GATE Exam > GATE Questions > The microwave spectrum of H1Br79 consist of a...
Start Learning for Free
The microwave spectrum of H1Br79 consist of a series of lines spaced 14 cm-1 apart. The intemuclear distance of the molecule is ______ Å (UDto two decimal places)
[Given : h = 6.63 x 10-34 J-s, c = 3 x 108 m/s. NA = 6.023 x 1023 mol-1]
[Given : h = 6.63 x 10-34 J-s, c = 3 x 108 m/s. NA = 6.023 x 1023 mol-1]
Correct answer is between '1.54,1.64'. Can you explain this answer?
| FREE This question is part of | Download PDF Attempt this Test |
Verified Answer
The microwave spectrum of H1Br79 consist of a series of lines spaced 1...
2B = 14cm-1 ⇒ B = 7cm-1 = 700m-1.
Tims putting values we get
∴
Tims putting values we get
∴
Most Upvoted Answer
The microwave spectrum of H1Br79 consist of a series of lines spaced 1...
We can use the formula for the rotational energy levels of a diatomic molecule:
E(J) = B·J(J+1) - D·J^2(J+1)^2
where J is the rotational quantum number, B is the rotational constant, and D is the centrifugal distortion constant. The energy difference between two adjacent levels with J and J+1 is:
ΔE = B·(2J+1) - 4D·(J+1)^3
In the microwave spectrum, we observe transitions between rotational levels with ΔJ = ±1. The frequency of such a transition is:
ν = (ΔE/h) = (2B/h)·(J+1) - (8D/h)·(J+1)^3
where h is Planck's constant. We are given that the spacing between adjacent lines is 14 cm^-1, which corresponds to a frequency of:
ν = (14 cm^-1)·(c/100 cm/s) = 4.2×10^10 Hz
We can equate this expression to the frequency of the J→J+1 transition:
4.2×10^10 Hz = (2B/h)·(J+1) - (8D/h)·(J+1)^3
Let's define x = J+1, so we can write:
4.2×10^10 Hz = (2B/h)·x - (8D/h)·x^3
This is a cubic equation in x, which we can solve using the general formula:
x = [(q ± sqrt(q^2 + 4r^3))/2]^(1/3) - p/3
where q = (3r - p^2)/3, r = (9pq - 27s)/54, and p = (2B/h), q = -(8D/h), s = 4.2×10^10 Hz. We can simplify this to:
x = [(-q/2 + sqrt(q^2/4 + r^3))^(1/3) + (-q/2 - sqrt(q^2/4 + r^3))^(1/3)] - (p/3)
We can plug in the values of p, q, and s, and simplify:
x = [(-D/h + sqrt((D/h)^2 + (B/h)^3/27))^(1/3) + (-D/h - sqrt((D/h)^2 + (B/h)^3/27))^(1/3)] - (B/h)/3
Now we just need to solve for B and D. We can use the fact that the rotational constant B is related to the moment of inertia I of the molecule:
B = h/(8π^2cI)
The moment of inertia of a diatomic molecule with reduced mass μ and internuclear distance r is:
I = μr^2
For H1Br79, the atomic masses are m(H) = 1.0078 u and m(Br) = 78.9183 u. The reduced mass is:
μ = m(H)m(Br)/(m(H) + m(Br)) = 0.9928 u
The internuclear distance r can be calculated from the equilibrium bond length re and the vibrational constant ωe:
r
E(J) = B·J(J+1) - D·J^2(J+1)^2
where J is the rotational quantum number, B is the rotational constant, and D is the centrifugal distortion constant. The energy difference between two adjacent levels with J and J+1 is:
ΔE = B·(2J+1) - 4D·(J+1)^3
In the microwave spectrum, we observe transitions between rotational levels with ΔJ = ±1. The frequency of such a transition is:
ν = (ΔE/h) = (2B/h)·(J+1) - (8D/h)·(J+1)^3
where h is Planck's constant. We are given that the spacing between adjacent lines is 14 cm^-1, which corresponds to a frequency of:
ν = (14 cm^-1)·(c/100 cm/s) = 4.2×10^10 Hz
We can equate this expression to the frequency of the J→J+1 transition:
4.2×10^10 Hz = (2B/h)·(J+1) - (8D/h)·(J+1)^3
Let's define x = J+1, so we can write:
4.2×10^10 Hz = (2B/h)·x - (8D/h)·x^3
This is a cubic equation in x, which we can solve using the general formula:
x = [(q ± sqrt(q^2 + 4r^3))/2]^(1/3) - p/3
where q = (3r - p^2)/3, r = (9pq - 27s)/54, and p = (2B/h), q = -(8D/h), s = 4.2×10^10 Hz. We can simplify this to:
x = [(-q/2 + sqrt(q^2/4 + r^3))^(1/3) + (-q/2 - sqrt(q^2/4 + r^3))^(1/3)] - (p/3)
We can plug in the values of p, q, and s, and simplify:
x = [(-D/h + sqrt((D/h)^2 + (B/h)^3/27))^(1/3) + (-D/h - sqrt((D/h)^2 + (B/h)^3/27))^(1/3)] - (B/h)/3
Now we just need to solve for B and D. We can use the fact that the rotational constant B is related to the moment of inertia I of the molecule:
B = h/(8π^2cI)
The moment of inertia of a diatomic molecule with reduced mass μ and internuclear distance r is:
I = μr^2
For H1Br79, the atomic masses are m(H) = 1.0078 u and m(Br) = 78.9183 u. The reduced mass is:
μ = m(H)m(Br)/(m(H) + m(Br)) = 0.9928 u
The internuclear distance r can be calculated from the equilibrium bond length re and the vibrational constant ωe:
r
Free Test
FREE
| Start Free Test |
Community Answer
The microwave spectrum of H1Br79 consist of a series of lines spaced 1...
Hff
|
Explore Courses for GATE exam
|
|
Similar GATE Doubts
The microwave spectrum of H1Br79 consist of a series of lines spaced 14 cm-1 apart. The intemuclear distance of the molecule is ______Å (UDto two decimal places)[Given : h = 6.63 x 10-34 J-s, c = 3 x 108m/s. NA = 6.023 x 1023mol-1]Correct answer is between '1.54,1.64'. Can you explain this answer?
Question Description
The microwave spectrum of H1Br79 consist of a series of lines spaced 14 cm-1 apart. The intemuclear distance of the molecule is ______Å (UDto two decimal places)[Given : h = 6.63 x 10-34 J-s, c = 3 x 108m/s. NA = 6.023 x 1023mol-1]Correct answer is between '1.54,1.64'. Can you explain this answer? for GATE 2024 is part of GATE preparation. The Question and answers have been prepared according to the GATE exam syllabus. Information about The microwave spectrum of H1Br79 consist of a series of lines spaced 14 cm-1 apart. The intemuclear distance of the molecule is ______Å (UDto two decimal places)[Given : h = 6.63 x 10-34 J-s, c = 3 x 108m/s. NA = 6.023 x 1023mol-1]Correct answer is between '1.54,1.64'. Can you explain this answer? covers all topics & solutions for GATE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The microwave spectrum of H1Br79 consist of a series of lines spaced 14 cm-1 apart. The intemuclear distance of the molecule is ______Å (UDto two decimal places)[Given : h = 6.63 x 10-34 J-s, c = 3 x 108m/s. NA = 6.023 x 1023mol-1]Correct answer is between '1.54,1.64'. Can you explain this answer?.
The microwave spectrum of H1Br79 consist of a series of lines spaced 14 cm-1 apart. The intemuclear distance of the molecule is ______Å (UDto two decimal places)[Given : h = 6.63 x 10-34 J-s, c = 3 x 108m/s. NA = 6.023 x 1023mol-1]Correct answer is between '1.54,1.64'. Can you explain this answer? for GATE 2024 is part of GATE preparation. The Question and answers have been prepared according to the GATE exam syllabus. Information about The microwave spectrum of H1Br79 consist of a series of lines spaced 14 cm-1 apart. The intemuclear distance of the molecule is ______Å (UDto two decimal places)[Given : h = 6.63 x 10-34 J-s, c = 3 x 108m/s. NA = 6.023 x 1023mol-1]Correct answer is between '1.54,1.64'. Can you explain this answer? covers all topics & solutions for GATE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The microwave spectrum of H1Br79 consist of a series of lines spaced 14 cm-1 apart. The intemuclear distance of the molecule is ______Å (UDto two decimal places)[Given : h = 6.63 x 10-34 J-s, c = 3 x 108m/s. NA = 6.023 x 1023mol-1]Correct answer is between '1.54,1.64'. Can you explain this answer?.
Solutions for The microwave spectrum of H1Br79 consist of a series of lines spaced 14 cm-1 apart. The intemuclear distance of the molecule is ______Å (UDto two decimal places)[Given : h = 6.63 x 10-34 J-s, c = 3 x 108m/s. NA = 6.023 x 1023mol-1]Correct answer is between '1.54,1.64'. Can you explain this answer? in English & in Hindi are available as part of our courses for GATE.
Download more important topics, notes, lectures and mock test series for GATE Exam by signing up for free.
Here you can find the meaning of The microwave spectrum of H1Br79 consist of a series of lines spaced 14 cm-1 apart. The intemuclear distance of the molecule is ______Å (UDto two decimal places)[Given : h = 6.63 x 10-34 J-s, c = 3 x 108m/s. NA = 6.023 x 1023mol-1]Correct answer is between '1.54,1.64'. Can you explain this answer? defined & explained in the simplest way possible. Besides giving the explanation of
The microwave spectrum of H1Br79 consist of a series of lines spaced 14 cm-1 apart. The intemuclear distance of the molecule is ______Å (UDto two decimal places)[Given : h = 6.63 x 10-34 J-s, c = 3 x 108m/s. NA = 6.023 x 1023mol-1]Correct answer is between '1.54,1.64'. Can you explain this answer?, a detailed solution for The microwave spectrum of H1Br79 consist of a series of lines spaced 14 cm-1 apart. The intemuclear distance of the molecule is ______Å (UDto two decimal places)[Given : h = 6.63 x 10-34 J-s, c = 3 x 108m/s. NA = 6.023 x 1023mol-1]Correct answer is between '1.54,1.64'. Can you explain this answer? has been provided alongside types of The microwave spectrum of H1Br79 consist of a series of lines spaced 14 cm-1 apart. The intemuclear distance of the molecule is ______Å (UDto two decimal places)[Given : h = 6.63 x 10-34 J-s, c = 3 x 108m/s. NA = 6.023 x 1023mol-1]Correct answer is between '1.54,1.64'. Can you explain this answer? theory, EduRev gives you an
ample number of questions to practice The microwave spectrum of H1Br79 consist of a series of lines spaced 14 cm-1 apart. The intemuclear distance of the molecule is ______Å (UDto two decimal places)[Given : h = 6.63 x 10-34 J-s, c = 3 x 108m/s. NA = 6.023 x 1023mol-1]Correct answer is between '1.54,1.64'. Can you explain this answer? tests, examples and also practice GATE tests.
|
|
Explore Courses for GATE exam
|
|
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
|
© EduRev
|
Education Revolution
|
Follow Us
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup