Use the de Broglie relationship to determine the wavelengths of the following objects: (c) a lithium atom moving at 2.5 × 10⁵ m/s (d) an ozone (O₃) molecule in the upper atmosphere moving at 550 m/s.

검증된 단계별 안내

Identify the de Broglie wavelength formula: \( \lambda = \frac{h}{mv} \), where \( \lambda \) is the wavelength, \( h \) is Planck's constant (\( 6.626 \times 10^{-34} \) Js), \( m \) is the mass of the object, and \( v \) is the velocity.

Determine the mass of the ozone molecule (\( \text{O}_3 \)). The molar mass of ozone is approximately 48.00 g/mol. Convert this to kilograms by dividing by 1000, resulting in 0.048 kg/mol.

Calculate the mass of a single ozone molecule by dividing the molar mass by Avogadro's number (\( 6.022 \times 10^{23} \) molecules/mol).

Substitute the values into the de Broglie equation: \( \lambda = \frac{6.626 \times 10^{-34} \text{ Js}}{m \times 550 \text{ m/s}} \), where \( m \) is the mass of a single ozone molecule calculated in the previous step.

Simplify the expression to find the wavelength \( \lambda \) of the ozone molecule.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

de Broglie Wavelength

The de Broglie wavelength is a concept in quantum mechanics that relates the wavelength of a particle to its momentum. It is given by the formula λ = h/p, where λ is the wavelength, h is Planck's constant, and p is the momentum of the particle. This relationship implies that all matter exhibits wave-like properties, especially at the atomic and subatomic levels.

Momentum

Momentum is a physical quantity defined as the product of an object's mass and its velocity (p = mv). In the context of the de Broglie relationship, momentum is crucial because it determines the wavelength of a moving particle. For an ozone molecule, knowing its mass and velocity allows us to calculate its momentum and subsequently its wavelength.

Planck's Constant

Planck's constant (h) is a fundamental constant in quantum mechanics, approximately equal to 6.626 x 10^-34 Js. It plays a key role in the quantization of energy and is essential in the de Broglie wavelength formula. Understanding Planck's constant is vital for calculating the wavelengths of particles, as it bridges the gap between classical and quantum physics.

Use the de Broglie relationship to determine the wavelengths of the following objects: (c) a lithium atom moving at 2.5 × 105 m/s (d) an ozone (O3) molecule in the upper atmosphere moving at 550 m/s.