A rectangular corral of widths L x = L and L y = 2 L , contains seven electrons. Assume that the electrons do not interact with one another and do not neglect spin. What multiple of h 2 8 mL 2 gives the energy of ( a ) the first excited state, ( b ) the second excited state, and ( c ) the third excited state of the system of seven electrons? ( d ) Construct an energy-level diagram for the lowest four energy levels.

(a) We find that the lowest five levels of the rectangular corral (with this specific "aspect ratio") have energies E 1 , 1 = 1 . 25 , E 1 , 2 = 2 . 00 , E 1 , 3 = 3 . 25 , E 2 , 1 = 4 . 25 , E 2 , 2 = 5 . 00 . (all of these understood to be in "units" of h 2 8 mL 2 ). It should be noted that the energy level we denote E 2 , 2 actually corresponds to two energy levels. ( E 2 , 2 & E 1 , 4 ). They are degenerate, but that will not affect our calculations in this problem. The configuration that provides the lowest system energy higher than that of the ground state has the first three levels filled, the fourth one empty, and the fifth one half-filled: E first excited = 2 E 1 , 1 + 2 E 1 , 2 + 2 E 1 , 3 + E 2 , 2 = 2 1 . 25 + 2 2 . 00 + 2 3 . 25 + 5 . 00 , which means (putting the "unit" factor back in) the energy of the first excited state is E first excited = 18 . 00 h 2 8 mL 2 . Thus, the multiple of h 2 8 mL 2 is 18 . 00 . (b) The configuration that provides the next higher system energy has the first two levels filled, the third one half-filled, and the fourth one filled: E second excited = 2 E 1 , 1 + 2 E 1 , 2 + E 1 , 3 + 2 E 2 , 1 = 2 1 . 25 + 2 2 . 00 + 3 . 25 + 2 4 . 25 , which means (putting the "unit" factor back in) the energy of the second excited state is E second excited = 18 . 25 h 2 8 mL 2 Thus, the multiple of h 2 8 mL 2 is 18 . 25 . (c) Now, the configuration that provides the next higher system energy has the first two levels filled, with the next three levels half-filled: E third excited = 2 E 1 , 1 + 2 E 1 , 2 + E 1 , 3 + E 2 , 1 + E 2 , 2 = 2 1 . 25 + 2 2 . 00 + 3 . 25 + 4 . 25 + 5 . 00 , which means (putting the "unit" factor back in) the energy of the third excited state is E third excited = 19 . 00 h 2 8 mL 2 . Thus, the multiple of h 2 8 mL 2 is 19 . 00 . (d) The energy states of this problem are suggested below: ________ third excited 19 . 00 h 2 8 mL 2 ________ second excited 18 . 25 h 2 8 mL 2 ________ first excited 18 . 00 h 2 8 mL 2 ________ ground state 17 . 25 h 2 8 mL 2

MEDIUM

JEE Main

IMPORTANT

Earn 100

A rectangular corral of widths $L_{x} = L$ and $L_{y} = 2 L$ contains seven electrons. What multiple of $h^{2} / 8 {mL}^{2}$ gives the energy of the ground state of this system? Assume that the electrons do not interact with one another, and do not neglect spin.

Important Questions on All About Atoms

EASY

JEE Main

IMPORTANT

Principles Of Physics International Student Version>Chapter 37 - All About Atoms>Problems>Q 45

Through what minimum potential difference must an electron in an

X

-ray tube be accelerated so that it can produce

X

-rays with a wavelength of

0.100 nm ?

HARD

JEE Main

IMPORTANT

Principles Of Physics International Student Version>Chapter 37 - All About Atoms>Problems>Q 46

A rectangular corral of widths $L_{x} = L$ and $L_{y} = 2 L$ , contains seven electrons. Assume that the electrons do not interact with one another and do not neglect spin.

What multiple of $\frac{h^{2}}{8 {mL}^{2}}$ gives the energy of $(a)$ the first excited state, $(b)$ the second excited state, and $(c)$ the third excited state of the system of seven electrons? $(d)$ Construct an energy-level diagram for the lowest four energy levels.

MEDIUM

JEE Main

IMPORTANT

Principles Of Physics International Student Version>Chapter 37 - All About Atoms>Problems>Q 47

Calculate the ratio of the wavelength of the

K_{α}

line for niobium

(Nb)

to that for gallium

(Ga) .

EASY

JEE Main

IMPORTANT

Principles Of Physics International Student Version>Chapter 37 - All About Atoms>Problems>Q 48

A hydrogen atom in its ground state actually has two possible, closely spaced energy levels, because the electron is in the magnetic field

\vec{B}

of the proton (the nucleus). Accordingly, an energy is associated with the orientation of the electron's magnetic moment

\vec{μ}

relative to

\vec{B},

and the electron is said to be either spin up (higher energy) or spin down (lower energy) in that field. If the electron is excited to the higher-energy level, it can de-excite by spin-flipping and emitting a photon. The wavelength associated with that photon is

21 cm .

(Such a process occurs extensively in the Milky Way galaxy, and reception of the

21 cm

radiation by radio telescopes reveals where hydrogen gas lies between stars) What is the effective magnitude of

\vec{B}

as experienced by the electron in the ground-state hydrogen atom?

HARD

JEE Main

IMPORTANT

Principles Of Physics International Student Version>Chapter 37 - All About Atoms>Problems>Q 49

An electron is in a state with

l = 3 .

(a)

What multiple of

ℏ

gives the magnitude of

\vec{L}

(b)

What multiple of

μ_{B}

gives the magnitude of

\vec{μ}

(c)

What is the largest possible value of

m_{l}

(d)

What multiple of

ℏ

gives the corresponding value of

L_{z}

(e)

What multiple of

μ_{B}

gives the corresponding value of

μ_{orb, z}

(f)

What is the value of the semi-classical angle

θ

between the directions of

L_{2}

and

\vec{L} ?

What is the value of angle

θ

for

(g)

the second largest possible value of

m_{l}

and

(h)

the smallest (that is, most negative) possible value of

m_{l}

EASY

JEE Main

IMPORTANT

Principles Of Physics International Student Version>Chapter 37 - All About Atoms>Problems>Q 50

When electrons bombard a molybdenum target, they produce both continuous and characteristic

x

rays, as shown in the figure. In that figure, the kinetic energy of the incident electrons is

35.0 keV .

If the accelerating potential is increased to

55.0 keV, (a)

what is the value of

λ_{\min}

, and

(b)

do the wavelengths of the

K_{α}

and

K_{β}

lines increase, decrease or remain the same?

MEDIUM

JEE Main

IMPORTANT

Principles Of Physics International Student Version>Chapter 37 - All About Atoms>Problems>Q 51

An electron in a hydrogen atom is in a state with

l = 5

. What is the minimum possible value of the semi-classical angle between

\vec{L}

and

L_{z}

MEDIUM

JEE Main

IMPORTANT

Principles Of Physics International Student Version>Chapter 37 - All About Atoms>Problems>Q 52

An electron is in a state with

n = 4

. What are

(a)

the number of possible values of

l, (b)

the number of possible values of

m_{l}

(c)

the number of possible values of

m_{s}, (d)

the number of states in the

n = 4

shell and

(e)

the number of sub-shells in the

n = 4

shell?

A rectangular corral of widths Lx=L and Ly=2L contains seven electrons. What multiple of h2/8 mL2 gives the energy of the ground state of this system? Assume that the electrons do not interact with one another, and do not neglect spin.

Important Questions on All About Atoms

Learn and Prepare for any exam you want !

A rectangular corral of widths $L_{x} = L$ and $L_{y} = 2 L$ contains seven electrons. What multiple of $h^{2} / 8 {mL}^{2}$ gives the energy of the ground state of this system? Assume that the electrons do not interact with one another, and do not neglect spin.