Relation Between Speed Of Sound In Gas And RMS Speed Of Gas Molecules

The premise behind the kinetic theory of an ideal gas is that the molecules of a gas confined in a container clash randomly with one other and with the container’s walls. These collisions are referred to as elastic collisions. The root mean square speed or vrms, is used to express the velocity of molecules in a gas. As a result, the RMS speed of gas molecules and the speed of sound in the gas are related. Let us first know the formula of RMS of gas molecules and speed of sound in air formula and then proceed to find out the relation between the two.

Root-Mean-Square Velocities of Gas Molecules

Gaseous particles, according to Kinetic Molecular Theory, are in a state of continual random motion; individual particles travel at varying speeds, colliding and changing directions frequently. The distribution of velocities does not vary despite the fact that the velocity of gaseous particles is continually changing. Because we can’t determine the velocity of each individual particle, we frequently think in terms of the average behaviour of the particles. The velocities of particles travelling in opposing directions have opposite signs. Since the movement of particles/molecules of a gas is largely random, one can conclude that there will be as many gas molecules moving in one direction as in its opposite direction; this will make the average velocity of this collection of gas particles zero; because this value is unhelpful, the average of velocities can be determined using a different method.

We overcome the “directional” component of velocity and concurrently obtain the particles’ average velocity by squaring the velocities and calculating the square root. We now call the value the average speed because it does not include the direction of the particles. 

Following equation shows the formula of RMS of gas molecules:

vrms=√(3RT/M)

Where, 

vrms=root-mean-square velocity

M= molar mass of gas (in kg per mole)

R= molar gas constant

T= temperature (in Kelvin).

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Speed of Sound in Gas

Sound, like all waves, travels at a certain speed and has frequency and wavelength parameters. While viewing a fireworks show, you may see visual proof of sound speed. You can see the flash of an explosion long before you hear the sound or feel the pressure wave, demonstrating that sound has a definite speed and is significantly slower than light.

The speed of sound in a medium is determined by the rate at which vibrational energy may be conveyed through it. As a result, the derivation of the speed of sound in a medium is dependent on the medium and its condition. In general, the square root of the restoring force, or the elastic property, divided by the inertial property determines the speed of a mechanical wave in a medium.

The equation for the speed of sound in gas (see The Kinetic Theory of Gases) is:

v=√(γRT/M)

Where R is the universal gas constant, T is the temperature of the gas, and M is the molecular mass of the gas.

The faster the sound travels mean, the more stiff (or less compressible) the medium is. This is similar to how the frequency of simple harmonic motion is related to the stiffness of the oscillating item as measured by k, the spring constant. The slower the speed of sound means the higher the density of the medium. This is similar to the fact that the frequency of a simple harmonic motion is inversely related to m, the oscillating object’s mass. Because air is highly compressible, the speed of sound in air is low.

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Relationship between Speed of Sound in Gas and Formula of RMS of Gas Molecules

Let us deduce the Relation between the speed of sound in gas and the RMS speed of gas molecules.

v=√(γRT/M) is the formula for the speed of sound in a gas, where R is the universal gas constant, T is the temperature of the gas, and M is the molecular mass of the gas.

vrms=√(3RT/M) is the RMS velocity of gas molecules, where R is the universal gas constant, T is the gas’s temperature, and M is the gas’s molecular mass.

The following is a complete step-by-step solution:

Step 1: To get a relationship between the velocity of sound in gas and the formula of RMS of gas molecules, express the velocity of sound in gas and the RMS velocity of gas molecules.

v=√(γRT/M) is the formula for the velocity of sound in a gas. —-(1)

Where γ= adiabatic constant

R= universal gas constant

T= temperature of the gas

M= molecular mass of the gas.

vrms=√(3RT/M)is the RMS velocity of the gas. —-(2)

Where γ= adiabatic constant

R= universal gas constant

T= temperature of the gas

M= molecular mass of the gas.

Step 2: By dividing equation (1) by (2), we get:

v/vrms = √(γRT/M)/ √(3RT/M) 

Step 3: By cancelling out the comparable terms in the numerator and denominator, we get the required relationship:

v/vrms = √(γ/3)

v=vrms (γ/3)1/2

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Conclusion

The adiabatic constant, the universal gas constant, the temperature of the gas, and the molecular mass of the gas all influence the sound velocity in a gas. The formula of RMS of gas molecules is determined by the universal gas constant, the temperature of the gas, and the molecular mass of the gas, according to the kinetic theory of an ideal gas. Taking the ratio of the two velocities yields a relationship between them.