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Sound

Speed of Sound in a Gas

An The speed of sound in a gas is primarily a function of its temperature. Air is a mixture of gases and includes water vapor. The relative humidity of air has a slight effect on the speed of sound, while changes in air pressure have no real effect on the speed. A simple equation can be used to approximate the speed of sound. A more complex equation takes into account factors and can be used with other gases. The speed of sound in a gas is a specific fraction of the average speed of the gas molecules at the given temperature.

Questions you may have about sound are:

• What is the approximate equation for the speed of sound in air?

• What is the equation for the speed of sound in any gas?

• What is the relationship between speed of sound and velocity of molecules?

Speed in air

The speed of sound in air is approximately 344 meters/second, 1130 feet/sec. or 770 miles per hour at room temperature of 20oC (70oF). This speed is primarily a function of temperature. The only other factor that has any effect on the speed of sound in air is the amount of humidity in the air.

Air is a mixture

Air is a relatively fixed mixture of gases, primarily consisting of Nitrogen, Oxygen, Argon and Carbon Dioxide. It also includes varying amounts of water vapor or humidity. An increase in the amount of humidity in the air increases the speed only small amount. Since humidity can vary so much, and since the amount of change of speed with an extreme change in humidity is less the 0.5%, the speed of sound in air is usually simply measured in dry air, neglecting the effect of humidity.

Counter to intuition, changes in air pressure have no real effect on the speed of sound. Sound travels slower at higher altitudes because the temperature and relative humidity are lower, not because the air pressure is lower higher altitudes.

Equation for speed in air

The speed of sound in dry air in meters per second (m/s) is approximately equal to:

s = 331.4 + 0.6TC m/s

where

s = the speed of sound

Note that sometimes the letter c is used for the speed of sound. We'll use s to avoid confusion with the speed of light, which also uses c.

TC is the Celsius temperature.

Thus, if TC = 0oC, then s = 331.4 m/s. You can check that answer with the calculator below.

Speed in any gas

The speed of sound in any gas is a function of temperature, the molecular structure of the gas and its molecular mass.

Equation for speed in gas

Advanced students can study the theoretical equation for speed of sound in a gas to see what factors are involved in making the determination. Less advanced students should still go over the material to gain some understanding.

The equation for the speed of sound in a gas is:

s = sqrt(γ*R*TK/M)

where

s = the speed of sound

sqrt is the square root of the number in the parentheses (γ*R*TK/M)

Note that usually the square root is written √(γ*R*TK/M), but since the square root sign doesn't show up well on some web pages, we'll use sqrt

γ (Greek letter gamma) is the adiabatic constant for the gas, which is related to the type of atom or molecule, its specific heat and its degrees of freedom when it moves

For a gas made of single atoms, such as Argon (Ar) gas, γ = 5/3 = 1.67

For a gas made up of molecules consisting of two atoms, such as Oxygen (O2), γ = 7/5 = 1.4

For a gas made up of molecules consisting of multiple atoms, such as Carbon Dioxide (CO2) or gaseous water (H2O), γ = 4/3 = 1.3

R is a number called the Universal Gas Constant

R = 8.314 J/mol K (Joules/moles-degrees Kelvin)

TK is the absolute temperature of the gas in degrees Kelvin

Note that you can use TK = (273.15 + temperature in degrees C)

M is the molecular mass of the gas in kg/mol

The molecular mass is the atomic weight of the molecule divided by 1000; For example, CO2 has an atomic weight of 44 since the atomic weight of C is 12 and O2 is 32; Thus M = 44/1000 = 0.044 for CO2

These values can then be used to calculate the speed of sound for any gas at any temperature.

Application of gas equation to speed in air

You can use the above equation and values to calculate the speed of sound in air.

Since air is primarily made up of molecules consisting of two atoms--Nitrogen (N2) and Oxygen (O2)--then γ = 1.4.

Molecular mass

The molecular mass of air is an average of the molecular masses of the gases that make up air. The chart below shows the various components of air, the percent of each in air, and the molecular mass of the gas.

Finding the Average Molecular Mass of Air

Gas Component Percent by Volume Molecular Mass
Nitrogen (N2) 78.08% 0.028013
Oxygen (O2) 20.95% 0.031998
Argon (Ar) 0.93% 0.039948
Carbon Dioxide (CO2) 0.03% 0.043999
Total 99.99% 0.02895 Average

The average molecular weight for dry air is the sum of the Percent times the Molecular Weight for each molecule or component of air. That would be (0.7808 X 0.028013) + (0.2095 X 0.031998) +... and so on.

The result for air is the average M = 0.02895 kg/mol

Calculation for speed of sound in air

Substituting the values:

γ = 1.4

R = 8.314 J/mol K 286 m2/s2/oK

M = 0.02895 kg/mol

into s = sqrt(γ*R*TK/M), you get s = sqrt(1.4 * 8.314 * TK / 0.02895) = sqrt(13830.74)* sqrt(TK)

Thus s = 20.03 * sqrt(TK) = 20.03 * sqrt(273.15 + TC).

If TC = 0oC, then s = 20.03 * sqrt(273.15) = 20.03 * 16.53 = 331 m/s.

Speed in various gases

The speed of sound for various gases at 0oC:

Gas Speed (meters/second)
Air
331
Carbon Dioxide
259
Oxygen
316
Helium
965
Hydrogen
1290

Speed limited by velocity of particles

An interesting characteristic of the speed of sound in a gas is that the speed is a distinct fraction of the average speed or velocity of the molecules or atoms making up the gas.

The following equation shows that relationship:

s = v * sqrt(γ/3)

where

s = the speed of sound

v = the average velocity of the molecules at some given temperature

γ = the adiabatic constant for the gas

Thus for air, s = v * sqrt(1.4/3) = 0.68v. In other words, the speed of sound in air is 0.68 times the speed of the air molecules at a given temperature.

In conclusion

The speed of sound in a gas is a function of its temperature. Air is a mixture of gases and includes water vapor. A simple equation can be used to approximate the speed of sound. A more complex equation takes into account factors and can be used with other gases. The speed of sound in a gas is a specific fraction of the average speed of the gas molecules at the given temperature.