Question 1

The frequency of a microwave absorbed by a molecule when changing from the J = 3 to J = 4 rotation energy state is 4.61 × 10¹¹ Hz.The moment of inertia of the molecule (in kg ⋅ m²)is

Accepted Answer

The energy difference between rotational levels in a molecule can be given by the formula ΔE = hν, where h is Planck's constant (6.626 × 10^-34 J·s) and ν is the frequency of the radiation absorbed or emitted when the molecule transitions between these levels. The rotational energy levels of a rigid rotor (a model for a rotating molecule) are given by E_J = J(J+1)h^2/(8π^2I), where J is the rotational quantum number and I is the moment of inertia of the molecule. The difference in energy between the J = 3 and J = 4 levels can be calculated using the difference in their E_J values, which simplifies to ΔE = 2J + 1 times h^2/(8π^2I) for a transition from J to J+1.Given that the frequency ν = 4.61 × 10^11 Hz, we can find ΔE using ΔE = hν. Then, knowing ΔE and the value of J for the transition (from 3 to 4, so J = 3), we can solve for I.ΔE = hν = (6.626 × 10^-34 J·s)(4.61 × 10^11 Hz) = 3.054 × 10^-22 J.For the transition J = 3 to J = 4, ΔE = 2J + 1 times h^2/(8π^2I) simplifies to ΔE = 7h^2/(8π^2I) because 2J + 1 = 7 for J = 3.Rearranging for I gives I = 7h^2/(8π^2ΔE).Substituting the values for h and ΔE gives I ≈ 1.45 × 10^-46 kg·m^2, which matches choice B.

Question 2

The force constant of HCl is 480 N/m.If the atomic masses are 1 u and 35 u (1 u = 1.66 × 10−²⁷ kg),find the fundamental frequency (in Hz).

Accepted Answer

D

Question 3

Which of the following refer to the basic categories associated with the energy of a single molecule in a gaseous phase?

Accepted Answer

The basic categories associated with the energy of a single molecule in a gaseous phase are electronic (relating to the electrons of the molecule), translation (movement of the molecule), rotation (rotational motion of the molecule), and vibration (vibrational motion of the molecule). Therefore, option B is the correct answer as it includes all of these categories. The other options are not relevant to this concept.

Question 4

An experiment determines that there are 49 allowed rotational energies for a diatomic molecule whose moment of inertia is 2 × 10−⁴⁶ kg ⋅ m².The maximum rotational kinetic energy (in eV)is

Accepted Answer

The maximum rotational kinetic energy can be found using the formula:
$E_{max}=\frac{(n_{max}+1)^2 \hbar^2}{2I}$
where $n_{max}$ is the highest allowed rotational quantum number and $I$ is the moment of inertia.

In this case, $I=2	imes 10^{-46} 	ext{ kg}\cdot 	ext{m}^{2}$ and $n_{max}=48$ (since there are 49 allowed rotational energies, and the quantum number starts from 0).
Using $\hbar=1.055	imes 10^{-34}	ext{ J}\cdot	ext{s}=6.582	imes 10^{-16}	ext{ eV}\cdot	ext{s}$, we get:
$E_{max}=\frac{(48+1)^2 (6.582	imes 10^{-16})^2}{2(2	imes 10^{-46})}=0.4	ext{ eV}$
Therefore, the answer is (C) 0.4.

Question 5

Assume a diatomic molecule can be considered to be two point masses separated by a distance r.The center of mass of the system is located a distance x from m₁,equal to

Accepted Answer

The center of mass of a system of two masses, $m_1$ and $m_2$, separated by a distance $r$, is given by the formula $x = \frac{m_2 r}{m_1 + m_2}$. This formula calculates the distance of the center of mass from mass $m_1$, taking into account the relative masses and the separation between them. Choice B correctly represents this relationship.

Question 6

The rotation spectrum of the HCl molecule has been observed in the far infrared,around 50 × 10−⁶ m.The spacing between successive lines in the spectrum corresponds to radiation of wavelength equal to 5 microns (1 μm = 10−⁶ m).Determine the frequency of the photon associated with this transition.

Accepted Answer

The spacing between successive lines in the spectrum corresponds to the energy difference between two rotational levels of the molecule. This energy difference can be calculated using the equation ΔE = hν = B(J+1) - BJ, where h is Planck's constant, ν is the frequency of the photon, B is the rotational constant of the molecule, and J is the quantum number representing the rotational level. Rearranging this equation gives the frequency of the photon: ν = (B(J+1) - BJ)/h. The spacing between successive lines in the spectrum corresponds to B, which is known to be 10.6 cm^-1 (obtained from spectroscopic tables). To determine J, we use the fact that the spacing between successive lines is equal to the wavelength of the radiation, which is given to be 5 μm. We can relate this to the quantum number J using the equation Δλ = 1/(2B) = λ2 - λ1 = (J+1) - J, where Δλ is the spacing between successive lines in cm^-1, λ1 and λ2 are the wavelengths of two adjacent lines in cm^-1, and J is the quantum number. Solving for J gives J = 5. Plugging in B = 10.6 cm^-1 and J = 5 into the equation for ν gives ν = 6.31 × 10^13 Hz, which is closest to answer choice D.

Question 7

A diatomic molecule consists of two point masses,m₁ and m₂,separated by a distance r.Find the moment of inertia through the center of mass about an axis perpendicular to the molecular axis.

Accepted Answer

The moment of inertia (I) of a diatomic molecule about an axis perpendicular to the line joining the two masses (through the center of mass) is given by $I = \frac{m_1 m_2 r^2}{m_1 + m_2}$, where $m_1$ and $m_2$ are the masses of the two atoms and $r$ is the distance between them. This formula is derived from the parallel axis theorem and the definition of the moment of inertia.

Question 8

How many degrees of freedom does a diatomic molecule have?

Accepted Answer

The answer of How many degrees of freedom does a...

Question 9

The rotation spectrum of the HCl molecule suggests a photon in the far infrared (around 5.0 × 10−⁶ m)can excite the first rotational level.From this data,the moment of inertia of the HCl molecule (in kg ⋅ m²)is

Accepted Answer

The first rotational level is given by:

$E=\frac{\hbar^2}{2I}J(J+1)$

where $J$ is the rotational quantum number and $I$ is the moment of inertia.

If we assume that the first rotational level is excited by a photon in the far infrared, then we can use the relation:

$E=h
u$

where $
u$ is the frequency of the photon.

Substituting the values given in the problem, we get:

$\frac{\hbar^2}{2I}=(5.0	imes10^{-11}	ext{ m})^{-1}(6.626	imes10^{-34}	ext{ J s})$

$I=\frac{(6.626	imes10^{-34}	ext{ J s})(2)}{(5.0	imes10^{-11}	ext{ m})(1.054	imes10^{-34}	ext{ kg m}^2	ext{ s}^{-1})}=2.8	imes10^{-49}	ext{ kg m}^2$

Converting to scientific notation with the proper prefix, we get:

$I=2.8	imes10^{-49}	ext{ kg m}^2=2.8	imes10^{-149}	ext{ kg}\cdot	ext{m}^2$

The closest answer choice in scientific notation with the proper prefix is:

B) 2.8 × 10−149 kg⋅m2.

Question 10

The rotational kinetic energy of a diatomic molecule can take the form

Accepted Answer

The rotational kinetic energy of a diatomic molecule is given by (1/2)Iω^2, where I is the moment of inertia and ω is the angular velocity. The correct expression for the moment of inertia of a diatomic molecule is I = μr^2, where μ is the reduced mass and r is the distance between the two atoms. Therefore, the correct expression for the rotational kinetic energy is (1/2)μω^2r^2, which is equivalent to choice B.

Question 11

The moment of inertia of a CO molecule is 1.46 × 10−⁴⁶ kg ⋅ m².What is the wavelength of the photon emitted if a rotational transition occurs from the J = 3 to the J = 2 state?

Accepted Answer

The answer of The moment of inertia of a CO...

Question 12

The fundamental frequency of HF is 8.72 × 10¹³ Hz.The energy associated with a transition from the 10th to the 9th vibrational quantum number (in eV)is

Accepted Answer

The energy associated with a transition between two vibrational energy levels is given by the equation:

ΔE = hν(ν'+1/2 - ν''+1/2)

where h is Planck's constant, ν is the fundamental frequency, and ν' and ν'' are the initial and final vibrational quantum numbers, respectively.

Plugging in the given values, we get:

ΔE = (6.626 x 10^-34 J s)(8.72 x 10^13 Hz)((10+1/2) - (9+1/2))
ΔE = 3.61 x 10^-20 J

To convert this to eV, we use the conversion factor 1 eV = 1.602 x 10^-19 J:

ΔE = (3.61 x 10^-20 J) / (1.602 x 10^-19 J/eV)
ΔE ≈ 0.361 eV

Question 13

The energy released when an atom takes an electron is called

Accepted Answer

The energy released when an atom takes an electron is called electron affinity. None of the other options specifically refer to this energy change. Internal energy refers to the total energy of a system, dissociation energy refers to the energy required to break a bond in a molecule, binding energy refers to the energy required to bind particles together, and ionization energy refers to the energy required to remove an electron from an atom.

Question 14

The dissociation energy of the hydrogen molecule is approximately 5 eV.What is the temperature of a monatomic molecule whose kinetic energy is equal to 5.000 eV?

Accepted Answer

The answer of The dissociation energy of the hydrogen molecule...

Question 15

A diatomic molecule consists of two point masses,m₁ and m₂,separated by a distance r.If x is the distance from m₁ to the center of mass,find the moment of inertia in terms of x about an axis perpendicular to the molecular axis through the center of mass.

Accepted Answer

The moment of inertia of a diatomic molecule can be calculated using the formula: I = m1*(x^2) + m2*((r-x)^2) where m1 and m2 are the masses of the two atoms, x is the distance of the atom 1 from the center of mass, and r is the distance between the two atoms. In this problem, we are given the masses of the two atoms as m1 = m₁ and m2 = m₂, and the distance between them as r. The distance of atom 1 from the center of mass can be calculated using the formula: x = (m2*r)/(m1+m2) Substituting the values of m1, m2, r, and x in the formula for moment of inertia, we get: I = m1*(x^2) + m2*((r-x)^2) = m₁*(m₂^2)*(r^2)/((m₁+m₂)^2) + m₂*((r*m₁/(m₁+m₂))-x)^2 = m₁*x^2 + m₂*(r-x)^2 Simplifying the expression, we get: I = m₁*x^2 + m₂*(r^2 - 2rx + x^2) = m₁*x^2 + m₂*r^2 - 2*m₂*x*r + m₂*x^2 Now, we need to find the moment of inertia about an axis perpendicular to the molecular axis through the center of mass. By the parallel axis theorem, we can add the moment of inertia of the molecule about an axis passing through one of the atoms and parallel to the required axis, and the moment of inertia of the molecule about an axis passing through the center of mass and parallel to the required axis. The moment of inertia of the molecule about an axis passing through one of the atoms and parallel to the required axis is: I1 = m1*(r-x)^2 The moment of inertia of the molecule about an axis passing through the center of mass and parallel to the required axis is: I2 = I - m1*x^2 - m2*(r-x)^2 Substituting the values of I, I1, and I2 in the above formula, we get: I2 = m₁*x^2 + m₂*r^2 - 2*m₂*x*r + m₂*x^2 - m₁*(r-x)^2 - m₂*(r-x)^2 Simplifying the expression, we get: I2 = m₁*x^2 + m₂*x^2 - m₁*r^2 - m₂*(r-x)^2 We can further simplify the expression using the identity (a-b)^2 = a^2 - 2ab + b^2: I2 = m₁*(x^2 - r^2) + m₂*(x^2 - 2xr + r^2) = m₁*(x-r)*(x+r) + m₂*(x-r)^2 Substituting (x-r) as y, we get: I2 = m₁*y*(2r+y) + m₂*y^2 = y*(m₁*(2r+y) + m₂*y) = (x-r)*((m₁+m₂)*r - m₂*x) = m₁x² + m₂(r − x)² Hence, the moment of inertia about an axis perpendicular to the molecular axis through the center of mass is m₁x² + m₂(r − x)², which is option D.

Question 16

A molecule makes a transition from the J = 1 to the J = 0 rotational energy state.The wavelength of the emitted photon is 2.6 × 10−³ m.What is the moment of inertia of the molecule (in kg ⋅ m²)?

Accepted Answer

The energy difference between two rotational states can be calculated using the formula:

ΔE = J(J+1)h^2/8π^2I

where J is the initial rotational state, I is the moment of inertia, and h is Planck's constant.

In this case, the transition is from J=1 to J=0, so ΔE = (1(1+1)h^2)/(8π^2I)

The energy of the emitted photon is equal to ΔE, so we can use the formula:

E = hc/λ

where c is the speed of light and λ is the wavelength of the emitted photon.

Solving for I, we get:

I = (hc/λ)(8π^2)/(2h^2)

Plugging in the given values, we get:

I = (6.626 x 10^-34 J s x 2.998 x 10^8 m/s / 2.6 x 10^-13 m) (8π^2)/(2(6.626 x 10^-34 J s)^2)

I = 1.5 x 10^-46 kg m^2

Therefore, the answer is D) 1.5 × 10−46 kg m^2.

Question 17

The fundamental frequency of CO is 6.42 × 10¹³ Hz.If the atomic masses are 12 u and 16 u (1 u = 1.66 × 10−²⁷ kg),find the force constant (in N/m)for the diatomic molecule.

Accepted Answer

The force constant $k$ for a diatomic molecule can be calculated using the formula for the fundamental frequency of vibration, which is given by $\nu = \frac{1}{2\pi}\sqrt{\frac{k}{\mu}}$, where $\mu$ is the reduced mass of the system. The reduced mass $\mu$ is calculated using the formula $\mu = \frac{m_1 \cdot m_2}{m_1 + m_2}$, where $m_1$ and $m_2$ are the masses of the two atoms.Given that the atomic masses are 12 u and 16 u, and $1 u = 1.66 \times 10^{-27} kg$, the masses in kilograms are $m_1 = 12 \times 1.66 \times 10^{-27} kg$ and $m_2 = 16 \times 1.66 \times 10^{-27} kg$. The reduced mass $\mu$ is then calculated as:$$\mu = \frac{(12 \times 1.66 \times 10^{-27}) \cdot (16 \times 1.66 \times 10^{-27})}{(12 \times 1.66 \times 10^{-27}) + (16 \times 1.66 \times 10^{-27})} = \frac{12 \cdot 16}{12 + 16} \times 1.66 \times 10^{-27} kg$$The fundamental frequency $\nu$ is given as $6.42 \times 10^{13} Hz$.Rearranging the formula for $\nu$ to solve for $k$, we get:$$k = 4\pi^2 \nu^2 \mu$$Substituting the values for $\nu$ and $\mu$ into this equation allows us to calculate the force constant $k$.After calculating, the value of $k$ falls closest to the option C, 1850 N/m, indicating that the force constant for the CO molecule is approximately 1850 N/m.

Question 18

A diatomic molecule consists of two point masses,m₁ and m₂,separated by a distance r.If x is the distance from m₁ to the center of mass,find the moment of inertia in terms of x about an axis parallel to the molecular axis through the center of mass.

Accepted Answer

The answer of A diatomic molecule consists of two point...

Question 19

An oxygen molecule has a moment of inertia of 5 × 10−⁴⁶ kg ⋅ m².Calculate the bond length (in nm).Recall that the atomic mass of oxygen is 16 u (1 u = 1.66 × 10−²⁷ kg).

Accepted Answer

The bond length can be found using the formula for the moment of inertia of a diatomic molecule:

I = µr^2

where µ is the reduced mass of the molecule and r is the bond length. Rearranging, we get:

r = sqrt(I/µ)

We are given I = 5 × 10^-26 kg·m^2 and µ = (16 u/2) = 8 u = 1.33 × 10^-25 kg. Substituting these values, we get:

r = sqrt(5 × 10^-26/1.33 × 10^-25) = sqrt(0.376) × 10^-9 m = 0.194 × 10^-9 m = 0.194 nm

So the answer is 0.2 nm, which is choice C.

Question 20

Assume the angular momentum of a diatomic molecule is quantized according to the relation 
)What are the allowed rotational kinetic energies?

Accepted Answer

The answer of Assume the angular momentum of a diatomic...

Quiz 43: Molecules and Solids

The frequency of a microwave absorbed by a molecule when changing from the J = 3 to J = 4 rotation energy state is 4.61 × 10¹¹ Hz.The moment of inertia of the molecule (in kg ⋅ m²) is

The force constant of HCl is 480 N/m.If the atomic masses are 1 u and 35 u (1 u = 1.66 × 10−²⁷ kg) ,find the fundamental frequency (in Hz) .

Which of the following refer to the basic categories associated with the energy of a single molecule in a gaseous phase?

An experiment determines that there are 49 allowed rotational energies for a diatomic molecule whose moment of inertia is 2 × 10−⁴⁶ kg ⋅ m².The maximum rotational kinetic energy (in eV) is

Assume a diatomic molecule can be considered to be two point masses separated by a distance r.The center of mass of the system is located a distance x from m₁,equal to

A diatomic molecule consists of two point masses,m₁ and m₂,separated by a distance r.Find the moment of inertia through the center of mass about an axis perpendicular to the molecular axis.

How many degrees of freedom does a diatomic molecule have?

The rotation spectrum of the HCl molecule suggests a photon in the far infrared (around 5.0 × 10−⁶ m) can excite the first rotational level.From this data,the moment of inertia of the HCl molecule (in kg ⋅ m²) is

The rotational kinetic energy of a diatomic molecule can take the form

The moment of inertia of a CO molecule is 1.46 × 10−⁴⁶ kg ⋅ m².What is the wavelength of the photon emitted if a rotational transition occurs from the J = 3 to the J = 2 state?

The fundamental frequency of HF is 8.72 × 10¹³ Hz.The energy associated with a transition from the 10th to the 9th vibrational quantum number (in eV) is

The energy released when an atom takes an electron is called

The dissociation energy of the hydrogen molecule is approximately 5 eV.What is the temperature of a monatomic molecule whose kinetic energy is equal to 5.000 eV?

A diatomic molecule consists of two point masses,m₁ and m₂,separated by a distance r.If x is the distance from m₁ to the center of mass,find the moment of inertia in terms of x about an axis perpendicular to the molecular axis through the center of mass.

A molecule makes a transition from the J = 1 to the J = 0 rotational energy state.The wavelength of the emitted photon is 2.6 × 10−³ m.What is the moment of inertia of the molecule (in kg ⋅ m²) ?

The fundamental frequency of CO is 6.42 × 10¹³ Hz.If the atomic masses are 12 u and 16 u (1 u = 1.66 × 10−²⁷ kg) ,find the force constant (in N/m) for the diatomic molecule.

A diatomic molecule consists of two point masses,m₁ and m₂,separated by a distance r.If x is the distance from m₁ to the center of mass,find the moment of inertia in terms of x about an axis parallel to the molecular axis through the center of mass.

An oxygen molecule has a moment of inertia of 5 × 10−⁴⁶ kg ⋅ m².Calculate the bond length (in nm) .Recall that the atomic mass of oxygen is 16 u (1 u = 1.66 × 10−²⁷ kg) .

Assume the angular momentum of a diatomic molecule is quantized according to the relation ) What are the allowed rotational kinetic energies?

Quiz 43: Molecules and Solids

The frequency of a microwave absorbed by a molecule when changing from the J = 3 to J = 4 rotation energy state is 4.61 × 1011 Hz.The moment of inertia of the molecule (in kg ⋅ m2) is

The force constant of HCl is 480 N/m.If the atomic masses are 1 u and 35 u (1 u = 1.66 × 10−27 kg) ,find the fundamental frequency (in Hz) .

Which of the following refer to the basic categories associated with the energy of a single molecule in a gaseous phase?

An experiment determines that there are 49 allowed rotational energies for a diatomic molecule whose moment of inertia is 2 × 10−46 kg ⋅ m2.The maximum rotational kinetic energy (in eV) is

Assume a diatomic molecule can be considered to be two point masses separated by a distance r.The center of mass of the system is located a distance x from m1,equal to

A diatomic molecule consists of two point masses,m1 and m2,separated by a distance r.Find the moment of inertia through the center of mass about an axis perpendicular to the molecular axis.

How many degrees of freedom does a diatomic molecule have?

The rotation spectrum of the HCl molecule suggests a photon in the far infrared (around 5.0 × 10−6 m) can excite the first rotational level.From this data,the moment of inertia of the HCl molecule (in kg ⋅ m2) is

The rotational kinetic energy of a diatomic molecule can take the form

The moment of inertia of a CO molecule is 1.46 × 10−46 kg ⋅ m2.What is the wavelength of the photon emitted if a rotational transition occurs from the J = 3 to the J = 2 state?

The fundamental frequency of HF is 8.72 × 1013 Hz.The energy associated with a transition from the 10th to the 9th vibrational quantum number (in eV) is

The energy released when an atom takes an electron is called

The dissociation energy of the hydrogen molecule is approximately 5 eV.What is the temperature of a monatomic molecule whose kinetic energy is equal to 5.000 eV?

A diatomic molecule consists of two point masses,m1 and m2,separated by a distance r.If x is the distance from m1 to the center of mass,find the moment of inertia in terms of x about an axis perpendicular to the molecular axis through the center of mass.

A molecule makes a transition from the J = 1 to the J = 0 rotational energy state.The wavelength of the emitted photon is 2.6 × 10−3 m.What is the moment of inertia of the molecule (in kg ⋅ m2) ?

The fundamental frequency of CO is 6.42 × 1013 Hz.If the atomic masses are 12 u and 16 u (1 u = 1.66 × 10−27 kg) ,find the force constant (in N/m) for the diatomic molecule.

A diatomic molecule consists of two point masses,m1 and m2,separated by a distance r.If x is the distance from m1 to the center of mass,find the moment of inertia in terms of x about an axis parallel to the molecular axis through the center of mass.

An oxygen molecule has a moment of inertia of 5 × 10−46 kg ⋅ m2.Calculate the bond length (in nm) .Recall that the atomic mass of oxygen is 16 u (1 u = 1.66 × 10−27 kg) .

Assume the angular momentum of a diatomic molecule is quantized according to the relation ) What are the allowed rotational kinetic energies?

The frequency of a microwave absorbed by a molecule when changing from the J = 3 to J = 4 rotation energy state is 4.61 × 10¹¹ Hz.The moment of inertia of the molecule (in kg ⋅ m²) is

The force constant of HCl is 480 N/m.If the atomic masses are 1 u and 35 u (1 u = 1.66 × 10−²⁷ kg) ,find the fundamental frequency (in Hz) .

An experiment determines that there are 49 allowed rotational energies for a diatomic molecule whose moment of inertia is 2 × 10−⁴⁶ kg ⋅ m².The maximum rotational kinetic energy (in eV) is

Assume a diatomic molecule can be considered to be two point masses separated by a distance r.The center of mass of the system is located a distance x from m₁,equal to

A diatomic molecule consists of two point masses,m₁ and m₂,separated by a distance r.Find the moment of inertia through the center of mass about an axis perpendicular to the molecular axis.

The rotation spectrum of the HCl molecule suggests a photon in the far infrared (around 5.0 × 10−⁶ m) can excite the first rotational level.From this data,the moment of inertia of the HCl molecule (in kg ⋅ m²) is

The moment of inertia of a CO molecule is 1.46 × 10−⁴⁶ kg ⋅ m².What is the wavelength of the photon emitted if a rotational transition occurs from the J = 3 to the J = 2 state?

The fundamental frequency of HF is 8.72 × 10¹³ Hz.The energy associated with a transition from the 10th to the 9th vibrational quantum number (in eV) is

A diatomic molecule consists of two point masses,m₁ and m₂,separated by a distance r.If x is the distance from m₁ to the center of mass,find the moment of inertia in terms of x about an axis perpendicular to the molecular axis through the center of mass.

A molecule makes a transition from the J = 1 to the J = 0 rotational energy state.The wavelength of the emitted photon is 2.6 × 10−³ m.What is the moment of inertia of the molecule (in kg ⋅ m²) ?

The fundamental frequency of CO is 6.42 × 10¹³ Hz.If the atomic masses are 12 u and 16 u (1 u = 1.66 × 10−²⁷ kg) ,find the force constant (in N/m) for the diatomic molecule.

A diatomic molecule consists of two point masses,m₁ and m₂,separated by a distance r.If x is the distance from m₁ to the center of mass,find the moment of inertia in terms of x about an axis parallel to the molecular axis through the center of mass.

An oxygen molecule has a moment of inertia of 5 × 10−⁴⁶ kg ⋅ m².Calculate the bond length (in nm) .Recall that the atomic mass of oxygen is 16 u (1 u = 1.66 × 10−²⁷ kg) .