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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?
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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
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Tims putting values we get
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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
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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?
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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?.
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