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 CP15.82, 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. The symmetric stretch frequency is known to be 4.00 X 10¹³ Hz. What is the spring constant of the C - O bond? Use 1 u = 1 atomic mass unit = 1.66 X 10⁻²⁷ kg to find the atomic masses in SI units. Interestingly, the spring constant is similar to that of springs you might use in the lab.
17. Periodic Motion
Spring Force (Hooke's Law)
Problem 82h
Textbook Question
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 CP15.82, 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. Use the frequency of the symmetric stretch to predict the frequency of the antisymmetric stretch. The measured frequency is 7.05 × 1013 Hz so your prediction is close but not perfect. The reason is that the bonds are not ideal springs but have a slight amount of anharmonicity. Nonetheless, you’ve learned a great deal about the CO₂ molecule from a simple model of oscillating masses.
Verified step by step guidance1
Step 1: Understand the system. The CO₂ molecule consists of two oxygen atoms (mass mₒ = 16 u) and one carbon atom (mass m_c = 12 u) arranged linearly. The bonds between the atoms are modeled as ideal springs with spring constant k. The system has two normal modes of vibration along the axis: symmetric stretch and antisymmetric stretch.
Step 2: Write the equations of motion for the system. Let x₁, x₂, and x₃ represent the displacements of the oxygen atom 1, carbon atom, and oxygen atom 3, respectively. Using Newton's second law, the forces on each mass due to the springs can be expressed as:
mₒ d²x₁/dt² = -k(x₁ - x₂) ,
m_c d²x₂/dt² = -k(x₂ - x₁) - k(x₂ - x₃) ,
mₒ d²x₃/dt² = -k(x₃ - x₂) .
Step 3: Solve for the normal modes. For the symmetric stretch, both oxygen atoms move in the same direction while the carbon atom moves in the opposite direction. Assume x₁ = x₃ and x₂ = -x₁. Substitute these into the equations of motion to find the frequency of the symmetric stretch. For the antisymmetric stretch, the oxygen atoms move in opposite directions while the carbon atom remains stationary. Assume x₁ = -x₃ and x₂ = 0. Substitute these into the equations of motion to find the frequency of the antisymmetric stretch.
Step 4: Relate the frequencies to the spring constant. The frequency of oscillation for a normal mode is given by ω = √(k_eff/m_eff), where k_eff is the effective spring constant and m_eff is the effective mass for the mode. Use the measured frequency of the symmetric stretch (7.05 × 10¹³ Hz) to calculate the spring constant k. Then use this value of k to predict the frequency of the antisymmetric stretch.
Step 5: Compare the predicted frequency of the antisymmetric stretch to the measured frequency. Note that the slight discrepancy arises due to anharmonicity in the bonds, which deviates from the ideal spring model. This demonstrates the limitations of the model and provides insight into the physical behavior of the CO₂ molecule.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Normal Modes of Vibration
Normal modes of vibration refer to the specific patterns in which a system oscillates when it is disturbed. In the case of the CO₂ molecule, there are two primary normal modes: symmetric and antisymmetric stretches. These modes describe how the atoms in the molecule move relative to each other, and understanding them is crucial for analyzing the molecule's vibrational behavior and its interaction with infrared radiation.
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The Normal Force
Spring Constant (k)
The spring constant (k) is a measure of a spring's stiffness, defined by Hooke's Law, which states that the force exerted by a spring is proportional to its displacement. In the context of the CO₂ molecule, the bonds between the carbon and oxygen atoms are modeled as springs, and the spring constant is essential for calculating the vibrational frequencies of the normal modes. A higher spring constant indicates a stiffer bond, leading to higher vibrational frequencies.
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Anharmonicity
Anharmonicity refers to the deviation of a system from the ideal harmonic oscillator model, where the restoring force is directly proportional to displacement. In real molecular systems, such as CO₂, anharmonic effects arise due to the non-linear nature of molecular interactions at larger displacements. This results in discrepancies between predicted and observed vibrational frequencies, as seen in the CO₂ molecule's vibrational modes, where the measured frequency differs from the ideal predictions.
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