## Introduction

The density of air varies significantly with both pressure and temperature, so how do you find the density of air at different pressures and temperatures as needed for applications such as drag force calculations, frictional loss for flow in pipes or ducts, or air velocity determination with a pitot tube. The ideal gas equation and the ideal gas constant, which express the ideal gas law provide a means for calculating air density for different pressures and temperatures. Although this article is primarily about determining the density of air, the density of other gases at known temperature and gas pressure can also be estimated using the ideal gas law in the same way.

## The Ideal Gas Law Equation

You may be familiar with the ideal gas law in the following commonly used form: PV = nRT. This equation gives the relationship among the pressure, P, volume, V, and temperature, T, of n moles of an ideal gas, using an ideal gas constant, R. We can introduce the density of the gas into the equation by making use of the fact that molecular weight (MW) has units of mass/mole, and thus that n = m/MW, and that the ideal gas law can be written as: PV = (m/MW)RT. Noting that m/V is density, ρ, the equation can be written as P(MW) = (m/V)RT = ρRT. Solving for density gives the following equation for the density of an ideal gas in terms of its MW, pressure and temperature.

ρ = (MW)P/RT

with commonly used U.S. units as follows:

ρ = density of the gas in slugs/ft^{3},

MW = molecular weight of the gas in slugs/slugmole (or kg/kgmole, etc.) (**NOTE: MW of air = 29**),

P = absolute gas pressure in psia (NOTE: Absolute pressure equals pressure measured by a gauge plus atmospheric pressure.),

T = absolute temperature of the gas in ^{o}R (NOTE: ^{o}R = ^{o}F + 459.67)

R = ideal gas constant = 345.23 psia-ft^{3}/slugmole-^{o}R.

If we can treat air as an ideal gas, then the ideal gas law in this form can be used to calculate the density of air at different pressures and temperatures.

An Excel spreadsheet works well for ideal gas law/gas density calculations. For a downloadable Excel template to calculate gas density with the ideal gas law, see the article, "Excel Templates for Venturi and Orifice Flow Meter Calculations."

## But, Can I Use the Ideal Gas Law to Calculate the Density of Air?

This is a good question to ask, because the air in a pipe friction loss, drag force, or pitot tube calculation, is indeed a real gas, not an ideal gas. Fortunately, however, many real gases behave almost exactly like an ideal gas over a wide range of temperatures and pressures. The ideal gas law doesn’t work well for very low temperatures (approaching the critical temperature of the gas) or very high gas pressures (approaching the critical pressure of the gas). For many practical, real situations, however, the ideal gas law gives quite accurate values for the density of air (and many other gases) at different pressures and temperatures.

## SI Units for the Ideal Gas Law

The units on the ideal gas constant, R, can be converted to any convenient set of units, making the ideal gas law useable with any consistent set of units. For SI units the ideal gas law parameters are as follows:

ρ = density in kg/m^{3},

P = absolute gas pressure in pascals (N/m^{2}),

T = absolute temperature in ^{o}K (NOTE: ^{o}K = ^{o}C + 273.15)

R = ideal gas constant = 8314.5 Joules/kgmole-K

## Example: Calculation of the Density of Air at Different Pressures and Temperatures

**Example 1**: Calculate the density of air at 75^{o}F and a pressure of 14.9 psia.

**Solution**: T = 75 + 459.67^{o}R = 534.25^{o}R; substituting values into the ideal gas law:

ρ = (MW)P/RT = (29)(14.9)/[(345.23)(534.25)] (slugs/slugmole)(psia)/[(psia-ft^{3}/slugmole-^{o}R)(^{o}R)]

= **0.002343 slugs/ft**^{3}

**Example 2: **Calculate the density of air at 45^{o}C and a pressure of 100,000 pascals absolute

**Solution:** T = 45 + 273.15 K = 318.15 K; substituting values into the ideal gas law:

p = (MW)P/RT = (29)(100,000)/(8314.5)(318.15) (kg/kgmole)(N/m^{2})/[(joules/kgmole-K)(K)] = **1.10 kg/m**^{3}

## References

For Further Information:

1. Bengtson, Harlan H., *Flow Measurement in Pipes and Ducts, *An online course for Professional Engineers, https://www.online-pdh.com/engcourses/file.php/90/Course_Content_-_Flow_Measurement_in_Pipes_and_Ducts-1.pdf

2. Munson, B. R., Young, D. F., & Okiishi, T. H., *Fundamentals of Fluid Mechanics, *4^{th} Ed., New York: John Wiley and Sons, Inc, 2002.

3. Applied Thermodynamics ebook, https://www.taftan.com/thermodynamics/