FIGURE 10.23 showed the potential-energy curve for the O₂ molecule. Consider a molecule with the energy E₁ shown in the figure. a. What is the maximum speed of an oxygen atom as it oscillates back and forth? Don't forget that the kinetic energy is the total kinetic energy of the system. The mass of an oxygen atom is 16 u, where 1u=1 atomic mass unit =1.66×10⁻²⁷ kg .

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1. Identify the total energy of the system, E1, as given in the problem. The total energy is the sum of the potential energy (U) and the kinetic energy (K). At the turning points of the oscillation, the kinetic energy is zero, and the total energy is equal to the potential energy.
2. Determine the point on the potential-energy curve where the potential energy (U) is at its minimum. This corresponds to the equilibrium position of the molecule. At this point, the kinetic energy (K) is at its maximum because the total energy (E1) is constant.
3. Use the relationship between total energy, potential energy, and kinetic energy: \( K = E_1 - U \). At the equilibrium position, \( U \) is at its minimum, so \( K \) is maximized. Substitute the values of \( E_1 \) and \( U \) to find the maximum kinetic energy.
4. Relate the maximum kinetic energy to the speed of the oxygen atom using the formula for kinetic energy: \( K = \frac{1}{2} m v^2 \), where \( m \) is the mass of the oxygen atom and \( v \) is its speed. Solve for \( v \) using \( v = \sqrt{\frac{2K}{m}} \).
5. Substitute the given mass of the oxygen atom (16 u, where \( 1 \text{ u} = 1.66 \times 10^{-27} \text{ kg} \)) and the calculated maximum kinetic energy into the equation for \( v \) to find the maximum speed of the oxygen atom.

Key Concepts

Potential Energy Curve

A potential energy curve represents the potential energy of a system as a function of the position of its components. In the context of molecular oscillations, it illustrates how the energy of a molecule changes as it moves between different positions, indicating stable and unstable configurations. The shape of the curve helps determine the energy states available to the molecule and is crucial for understanding oscillatory motion.

Kinetic Energy

Kinetic energy is the energy possessed by an object due to its motion, calculated using the formula KE = 1/2 mv², where m is mass and v is velocity. In the context of oscillating molecules, the total kinetic energy of the system is related to the maximum speed of the atoms as they move back and forth. Understanding the relationship between kinetic and potential energy is essential for analyzing the motion of the molecule.

Mass and Atomic Units

Mass in physics is a measure of the amount of matter in an object, typically expressed in kilograms. In this question, the mass of an oxygen atom is given in atomic mass units (u), where 1 u equals 1.66 × 10⁻²⁷ kg. Converting between these units is important for calculations involving kinetic energy and speed, as it allows for consistent application of physical formulas.