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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 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, B. dt² Submit 1977| ΑΣΦΑ -2α²x₁ + a²x₂ Previous Answers Request Answer X Incorrect; Try Again; 3 attempts remaining ?

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 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, B. dt² Submit 1977| ΑΣΦΑ -2α²x₁ + a²x₂ Previous Answers Request Answer X Incorrect; Try Again; 3 attempts remaining ?

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NOTE: Part A's answer previously given to me of -2α2x1 + α2x2 is incorrect. Part B's answer previously given to me of -2β2x2 + β2x1 + β2x3 is also incorrect. Can I please be given the correct answer this time?

 
**CO₂ Molecule and Vibrational Modes** Carbon dioxide (CO₂) is a greenhouse gas that strongly absorbs infrared radiation when its vibrational normal modes are excited by light at their normal-mode frequencies. As a linear triatomic molecule, CO₂ consists of oxygen atoms of mass \( m_O \) bonded to a central carbon atom of mass \( m_C \), as illustrated in Figure 1. **Chemical Context:** - Atomic masses: Carbon = 12, Oxygen = 16. - Bond: Assumed to be an ideal spring with spring constant \( k \). The system has two normal modes for oscillations along the axis (neglecting additional bending modes). This exercise involves finding these normal modes and using experimental data to determine the bond spring constant. **Figure Description:** - The figure shows a simplified model of a CO₂ molecule: - Two oxygen atoms labeled as \(1\) and \(3\) with mass \(m_O\). - A central carbon atom labeled as \(2\) with mass \(m_C\). - Springs symbolize the bonds with spring constant \(k\). - Positions \(x_1\), \(x_2\), \(x_3\) represent displacements from equilibrium. **Problem Statement:** **Part A:** - Given: - Let \(x_1\), \(x_2\), and \(x_3\) be the positions of the atoms measured from their equilibrium positions. - Use Hooke's law to write the net force expression for each atom. - For oxygen: \( m_O \frac{d^2 x}{dt^2} \). - For carbon: Different mass means the net force is \( m_C \frac{d^2 x}{dt^2} \). - Define: - \( \alpha^2 = \frac{k}{m_O} \) - \( \beta^2 = \frac{k}{m_C} \) - Objective: Find the equation for the second derivative of the \( x_1 \) coordinate. - Express your answer in terms of \( x_1, x_2, x_3 \), and constants \( \alpha, \beta \). _Example Attempt:_ \(-2\alpha^2 x_1 + \alpha^2 x_2\) **Response:** - Feedback: Incorrect response; 3 attempts remaining. **
Transcribed Image Text:**CO₂ Molecule and Vibrational Modes** Carbon dioxide (CO₂) is a greenhouse gas that strongly absorbs infrared radiation when its vibrational normal modes are excited by light at their normal-mode frequencies. As a linear triatomic molecule, CO₂ consists of oxygen atoms of mass \( m_O \) bonded to a central carbon atom of mass \( m_C \), as illustrated in Figure 1. **Chemical Context:** - Atomic masses: Carbon = 12, Oxygen = 16. - Bond: Assumed to be an ideal spring with spring constant \( k \). The system has two normal modes for oscillations along the axis (neglecting additional bending modes). This exercise involves finding these normal modes and using experimental data to determine the bond spring constant. **Figure Description:** - The figure shows a simplified model of a CO₂ molecule: - Two oxygen atoms labeled as \(1\) and \(3\) with mass \(m_O\). - A central carbon atom labeled as \(2\) with mass \(m_C\). - Springs symbolize the bonds with spring constant \(k\). - Positions \(x_1\), \(x_2\), \(x_3\) represent displacements from equilibrium. **Problem Statement:** **Part A:** - Given: - Let \(x_1\), \(x_2\), and \(x_3\) be the positions of the atoms measured from their equilibrium positions. - Use Hooke's law to write the net force expression for each atom. - For oxygen: \( m_O \frac{d^2 x}{dt^2} \). - For carbon: Different mass means the net force is \( m_C \frac{d^2 x}{dt^2} \). - Define: - \( \alpha^2 = \frac{k}{m_O} \) - \( \beta^2 = \frac{k}{m_C} \) - Objective: Find the equation for the second derivative of the \( x_1 \) coordinate. - Express your answer in terms of \( x_1, x_2, x_3 \), and constants \( \alpha, \beta \). _Example Attempt:_ \(-2\alpha^2 x_1 + \alpha^2 x_2\) **Response:** - Feedback: Incorrect response; 3 attempts remaining. **
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