 = 0 `
`Also, V^2 - (4Delta)/m`
`∴ v^2 = (4DeltaE)/m`
= `(4xx13.6xx1.6xx10^-19)/(1.67xx10^-27)`
`v =sqrt(4xx13.6xx1.6xx10^-19)/(1.67xx10^-27)`
= `10^4 sqrt((4xx13.6xx1.6)/(1.67))`
= 7.2 × 104 m/s
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.
Which wavelengths will be emitted by a sample of atomic hydrogen gas (in ground state) if electrons of energy 12.2 eV collide with the atoms of the gas?
In which of the following transitions will the wavelength 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?
A hydrogen atom in ground state absorbs 10.2 eV of energy. The orbital angular momentum of the electron is increased by
An electron with kinetic energy 5 eV is incident on a hydrogen atom in its ground state. The collision
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
Ionization energy of a hydrogen-like ion A is greater than that of another hydrogen-like ion B. Let r, u, E and L represent the radius of the orbit, speed of the electron, energy of the atom and orbital angular momentum of the electron respectively. In ground state
Calculate the smallest wavelength of radiation that may be emitted by (a) hydrogen, (b) He+ and (c) Li++.
Find the binding energy of a hydrogen atom in the state n = 2.
(a) Find the first excitation potential of He+ ion. (b) Find the ionization potential of Li++ion.
Find the maximum Coulomb force that can act on the electron due to the nucleus in a hydrogen atom.
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?
A gas of hydrogen-like ions is prepared in a particular excited state A. It emits photons having wavelength equal to the wavelength of the first line of the Lyman series together with photons of five other wavelengths. Identify the gas and find the principal quantum number of the state A.
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.
Find the temperature at which the average thermal kinetic energy is equal to the energy needed to take a hydrogen atom from its ground state to n = 3 state. Hydrogen can now emit red light of wavelength 653.1 nm. Because of Maxwellian distribution of speeds, a hydrogen sample emits red light at temperatures much lower than that obtained from this problem. Assume that hydrogen molecules dissociate into atoms.
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?
In the Auger process an atom makes a transition to a lower state without emitting a photon. The excess energy is transferred to an outer electron which may be ejected by the atom. (This is called an Auger electron). Assuming the nucleus to be massive, calculate the kinetic energy of an n = 4 Auger electron emitted by Chromium by absorbing the energy from a n = 2 to n = 1 transition.
A hydrogen atom makes a transition from n = 5 to n = 1 orbit. The wavelength of photon emitted is λ. The wavelength of photon emitted when it makes a transition from n = 5 to n = 2 orbit is ______.
