Atoms

(a) The correct alternative is: no different from

Explanation: The size of the atom in the thomson’s model and the size of the atom in Rutherford’s model of an atom were of same magnitude.

(b) The correct alternative is: Thomson’s model, Rutherford’s model

Explanation: This is because when the electrons are in the ground state in case of Thomson’s model, they are in stable equilibrium whereas in case of Rutherford’s model of an atom, when electrons are at ground state, they experience a net force.

(c) The correct alternative is: Rutherford’s model

Explanation: A classical atom based on Rutherford’s model is doomed to collapse.

(d) The correct alternative is: Thomson’s model, Rutherford’s model

Explanation: The mass distribution of an atom is nearly continous in case of Thomson’s model of an atom, whereas in case of Rutherford’s model of an atom, the mass distribution is not uniform or continuous.

(e) The correct alternative is: both the models.

Explanation: In both Rutherford and Thomson’s model of an atom, the most of the mass is due to the positive part of the atom i.e. protons.

In alpha-particle scattering experiment, if we use a thin sheet of solid hydrogen in place of the gold foil, then the mass of Hydrogen will be lesser than that of incident alpha particle and the scattering angle will be not large enough. As mass of the scattering particle is more than the target nucleus, the alpha particles will not bounce while using solid Hydrogen.

The shortest wavelength present in the Paschen series of spectral lines is given for n₁ = 3 and n₂ = ∞

The diagram is shown below:

Rydberg’s formula is given as:

where

Z is atomic number

R is Rydberg’s constant, R = 1.9074 x 10⁷ m-1

h is planck’s constant and is equal to 6.64 × 10^-34 Js

c is speed of light and is equal to 3.0 × 10⁸ m/s

substituting the values we get,

⇒

⇒

⇒

⇒

⇒ λ = 8.2012 × 10^-7 m ~ 820.12 × 10^-9 m ~

Or λ = 820 nm

(∵ 10^-9 m = 1nm)

Given: energy = 2.3 eV

In joules, Energy E = 2.3 × 1.6 × 10^-19 = 3.68 × 10^-19J

Energy is given as:

E = hv

Where h is Planck’s constant

v is frequency of the radiation

Or

On calculating, we get

Frequency v =0.55 × 10^-15

Or Frequency v = 5.55 × 10^-14 Hz.

Given: Total energy of the Hydrogen atom = Ground state energy of hydrogen atom is –13.6 eV

Since, the kinetic energy is equal to the negative of the total energy

∴ Kinetic energy of the hydrogen atom = - (-13.6 eV) = 13.6 eV

The potential energy is equal to the twice of the negative of the total energy.

∴ Potential energy of hydrogen atom =-2 × - (-13.6 eV)

Potential energy = -27.2 eV.

For ground level: n₁ =4

Energy for the ground level of hydrogen atom, E₁ = -13.6/n₁²

Substituting the value, we get

E₁ = -13.6/1² = -13.6 eV

When the hydrogen atom is excited to n₂= 4, then the energy absorbed by the photon is given as:

E₂ = -13.6/4²

E₂= -13.6/16

The energy absorbed by the photon can be calculated by the following:

E = E₂- E₁

Or, E = (-13.6/16)- (-13.6/1)

On calculating, we get

E = 12.75 eV

In Joules, Energy is = 12.75 × 1.6 × 10^-19 Joules

Or E = 2.04 × 10^-18 J

Energy is given by the relation, E =hc/λ

Or λ = hc/E

Where h is Planck’s constant and is equal to 6.64 × 10^-34 Js

c is speed of light and is equal to 3.0 × 10⁸ m/s

substituting the values we get

On calculating, we get

λ = 9.7 × 10^-8 m

or λ = 97 nm

Frequency can be calculated as follows:

v = c/λ

substituting the values, we get

v= 3.1 × 10¹⁵ Hz

The orbital speed of an electron in a hydrogen atom, is given by the relation

v₁ =

or v₁ =

where ε₀ is the absolute permittivity of free space and is given as = 8.854 × 10^-12 N^-1C²m^-2

h is Planck’s constant and is given as 6.64 × 10^-34 Js

substituting the values, we get

v₁ =

On calculating, we get

v₁ = 0.0218 × 10⁸ m/s

Or v₁ = 2.18 × 10⁶ m/s

For n = 2

or v₂ =

substituting the values, we get

On calculating, we get

v₂ = 1.09 × 10⁶ m/s

Similarly, for n = 3, we get

or v₃ =

substituting the values, we get

On calculating, we get

v₃ = 7.27 × 10⁶ m/s

Orbital period can be calculated as:

T₁ = (2πr₁)/v₁

Where r₁is the radius of the orbit and is given as follows:

Where ε₀ is the absolute permittivity of free space and is given as = 8.854 × 10^-12 N^-1C²m^-2

h is Planck’s constant and is given as 6.64 × 10^-34 Js

e is charge on an electron and is equals to 1.6 × 10^-19C

m is the mass of an electron, and is equals to 9.1 × 10^-31 Kg

substituting the values, we get

⇒

On calculating, we get

T₁ = 15.27 × 10^-16 seconds

For n = 2, we get

⇒

On calculating, we get

T₂ = 1.22 × 10^-15 seconds

For n = 3, we get

⇒

On calculating, we get

T₃ = 4.12 × 10^-15 seconds.

Given, radius of innermost orbit, r₁ = 5.3 × 10^–11 m

Then, let r₂ be the radius of orbit at n = 2.

Then, r₂ = (n)²r₁

As shown in the diagram:

⇒ r₂ = 2² × 5.3 × 10^–11 m

⇒ r₂ = 2.12 × 10^–10 m

For n = 3, we have,

⇒ r₃ = (n)² r₁

⇒ r₃ = 3² × 5.3 × 10^–11 m

⇒ r₃ = 4.77 × 10^–10 m

Hence, the radii of an electron for n = 2 and n=3 orbits are 2.12 × 10^–10 m and 4.77 × 10^–10 m

Energy of the electron used for bombardment of gaseous hydrogen at room temperature = 12.5 eV.

Energy of gaseous hydrogen at ground state = -13.6 eV

When gaseous hydrogen is bombarded with electron beam, then energy of hydrogen will be =

(-13.6 + 12.5) eV= -1.1 eV

Orbital energy is given by the relation E= -13.6/n²

For n = 3, orbital energy will be E= -13.6/3²

E = - 1.5eV

Since the energy is equal to the energy of gaseous hydrogen. Therefore, the electron has made it transition from n= 1 to n= 3

During deexcitation of the hydrogen gas, the electron will return back from n =3 to n= 1 and forms a line of Lyman series of the hydrogen spectrum.

The wave number of the lyman series is given by the relation:

1/λ = R_y (1/1² – 1/n²)

Where R_y is the Rydberg’s constant and is equals to 1.097 × 10⁷m^-1

For n =3, substituting the values, we get

1/λ = 1.097 × 10⁷ (m^-1)[(1 – 1/3²)]

On calculating, we get

λ = 102.55 nm

When electron makes transition from n = 2 to n =1 level

1/λ = 1.097 × 10⁷ (m^-1)[(1 – 1/2²)]

On calculating, we get

λ = 121.54 nm

When the electron makes transition from n=3 to n = 2

1/λ = 1.097 × 10⁷ (m^-1)[(1/4 – 1/3²)]

On calculating, we get

λ = 656.33 nm

∴ In Lyman series, two wavelengths 102.55 nm and 121.54 nm were emitted, and in Balmer series, one wavelength 656.33 nm is emitted.

Angular momentum is given by the relation:

⇒

Where M is the mass of Earth and is equal to 6.0 × 10²⁴ kg.

v is the orbital speed of Earth = 3 × 10⁴ m/s.

r is the orbital radius of Earth= 1.5 × 10¹¹ m

n is the quantum number

substituting the values, we get

on calculating, we get

n =2.6 × 10⁷⁴

(a) The average angle of deflection of α-particles by a thin gold foil predicted by Thomson’s model about the same as that predicted by Rutherford’s model. This is due to the reason that, in both Thomson and Rutherford model of an atom, the average angle of deflection was taken.

(b) The probability of backward scattering (i.e., scattering of α-particles at angles greater than 90°) predicted by Thomson’s model much less than that predicted by Rutherford’s model.

(c) With increase in the thickness of, the number of target atoms that will increase. This will lead to increase in the collisions, and as we know, scattering is due to single collision. Therefore, with increase in thickness, the chances of collisions increase.

The radius of first Bohr orbit is given by the relation:

……………(1)

Where ε₀ is the absolute permittivity of free space and is given as = 8.854 × 10^-12 N^-1C²m^-2

h is Planck’s constant and is given as 6.64 × 10^-34 Js

e is charge on an electron and is equals to 1.6 × 10^-19C

m_e is the mass of an electron, and is equals to 9.1 × 10^-31 Kg

m_p is the mass of proton and is equals to 1.67 × 10^-27 Kg

The forces of coulomb attraction between an electron and proton is given by the relation:

Gravitational forces of attraction between an electron and proton is given by the relation

Where G is the Gravitational constant and is equal to G = 6.67 × 10^-11 N m² Kg²

When coulomb force and Gravitational forces of attraction are equal

Or
…………(1)

Substituting the values of equation (1) in equation (2), we get

Substituting the values, we get

On calculating, we get

∴ r₁= 1.21 × 10²⁹ m

Since Universe is 150billion light years wide, which concludes that the radius of first Bohr orbit is much greater than the size of the universe.

The energy of radiation at level n is given by the relation:

…(i)

Where v₁ is the frequency of radiation at level n

H is Planck’s constant

m is mass of hydrogen atom

e is charge on electron

ε₀ is permittivity of free space

The energy of radiation at level (n-1) is given by the relation:

……………(ii)

Where v₂ is the frequency of radiation at level n-1

Due to deexcitation of the electrons, the energy released is:

E =E₂- E₁

hv = E₂- E₁…………………..(iii)

Putting equation (i) and (ii) in equation (iii), we get

v =

v =

For large values of n, (n-1)≈n

v =

Frequency of revolution of an electron is given by

V_c = v/2πr……………..(vi)

Velocity of electron in nth orbit is given by the following relation

…………………….(iv)

The radius of nth orbit is given by the following relation

r =………………….(v)

Substituting the values of equation (iv) and (v) in equation (vi), we get

(a) Given: charge on an electron and is equals to 1.6 × 10^-19C

m_e is the mass of an electron, and is equals to 9.1 × 10^-31 Kg

c is speed of light and is equal to 3.0 × 10⁸ m/s

The quantity involving the above quantities is as follows:

1/4πε₀ = 9 × 10⁹Nm²C^-2

Substituting the values we get,

On calculating, we get

= 2.81 × 10^-15m

Therefore, the numerical quantity is much smaller than the size of the atom.

(b) Given: charge on an electron and is equals to 1.6 × 10^-19C

m_e is the mass of an electron, and is equals to 9.1 × 10^-31 Kg

Planck’s constant = 6.62 × 10^-34Js

The quantity involving the above quantities is as follows:

1/4πε₀ = 9 × 10⁹Nm²C^-2

Substituting the values we get

On calculating, we get

= 2.81 × 10^-15m

The numerical quantity is of order of the atomic size.

(a) Given: The total energy of an electron in the first excited state of the hydrogen atom is about –3.4 eV.

Kinetic energy of the electron in this state = negative of the total energy = -E

Kinetic energy of the electron in this state =-(-3.4)eV = + 3.4 eV

(b) Potential energy is given as the negative of the twice of the kinetic energy

U = -2 (3.4) eV

U = -6.8eV

(c) If the choice of the zero of potential energy is changed, then the value of potential energy of the system also changes and as we know the total energy is sum of kinetic energy as well as potential energy. Therefore, the potential energy will also changes.

The angular momentum associated with planetary motion is large relative to the Planck’s constant. As the angular momentum of Earth in its orbit is of order of 1070 h which results in higher value of quantum level of order of 1070. The more is the value of n, the less will be the angular momentum and energies. Therefore, the quantum levels of planetary motion are considered as continuous.

Given: Mass of negatively charged muon mμ^– = 207m_e

The Bohr’s radius is given by the relation:

r_e ∝ (1/m_c)

Energy of ground state electronic hydrogen atom E_e∝ m

The value of the first Bohr orbit is given as r_e = 0.53A

Converting into metres, r_e =0.53 × 10^-10m

Let us consider r_μ is the radius of muonic hydrogen atom

At equilibrium

m_μ r_μ = m_e r_e

207m_e × r_μ = m_e r_e

r_μ = (0.53 × 10^-10m)/207

r_μ =2.56 × 10^-13m

The ratio of the energies is given as:

E_e/E_μ = m_e/m_μ = m_e/207m_e

We get

E_μ = 207E_E

Substituting the value of energy, we get

E_μ = 207 × (-13.6)

E_μ =-2.81 keV