Question

At what frequency would the wavelength of sound be on the order of the mean free path in nitrogen at \(1.02\) atm pressure and \(18.0^{\circ} \mathrm{C} ?\) Take the diameter of the nitrogen molecule to be \(315 \mathrm{pm}\)

Short answer

The frequency (\( f \)) of the sound can be ultimately calculated using the formula \( f = v/ \lambda \), once both \( v \) (velocity of sound in nitrogen) and \( \lambda \) (mean free path in nitrogen) are determined. The final answer will depend on these calculated values. Be aware that this is a relatively complex problem requiring knowledge of several physics and chemistry concepts.

Step-by-step solution

Calculate the Mean Free Path

The mean free path \( \lambda \) of a particle can be calculated using the following formula: \( \lambda = \frac{kT}{\sqrt{2} \pi d^2 P} \) where \( k \) is Boltzmann constant (\(1.38 \times 10^{-23} J/K\)), \( T \) is the temperature in Kelvin, \( d \) is the diameter of the molecule, and \( P \) is the pressure. First, convert the given pressure from atm to Pascals (1 atm = 101325 Pa) and the temperature from Celsius to Kelvin (18°C = 291.15K). With \( d = 315 \times 10^{-12} m \), \( P = 1.02 \times 101325 Pa \), and \( T = 291.15K \), substitute these values into the formula to calculate the mean free path.

Calculate the Speed of Sound in Nitrogen

The speed of sound in an ideal gas is given by \( v = \sqrt{\gamma RT/M} \), where \( \gamma \) is the adiabatic index (1.4 for diatomic gases like nitrogen), \( R \) is the universal gas constant (\(8.314J/(mol*K)\)), \( T \) is the temperature in Kelvin, and \( M \) is the molar mass of nitrogen (\( 28 \times 10^{-3} kg/mol \)). Substitute the known values into the formula to find the speed of sound in nitrogen.

Find the Frequency

The frequency \( f \) of the sound is given by \( f = v/ \lambda \), where \( v \) is the speed of sound and \( \lambda \) is the mean free path calculated in Step 1. Substituting the calculated values into this formula will yield the frequency of sound.

Key concepts

Mean Free Path

In gas theory, the concept of mean free path, denoted as \( \lambda \), refers to the average distance a molecule travels between collisions. Imagine the air around you as a dance floor where molecules are dancers, moving and twirling; the mean free path would be the average 'steps' a dancer takes before bumping into another dancer. This property is crucial in understanding phenomena like diffusion and viscosity, as well as in the context of sound waves, where it can influence sound propagation.
The mean free path depends on the conditions of the gas, such as pressure, temperature, and molecular size. In the textbook exercise, the mean free path is calculated using Boltzmann's equation, which relates these variables. As the pressure increases or the temperature decreases, the mean free path shortens since molecules collide more frequently. This direct relationship between mean free path and the properties of a gas helps us understand how sound waves interact at the molecular level and plays a role in determining sound frequency when the mean free path is comparable to the wavelength of sound being studied.

Speed of Sound in Nitrogen

The speed of sound isn't a one-size-fits-all number; it varies with the medium through which the sound travels. For gases like nitrogen, which is abundant in our atmosphere, the speed of sound can be determined using a specific formula. This speed is a measure of how fast sound waves can move through nitrogen gas.
To understand how this speed is calculated, think of a wave moving through a spring. The properties of the spring, such as its tension and mass, dictate how quickly a wave travels along its length. Similarly, the speed of sound in a gas like nitrogen is influenced by factors including the temperature of the gas and its molecular composition. The exercise reveals that this speed hinges on the adiabatic index, gas constant, temperature, and molar mass of nitrogen. These factors combine in a precise formula that outputs the speed at which sound waves move in a nitrogen atmosphere. Knowing the speed of sound in nitrogen is fundamental for a range of practical applications, from designing acoustic equipment to understanding environmental acoustics.

Sound Frequency Calculation

The frequency of a sound wave, denoted as \( f \), is essentially the number of wave crests that pass a given point per second. Imagine the oscillation of a guitar string; each vibration creates a wave that we hear as a note. The number of vibrations per second is the frequency of the note, which determines its pitch. In acoustics, this ties directly to the concept of wavelength, which is the distance between successive crests of a wave.
The textbook exercise poses a fascinating question: at what frequency does the wavelength of sound match the mean free path of nitrogen molecules? To find this frequency, one must first understand the relationship between speed of sound, wavelength, and frequency. The formula \( f = v/ \lambda \) connects these concepts and tells us that to produce a sound wave with a wavelength equal to the mean free path, the frequency should be the speed of sound divided by the mean free path. By calculating the speed of sound in nitrogen and knowing the mean free path, one can determine the frequency of sound that will have a wavelength comparable to the average distance molecules in nitrogen travel before colliding. This is a unique situation where molecular physics meets wave physics, showcasing the interconnectedness of scientific principles.