Question

If the velocity of sound wave in humid air is \(\mathrm{v}_{\mathrm{m}}\) and that in dry air is \(\mathrm{v}_{\mathrm{d}}\), then \(\ldots \ldots\) (A) \(\mathrm{v}_{\mathrm{m}}>\mathrm{v}_{\mathrm{d}}\) (B) \(\mathrm{v}_{\mathrm{m}}<\mathrm{v}_{\mathrm{d}}\) (C) \(\mathrm{v}_{\mathrm{m}}=\mathrm{v}_{\mathrm{d}}\) \((\mathrm{D}) \mathrm{v}_{\mathrm{m}} \gg \mathrm{v}_{\mathrm{d}}\)

Short answer

The short answer based on the step-by-step solution is: The velocity of sound wave in humid air, \(v_m\), is greater than that in dry air, \(v_d\) (A) \(v_m > v_d\).

Step-by-step solution

Understand the speed of sound in air formula

The speed of sound in air depends on temperature, pressure, and humidity. One formula that takes these factors into account is: \[ v = \sqrt{\frac{\gamma R T}{M}} \] Where: - \(v\) is the speed of sound - \(\gamma\) is the adiabatic index (usually 1.4 for air) - \(R\) is the specific gas constant (287 J/(kg·K) for air) - \(T\) is the absolute temperature in Kelvin - \(M\) is the molar mass of the gas in kg/mol

Compare vm and vd using the formula

Since both vm and vd are velocities of sound in air, they will have the same values for \(\gamma\) and \(R\). The only difference between them is the molar mass, which will be affected by the humidity. Humid air has a lower molar mass than dry air because water vapor (molar mass = 18 g/mol) is lighter than the gases that make up dry air (mostly nitrogen and oxygen, with molar masses of 28 g/mol and 32 g/mol, respectively). We can rewrite the formula for vm and vd as follows: \[ v_m = \sqrt{\frac{\gamma R T}{M_m}} \] \[ v_d = \sqrt{\frac{\gamma R T}{M_d}} \]

Determine the relationship between vm and vd

Since humid air has a lower molar mass (Mm) than dry air (Md), the denominator in the formula for vm will be smaller than the denominator for vd. When you divide by a smaller number, the result is larger. Therefore: \[ v_m > v_d \] So the correct answer is: (A) \(v_m > v_d\)

Key concepts

Humidity Effect on Sound

Humidity can significantly influence the speed of sound in air. When the air is humid, it contains more water vapor. Water vapor is composed of lighter molecules compared to the primary components of dry air, nitrogen and oxygen. This difference in molecular weight affects how sound waves travel through the air.

• Humid air has water vapor, which has a molar mass of 18 g/mol.
• Dry air is mostly nitrogen and oxygen, with molar masses of 28 g/mol and 32 g/mol, respectively.

As a result, when air is more humid, the "average" molar mass of the air decreases. This reduction in molar mass means that sound waves can travel faster through humid air compared to dry air. Hence, we find that the velocity of sound in humid air (\( v_m \)) is greater than the velocity in dry air (\( v_d \)).

Molar Mass and Gas Composition

The molar mass and composition of gases directly affect sound propagation. Sound travels through a medium by vibrating the molecules within that medium. The speed at which these vibrations move depends on the medium's molecular characteristics.

• Lighter molecules (like water vapor) move faster and allow sound to travel quicker.
• Heavier molecules (like nitrogen and oxygen) slow down the movement of sound.

In the case of air, adding water vapor (with its lower molar mass) reduces the average molar mass of the gaseous mixture. This lightening effect enables sound to travel faster through humid air, as the lighter molecules offer less resistance to the wave's energy transfer. Therefore, when comparing different gases, such as humid versus dry air, it is crucial to consider their molecular makeup, which affects their density and thus the speed of sound.

Adiabatic Index in Gases

The adiabatic index, represented by \(\gamma\), plays a crucial role in determining the speed of sound, particularly in gases. This index is the ratio of specific heat at constant pressure to specific heat at constant volume: \(\gamma = \frac{C_p}{C_v}\). It describes how much the gas resists compression, which is integral to sound wave propagation.

For gases like air, this index typically has a value of 1.4. This value remains constant for both humid and dry conditions since it primarily depends on the type of gas and not on temporary atmospheric conditions like humidity.

• A higher adiabatic index means the gas compresses less under the sound wave, leading to a faster speed of sound.
• A lower adiabatic index means the gas compresses more, slowing down the sound wave.

It's important to note that while humidity and molar mass have a more pronounced effect on the speed of sound, the adiabatic index is a constant that influences how effectively sound energy is transmitted in the gas environment. By considering these factors, we can better understand the dynamics of how sound travels through different gaseous mediums.