Problem 124 - Advanced Problems in Physical Chemistry for Competitive Examinations

Question

The momentum of a photon of wavelength \(6626 \mathrm{~nm}\) will be (a) \(10^{-28} \mathrm{~kg} \mathrm{~ms}^{-1}\) (b) \(10^{-25} \mathrm{~kg} \mathrm{~ms}^{-1}\) (c) \(10^{31} \mathrm{~kg} \mathrm{~m}^{-1}\) (d) zero

Step-by-Step Solution

Answer

The momentum of the photon is closest to option (b) 10^{-25} kg ms^{-1}.

Step 1: Understanding the Concept of Photon Momentum

The momentum p of a photon is given by the equation p = h/λ, where h is Planck's constant (approximately 6.626 x 10^-34 Js) and λ is the wavelength of the photon.

Step 2: Converting Wavelength from Nanometers to Meters

First, we need to convert the wavelength from nanometers to meters to use in the equation. We do this by multiplying by 10^-9, since there are 10^9 nanometers in a meter.

Step 3: Calculating the Momentum of the Photon

Next, we substitute the given wavelength, after conversion, and the value of Planck's constant into the momentum equation to find the photon's momentum.

Step 4: Comparing the Calculated Momentum to the Given Options

Compare the calculated momentum with the given options to determine which one is correct.

Key Concepts

Planck's Constant

Planck's constant (\( h \)) is a fundamental quantity in quantum mechanics, with a value of approximately 6.626 x 10^-34 Joule seconds (Js). This constant plays a pivotal role in the quantum theory of light and matter, linking the energy (\( E \) of a photon to its frequency (\( u \)) through the formula \( E = hu \). Understanding the significance of Planck's constant is crucial for a number of quantum phenomena, including the photoelectric effect and the particle-like behavior of photons.

The concept of a 'quantum' as introduced by Max Planck suggests that energy is exchanged in discrete amounts, or quanta, rather than in a continuous manner. This has profound implications for how light and matter interact at microscopic scales. Planck's constant also forms the basis of Heisenberg's uncertainty principle, which speaks to the limits of precision with which certain pairs of physical properties, like position and momentum, can be known simultaneously.

Wavelength Conversion

Wavelength conversion is a necessary step when working with photon-related equations as it ensures compatibility with constants that are usually defined in standard units. Typically, a photon's wavelength is measured in nanometers (nm), especially in the context of visible light. However, most fundamental physical constants, including Planck's constant, are defined in terms of meters. Therefore, it is essential to convert from nanometers to meters for calculations.

To convert the wavelength from nanometers to meters, you multiply by \(10^{-9}\) because one meter is equivalent to \(10^9\) nanometers. For example, a wavelength of \(6600\) nm, when converted to meters, becomes \(6600 \times 10^{-9}\) meters or \(6.6 \times 10^{-6}\) meters. Memorizing common conversion factors or understanding the relationship between the units can simplify these conversions in various physics problems.

Momentum Calculation

text)) should both be in meters to obtain the momentum in the correct units of \( kg \times m/s \)). Calculating the momentum of a photon involves understanding the interplay of these quantum mechanical concepts and carefully applying them to the specific parameters of the problem at hand.