Problem 37 - Master Resource Book in JEE Main Physics

Question

What is the de-Broglie wavelength of a nitrogen molecule in air at \(300 \mathrm{~K}\) ? Assume that the molecule is moving with the root-mean-square speed of molecules at this temperature. (Atomic mass of nitrogen \(=14.0076 \mathrm{u}\) ) \(\quad\) [NCERT] (a) \(0.01 \mathrm{~nm}\) (b) \(0.09 \mathrm{~nm}\) (c) \(0.03 \mathrm{~nm}\) (d) \(0.2 \mathrm{~nm}\)

Step-by-Step Solution

Answer

de Broglie wavelength is approximately 0.03 nm, so the answer is (c).

Step 1: Understand the problem

We need to find the de Broglie wavelength of a nitrogen molecule moving with its root-mean-square (rms) speed at 300 K. For a molecule, the rms speed is calculated using the formula related to kinetic theory of gases, and the de Broglie wavelength is given by a separate formula.

Step 2: Calculate the root-mean-square speed

The rms speed is calculated using \[ v_{rms} = \sqrt{\frac{3kT}{m}} \]where \( k = 1.38 \times 10^{-23} \ \mathrm{J/K} \) is the Boltzmann constant, \( T = 300 \ \mathrm{K} \) is the temperature, and \( m \) is the mass of the nitrogen molecule. Since a nitrogen molecule (\(N_2\)) has two nitrogen atoms, its mass is \(2 \times 14.0076 \ \mathrm{u} \), and 1 amu = \(1.66 \times 10^{-27} \ \mathrm{kg}\).

Step 3: Convert atomic mass to kilograms

We convert the mass of the nitrogen molecule from atomic mass units (amu) to kilograms.\[ m = 2 \times 14.0076 \times 1.66 \times 10^{-27} \ \mathrm{kg} \approx 4.65 \times 10^{-26} \ \mathrm{kg} \]

Step 4: Substitute values to find rms speed

Plugging the values in:\[ v_{rms} = \sqrt{\frac{3 \times 1.38 \times 10^{-23} \times 300}{4.65 \times 10^{-26}}} \]Solving this gives us \( v_{rms} \approx 517 \ \mathrm{m/s} \).

Step 5: Use the de Broglie wavelength formula

The de Broglie wavelength is given by \[ \lambda = \frac{h}{mv} \]where \( h = 6.626 \times 10^{-34} \ \mathrm{Js} \) is Planck's constant, \( m \) is the mass of the molecule, and \( v \) is the velocity calculated as rms speed (\( v_{rms} \)).

Step 6: Substitute values to find de Broglie wavelength

Using the formula:\[ \lambda = \frac{6.626 \times 10^{-34}}{4.65 \times 10^{-26} \times 517} \]Solving this results in \( \lambda \approx 0.029 \times 10^{-9} \ \mathrm{m} = 0.029 \ \mathrm{nm} \).

Step 7: Choose the nearest answer option

Comparing the calculated wavelength \(0.029 \ \mathrm{nm}\) with the provided options, it is closest to option (c) \(0.03 \ \mathrm{nm}\).

Key Concepts

Root-Mean-Square Speed

Root-mean-square speed, often abbreviated as rms speed, is a concept in physics that helps us understand how fast molecules in a gas are moving on average. It is a part of the kinetic theory of gases, which connects the motion of molecules to temperature. The formula for rms speed is:

\[v_{rms} = \sqrt{\frac{3kT}{m}}\]

• \(k\) is the Boltzmann constant \((1.38 \times 10^{-23} \ \mathrm{J/K})\).
• \(T\) represents the absolute temperature in Kelvin.
• \(m\) is the mass of one molecule of the gas.

This speed tells us the square root of the average of the squares of the speeds of all molecules present. It’s important when you want to find how quickly particles are moving in a substance at a given temperature. In this context, knowing the rms speed helps in calculating the de Broglie wavelength.

Kinetic Theory of Gases

The kinetic theory of gases is a fundamental theory that explains the macroscopic properties of gases, such as pressure and temperature, by considering their molecular composition and the motions of individual gas molecules.

Key points of this theory include:

• Molecules in a gas move in constant, random motion.
• Collisions between molecules and with the walls of their container are perfectly elastic, meaning they don’t lose energy during collisions.
• The average kinetic energy of gas molecules is proportional to the temperature of the gas in Kelvin.

The understanding of gas behavior through this theory is essential for calculating properties like rms speed and predicting how changes in temperature and mass affect such properties. It forms the backbone to derive the equations used in determining the de Broglie wavelength.

Planck's Constant

Planck's constant is a fundamental quantity in quantum mechanics, symbolized by \(h\), that relates the energy of photons to the frequency of their electromagnetic waves. It is used in various physical formulas, one of the most famous being the de Broglie wavelength equation.

The constant \(h\) has a value of \(6.626 \times 10^{-34} \ \mathrm{Js}\). In our exercise, Planck's constant is crucial for determining the wavelength linked to a moving nitrogen molecule.

In essence, the de Broglie wavelength equation \(\lambda = \frac{h}{mv}\) uses \(h\) to relate the mass \(m\) and velocity \(v\) of a particle to its wave-like properties. This powerful idea bridges classical physics and quantum mechanics, highlighting the dual particle-wave nature of matter.

Boltzmann Constant

The Boltzmann constant, represented as \(k\), is a bridge between macroscopic and microscopic physics. It connects the thermal energy of physical systems at the macroscopic scale to the energy of individual particles at the microscopic scale.

Its value is \(1.38 \times 10^{-23} \ \mathrm{J/K}\), and it appears in various expressions including the one for rms speed:
\[v_{rms} = \sqrt{\frac{3kT}{m}}\]
This constant is fundamental to statistics and thermodynamics. It provides a way to translate between the energy per molecule, expressed in Joules, and temperature, expressed in Kelvin. In solving the problem, we use the Boltzmann constant to calculate how fast nitrogen molecules are moving on average in a gas at a specific temperature, leading us to find the de Broglie wavelength. This highlights the Boltzmann constant's role in linking statistical physics with experimental measurements of macroscopic properties.