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The pure rotational (microwave) spectrum of the gaseous molecule CN consists of a series of equally spaced line separated by 3.7978 cm–1. The inter nuclear distance of the molecule is [Molar masses are 12C=12.011 and 14N=14.007 g mol–1]:
- a)130 pm
- b)117 pm
- c)150 pm
- d)93 pm
Correct answer is option 'B'. Can you explain this answer?
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The pure rotational (microwave) spectrum of the gaseous molecule CN co...
Given:
- Equally spaced lines in pure rotational (microwave) spectrum of CN
- Separation between the lines = 3.7978 cm^-1
- Molar masses of C and N = 12.011 g/mol and 14.007 g/mol, respectively
To find:
- Inter-nuclear distance of CN molecule
Formula used:
- The separation between the rotational energy levels (ΔE) of a diatomic molecule is given by:
ΔE = hB,
where h is Planck's constant and B is the rotational constant.
- The rotational constant of a diatomic molecule is given by:
B = h / (8π^2μr_e^2),
where μ is the reduced mass of the molecule, and r_e is the equilibrium bond length.
- The reduced mass (μ) of a diatomic molecule is given by:
μ = m_1m_2 / (m_1 + m_2),
where m_1 and m_2 are the masses of the two atoms in the molecule.
- The equilibrium bond length of a diatomic molecule can be estimated using the formula:
r_e = (r_1 + r_2) / 2,
where r_1 and r_2 are the covalent radii of the two atoms in the molecule.
Solution:
1. Calculate the rotational constant (B) of CN molecule using the given separation between the lines:
ΔE = hB
⇒ B = ΔE / h
⇒ B = 3.7978 cm^-1 / (6.626 x 10^-34 J s)
⇒ B = 5.72 x 10^-4 cm^-1
2. Calculate the reduced mass (μ) of CN molecule:
μ = m_1m_2 / (m_1 + m_2)
⇒ μ = 12.011 x 14.007 / (12.011 + 14.007) g/mol
⇒ μ = 6.68 g/mol
3. Estimate the equilibrium bond length (r_e) of CN molecule using the covalent radii of carbon and nitrogen:
r_e = (r_1 + r_2) / 2
⇒ r_e = (0.77 Å + 0.75 Å) / 2
⇒ r_e = 0.76 Å = 0.76 x 10^-10 m
4. Calculate the inter-nuclear distance (r) of CN molecule using the rotational constant (B) and reduced mass (μ):
B = h / (8π^2μr_e^2)
⇒ r = √(h / (8π^2μB))
⇒ r = √((6.626 x 10^-34 J s) / (8π^2 x 6.68 x 10^-3 kg/mol x 5.72 x 10^-4 cm^-1))
⇒ r = 117 pm
Therefore, the inter-nuclear distance of CN molecule is 117 pm.
- Equally spaced lines in pure rotational (microwave) spectrum of CN
- Separation between the lines = 3.7978 cm^-1
- Molar masses of C and N = 12.011 g/mol and 14.007 g/mol, respectively
To find:
- Inter-nuclear distance of CN molecule
Formula used:
- The separation between the rotational energy levels (ΔE) of a diatomic molecule is given by:
ΔE = hB,
where h is Planck's constant and B is the rotational constant.
- The rotational constant of a diatomic molecule is given by:
B = h / (8π^2μr_e^2),
where μ is the reduced mass of the molecule, and r_e is the equilibrium bond length.
- The reduced mass (μ) of a diatomic molecule is given by:
μ = m_1m_2 / (m_1 + m_2),
where m_1 and m_2 are the masses of the two atoms in the molecule.
- The equilibrium bond length of a diatomic molecule can be estimated using the formula:
r_e = (r_1 + r_2) / 2,
where r_1 and r_2 are the covalent radii of the two atoms in the molecule.
Solution:
1. Calculate the rotational constant (B) of CN molecule using the given separation between the lines:
ΔE = hB
⇒ B = ΔE / h
⇒ B = 3.7978 cm^-1 / (6.626 x 10^-34 J s)
⇒ B = 5.72 x 10^-4 cm^-1
2. Calculate the reduced mass (μ) of CN molecule:
μ = m_1m_2 / (m_1 + m_2)
⇒ μ = 12.011 x 14.007 / (12.011 + 14.007) g/mol
⇒ μ = 6.68 g/mol
3. Estimate the equilibrium bond length (r_e) of CN molecule using the covalent radii of carbon and nitrogen:
r_e = (r_1 + r_2) / 2
⇒ r_e = (0.77 Å + 0.75 Å) / 2
⇒ r_e = 0.76 Å = 0.76 x 10^-10 m
4. Calculate the inter-nuclear distance (r) of CN molecule using the rotational constant (B) and reduced mass (μ):
B = h / (8π^2μr_e^2)
⇒ r = √(h / (8π^2μB))
⇒ r = √((6.626 x 10^-34 J s) / (8π^2 x 6.68 x 10^-3 kg/mol x 5.72 x 10^-4 cm^-1))
⇒ r = 117 pm
Therefore, the inter-nuclear distance of CN molecule is 117 pm.
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Community Answer
The pure rotational (microwave) spectrum of the gaseous molecule CN co...
Given is 2B and we have to find out r.
use the formula of B and then convert the answer in pm.
use the formula of B and then convert the answer in pm.
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The pure rotational (microwave) spectrum of the gaseous molecule CN consists of a series of equally spaced line separated by 3.7978 cm–1. The inter nuclear distance of the molecule is [Molar masses are 12C=12.011 and 14N=14.007 g mol–1]:a)130 pmb)117 pmc)150 pmd)93 pmCorrect answer is option 'B'. Can you explain this answer?
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