Question

The density of a sample of air is \(1.205 \mathrm{~kg} / \mathrm{m}^{3}\), and the bulk modulus is \(1.42 \cdot 10^{5} \mathrm{~N} / \mathrm{m}^{2}\) a) Find the speed of sound in the air sample. b) Find the temperature of the air sample.

Short answer

Answer: The speed of sound in the air sample is approximately 331.25 m/s, and the temperature is approximately 273.1 K.

Step-by-step solution

Find the speed of sound in the air sample

To find the speed of sound in the air sample, we will use the following formula: v = sqrt(B/ρ) Where v is the speed of sound, B is the bulk modulus, and ρ is the density. Given values: B = 1.42 * 10^5 N/m^2 ρ = 1.205 kg/m^3 Plug in the given values into the formula: v = sqrt((1.42 * 10^5 N/m^2) / (1.205 kg/m^3)) Now, calculate the speed of sound (v): v ≈ 331.25 m/s The speed of sound in the air sample is approximately 331.25 m/s.

Find the temperature of the air sample

To find the temperature of the air sample, we will use the following formula: v = sqrt(γRT / ρ) Where v is the speed of sound, γ is the adiabatic index, R is the specific gas constant for air, T is the temperature, and ρ is the density. Given values: v = 331.25 m/s γ = 1.4 (adiabatic index for air) R = 287 J/(kg*K) (specific gas constant for air) ρ = 1.205 kg/m^3 Rewrite the formula to isolate T: T = (v^2 * ρ) / (γR) Plug in the known values into the formula: T = ((331.25 m/s)^2 * (1.205 kg/m^3)) / (1.4 * 287 J/(kg*K)) Now, calculate the temperature (T): T ≈ 273.1 K The temperature of the air sample is approximately 273.1 K.

Key concepts

Bulk Modulus

The bulk modulus is a fundamental concept in understanding how materials compress under pressure. It measures a material's resistance to uniform compression. Essentially, it tells us how difficult it is to change the volume of a substance when applying pressure. In the context of sound waves traveling through a medium like air, the bulk modulus is crucial.

• When sound waves travel, they create regions of compression and rarefaction in the medium.
• The bulk modulus determines how these regions affect the medium's volume.
• A high bulk modulus means the medium is less compressible and can support faster sound waves.

In the original exercise, the given bulk modulus of the air sample is used to calculate the speed of sound, illustrating its direct impact on sound propagation.

Density of Air

Density is a measure of how much mass is present in a given volume. In air, density affects how sound moves through it. In simple terms, the denser the air, the closer together its molecules are, impacting the speed of sound.

• More molecules in a given space lead to more collisions, potentially slowing the sound down.
• However, increased density also means more energy is transferred per collision, offsetting the slowing effect.

In the provided exercise, we have the air density at 1.205 kg/m³. This value is plugged into the formula for sound speed, showing how density works hand in hand with bulk modulus to determine the speed. Understanding air density is crucial to predicting and modeling sound propagation in different atmospheres.

Adiabatic Index

The adiabatic index, commonly represented as \( \gamma \) (gamma), is vital in thermodynamics and plays a role in sound speed calculations. It represents the ratio of specific heats at constant pressure and volume. For air, \( \gamma \) is typically 1.4, but here's why it matters:

• It reflects how air pressure changes with volume without heat transfer (adiabatic process).
• This index is crucial when calculating sound speed because sound propagation is an adiabatic process.
• The adiabatic index impacts how efficiently pressure changes translate to volume changes in sound waves.

In the exercise, \( \gamma \) is used to find the air temperature through the relationship with speed, revealing its tie to both thermodynamics and acoustic dynamics.

Specific Gas Constant

The specific gas constant, denoted as \( R \), ties temperature, volume, and pressure in a gas. Each distinct gas has its specific gas constant, and for air, this value is 287 J/(kg*K). This constant is crucial when determining properties like temperature via the ideal gas law and sound speed estimation.

• \( R \) represents energy per degree increase for one mole of the gas.
• In terms of sound, it helps relate the speed of sound to temperature through the equation \( v = \sqrt{\gamma RT / \rho} \).
• This relationship suggests that as temperature increases, the speed of sound increases, due to higher kinetic energy of air molecules.

Understanding the specific gas constant is essential for linking thermodynamic properties to acoustic phenomena, as shown in calculating the temperature from sound speed in the exercise.