The greenhouse-gas carbon dioxide molecule CO2 strongly absorbs infrared radiation when its vibrational normal modes are excited by light at the normal-mode frequencies. CO₂ is a linear triatomic molecule, as shown in (Figure 1), with oxygen atoms of mass mo bonded to a central carbon atom of mass mc. You bonded to care know from chemistry that the atomic masses of carbon pective 12 and 16 and oxygen are, respectively, 12 and 16. Assume that the bond is an ideal spring with spring constant k There are two normal modes of this system for which oscillations take place along the axis. (You can ignore additional bending modes.) In this problem, you will find the normal modes and then use experimental data to determine the bond spring constant. Figure O mo C 2 mc. < 1 of 1 0 3 mo www. 1 X3 > Part A Let ₁, 2, and 3 be the atoms' positions measured from their equilibrium positions. First, use Hooke's law to write the net force on each atom. Pay close attention to signs! For each oxygen, the net force equals mod²x/dt². Carbon has a different mass, so it net force is mcd²x/dt². Define a² = k/mo and 32 = k/mc. Find an equation for the second derivative of ₁ coordinate. Express your answer in terms of some, all, or none of the variables ₁, 2, 3, and the constants a, B. 195] ΑΣΦ dt² Submit ▾ Part B Request Answer d²x₂ dt2 Find an equation for the second derivative of 2 coordinate. Express your answer in terms of some, all, or none of the variables ₁, 2, 3, and the constants a, B. ? VE] ΑΣΦ ?
The greenhouse-gas carbon dioxide molecule CO2 strongly absorbs infrared radiation when its vibrational normal modes are excited by light at the normal-mode frequencies. CO₂ is a linear triatomic molecule, as shown in (Figure 1), with oxygen atoms of mass mo bonded to a central carbon atom of mass mc. You bonded to care know from chemistry that the atomic masses of carbon pective 12 and 16 and oxygen are, respectively, 12 and 16. Assume that the bond is an ideal spring with spring constant k There are two normal modes of this system for which oscillations take place along the axis. (You can ignore additional bending modes.) In this problem, you will find the normal modes and then use experimental data to determine the bond spring constant. Figure O mo C 2 mc. < 1 of 1 0 3 mo www. 1 X3 > Part A Let ₁, 2, and 3 be the atoms' positions measured from their equilibrium positions. First, use Hooke's law to write the net force on each atom. Pay close attention to signs! For each oxygen, the net force equals mod²x/dt². Carbon has a different mass, so it net force is mcd²x/dt². Define a² = k/mo and 32 = k/mc. Find an equation for the second derivative of ₁ coordinate. Express your answer in terms of some, all, or none of the variables ₁, 2, 3, and the constants a, B. 195] ΑΣΦ dt² Submit ▾ Part B Request Answer d²x₂ dt2 Find an equation for the second derivative of 2 coordinate. Express your answer in terms of some, all, or none of the variables ₁, 2, 3, and the constants a, B. ? VE] ΑΣΦ ?
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Transcribed Image Text:The greenhouse-gas carbon dioxide molecule CO2
strongly absorbs infrared radiation when its vibrational
normal modes are excited by light at the normal-mode
frequencies. CO₂ is a linear triatomic molecule, as
shown in (Figure 1), with oxygen atoms of mass mo
bonded to a central carbon atom of mass mc. You
know from chemistry that the atomic masses of carbon
and oxygen are, respectively, 12 and 16. Assume that
the bond is an ideal spring with spring constant k.
There are two normal modes of this system for which
oscillations take place along the axis. (You can ignore
additional bending modes.) In this problem, you will find
the normal modes and then use experimental data to
determine the bond spring constant.
Figure
O
1
mo
1x₁
2
mc
1 of 1
3
mo
1Xz
Part A
Let x₁, x2, and 3 be the atoms' positions measured from their equilibrium positions. First, use Hooke's law to write the net force
on each atom. Pay close attention to signs! For each oxygen, the net force equals mod²x/dt². Carbon has a different mass, so its
net force is mcd²x/dt². Define a² = k/mo and 3² = k/mc. Find an equation for the second derivative of ₁ coordinate.
Express your answer in terms of some, all, or none of the variables ₁, 2, 3, and the constants a, ß.
17 ΑΣΦ
d²x₁
dt²
Submit
Part B
d²x2
dt²
Request Answer
Find an equation for the second derivative of 2 coordinate.
Express your answer in terms of some, all, or none of the variables ₁, 2, 3, and the constants a, B.
=
?
——| ΑΣΦ
?
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