How to Calculate Air Density: Steps, Examples & Practice Problems
The density of a body is the mass of the body per unit volume. Therefore, density of air is the mass of air contained in a unit volume of a container. There are some factors that the density of the air depends upon. In the case of the atmosphere, the density of air varies from place to place.
Here, we will discuss how to calculate the value of air density at different temperatures and pressures using the density of air formula.
Amedeo Avogadro
Lorenzo Romano Amedeo Carlo Avogadro (9 August 1776 – 9 July 1856) was an Italian scientist best known for his contribution to molecular theory, which is now known as Avogadro's law, which states that equal volumes of gases at the same temperature and pressure contain equal numbers of molecules.
Amadeo Avogadro, discovered it in the early 1800s. In layman's terms, he discovered that no matter what gas is in the container, a fixed volume of gas, say one cubic metre, at the same temperature and pressure, will always have the same number of molecules. The majority of first-year chemistry textbooks describe how this works.
What is Density of Air ?
The density of air is defined as the mass per unit volume of air. The S.I unit of density of air is kg/m3. Consider an air taken in a container of volume 1 m3, then the mass of the air inside that container having volume 1 m3 is equal to the density of air.There are some factors that the density of air depends upon.
Variation of Density of Air at Different Altitudes
As we know the density of air in the atmosphere changes its value from place to place. As we are moving up away from the sea level, the density of air decreases. So the density of air is maximum at sea level. That is why mountaineers carry oxygen cylinders because the density of air is less at higher altitudes. So they need oxygen cylinders for easy breathing.
Derivation of Density of Air Formula
To derive the density of air formula , we use the ideal gas equation given by
$\Rightarrow PV=nRT$ …(1)
Where,
P- Pressure of the the gas
V- Volume of the gas
n - Number of moles of the gas
R - Universal gas constant
T - Temperature of the gas.
The formula to calculate the number of moles is given by ,
$\Rightarrow n=\dfrac{M}{M_0}$
Where,
n- Number of moles of the the gas
M- Mass of the the gas
M0- Molecular mass of the gas.
Substitute the equation for number of moles (n) in the ideal gas equation(1) and simplify
$\Rightarrow PV=nRT$
$\Rightarrow PV=\dfrac{M}{M_0}RT$
$\Rightarrow PM_0=\left(\dfrac{M}{V}\right)RT$…(2)
The formula to calculate the density of air is given by,
$\Rightarrow D=\dfrac{M}{V}$
Where,
D - Density of the the gas
M- Mass of the the gas
V- Volume of the the gas
So, we can substitute density of the gas (D) for the expression M/V in the equation (2) and simplify to get the formula for density of the gas.
$\Rightarrow PM_0=DRT$
$\Rightarrow \dfrac{PM_0}{RT}=D$
$\Rightarrow D=\dfrac{PM_0}{RT}$
So, we obtain the air density formula as,
$\Rightarrow D=\dfrac{PM_0}{RT}$
Effect of Temperature and Pressure on Density of air
We have derived the air density formula given by,
$\Rightarrow D=\dfrac{PM_0}{RT}$
So, whenever the pressure and temperature of air changes, the density of the air changes. The density of air is inversely proportional to the temperature of the air. Therefore, whenever the temperature of the air increases, the density of the air decreases and whenever the temperature of the air decreases,
From the air density formula, we can understand that density of the air is directly proportional to the pressure of the gases. That means that when the pressure of the air increases, then the air density value also increases. Also, when the pressure of the air decreases, the density of the air decreases.
The density of air at standard temperature and pressure STP ( 00 C and 1 atm ) is equal to 1.293 kg/m3.
Density of Air at Room Temperature
Let us calculate the air density value at room temperature using the air density formula at sea level. The room temperature is taken as 25oC and the atmospheric pressure is taken as 110325 Pa. The formula to calculate the density of the air is given by,
$\Rightarrow D=\dfrac{PM_0}{RT}$
Where,
D- Density of the the gas
M0- Molecular mass of the air
P- Pressure of the the gas
R - Universal gas constant
T - Temperature of the gas.
The molecular mass of air is 28.96 g/mol and the value of universal gas constant is 8.314 JK-1mol-1.
Substitute each value in the air density formula to calculate the density of air.
$\Rightarrow D=\dfrac{PM_0}{RT}$
$\Rightarrow D=\dfrac{110325 ~Pa \times 28.96~ g/mol \times 10^{-3}}{8.314 ~J/Kmol \times 298~ K}$
$D=1.289~kg/m^3$
Example:
What is the density of air at 320 K and 2105 Pa? The molecular mass of the air is taken as 28.96 g/mol.
Sol: The formula to calculate the density of air is given by,
$\Rightarrow D=\dfrac{PM_0}{RT}$
Where,
D- Density of the the gas
M0- Molecular mass of the air
P- Pressure of the the gas
R - Universal gas constant
T - Temperature of the gas.
The value of the universal gas constant is 8.314 JK-1mol-1. Substitute the values for the terms in the air density formula and simplify to calculate the density of air.
$\Rightarrow D=\dfrac{PM_0}{RT}$
$\Rightarrow D=\dfrac{2\times10^5~Pa \times 28.96~ g/mol \times 10^{-3}}{8.314 ~J/Kmol \times 320~ K}$
$D=2.178~kg/m^3$
What is the density of air when the pressure of the air is doubled keeping the temperature constant?
Sol: We know the relationship between density of air and pressure given by the formula,
$\Rightarrow D=\dfrac{PM_0}{RT}$
Where,
D- Density of the the gas
M0- Molecular mass of the air
P- Pressure of the the gas
R - Universal gas constant
T - Temperature of the gas.
Therefore , density of air is directly proportional to the pressure of the air.So, when the the pressure of the air is doubled keeping the temperature constant, then the new density of air is given by,
$\Rightarrow D'=\dfrac{(2P)M_0}{RT}$
Where,
D’ - the new density of air when pressure is doubled at constant temperature.
$\Rightarrow D'=2\times\dfrac{PM_0}{RT}$
Therefore, the new density of the air becomes doubled when the pressure of the air is doubled keeping the temperature constant.
Interesting Facts
Objects traveling through denser, or "heavier," air will slow down more because the object must, in effect, push away more or heavier molecules.
The term for this type of air resistance is "drag," and it increases as air density rises. Home runs travel farther in the less dense air in high-altitude Denver than in lower-altitude ballparks, according to baseball players. Reduced drag causes the ball to slow down more slowly, allowing it to go further.
A race car is slowed by cool, dense air, yet some race cars benefit from it. Cars built from the ground up for racing are essentially upside-down airplane wings that push down on the track as the air drives them around turns, improving their grip. The more dense the air, the harder it pushes them down.
When the density of the air falls, aircraft pilots perform worse than baseball players.
Conclusion
The density of air is the mass of the air for a unit volume. The density of air depends upon the temperature and pressure of the air. When the temperature of the air increases, the density of the air decreases. The density of air increases when the pressure of the air increases. The density of air at room temperature and normal atmospheric pressure is 1.289 kg/m3.
FAQs on Density of Air: Key Concepts, Formulas & Importance
1. What does the term 'air density' mean in Physics?
Air density is a fundamental physical property that measures the mass of air molecules contained within a specific unit of volume. In simpler terms, it tells us how tightly packed the air is. It is typically represented by the Greek letter rho (ρ). A higher density means there are more air molecules in a given space, while a lower density means there are fewer.
2. What is the standard value for the density of air?
According to the International Standard Atmosphere (ISA), the standard density of dry air at sea level is approximately 1.225 kilograms per cubic meter (kg/m³). This value is measured at a standard temperature of 15°C (59°F) and a standard pressure of 1013.25 millibars. In other units, this is equivalent to 0.001225 grams per cubic centimetre (g/cm³).
3. How is the density of air calculated using the ideal gas law?
For dry air, its density can be calculated using a form of the ideal gas law. The formula is: ρ = P / (R_specific × T). Here is what each variable represents:
- ρ (rho) is the air density in kg/m³.
- P is the absolute pressure in Pascals (Pa).
- R_specific is the specific gas constant for dry air, which is approximately 287.05 J/(kg·K).
- T is the absolute temperature in Kelvin (K).
4. What are the main factors that affect the density of air?
The density of air is not constant and is primarily affected by three main factors:
- Temperature: Air density is inversely proportional to temperature. When air is heated, its molecules move faster and spread out, making it less dense. This is why hot air rises.
- Pressure: Air density is directly proportional to pressure. At higher pressures, air molecules are forced closer together, increasing the mass within a given volume and thus increasing density.
- Humidity (Water Vapour): The addition of water vapour makes air less dense. This is because a water molecule (H₂O) has less mass than the average nitrogen (N₂) or oxygen (O₂) molecule it displaces.
5. Why does air density decrease with an increase in altitude?
Air density decreases as altitude increases mainly due to two reasons. First, at higher altitudes, there is less overlying air pushing down, which results in lower atmospheric pressure. Since density is directly proportional to pressure, lower pressure leads to lower density. Second, Earth's gravity pulls air molecules towards the surface, causing a higher concentration of them at lower altitudes, making the air denser at sea level compared to on a mountain top.
6. How does humidity make air less dense?
This is a common point of confusion. Moist air is actually less dense than dry air at the same temperature and pressure. Air is mostly composed of nitrogen (N₂, molar mass ≈ 28 g/mol) and oxygen (O₂, molar mass ≈ 32 g/mol). When humidity increases, lighter water vapour molecules (H₂O, molar mass ≈ 18 g/mol) displace some of the heavier nitrogen and oxygen molecules. This substitution reduces the total mass in a given volume of air, thereby decreasing its overall density.
7. Why is understanding air density important in fields like meteorology and aviation?
Understanding air density is critical for practical applications:
- In Aviation: Air density directly impacts an aircraft's performance. Denser air provides more lift for the wings and allows engines to produce more thrust. Pilots must account for changes in density due to altitude and temperature for safe takeoff and flight.
- In Meteorology: Differences in air density drive weather patterns. Warm, less dense air rises, while cold, denser air sinks. This movement creates convection currents, leading to wind, clouds, and storms.
8. What is the density of dry air at STP?
At Standard Temperature and Pressure (STP), which is defined as a temperature of 0°C (273.15 K) and a pressure of 1 standard atmosphere (101.325 kPa), the density of dry air is approximately 1.293 kg/m³. This value is slightly higher than the ISA standard density because the temperature at STP is lower (0°C vs 15°C).
9. If a hot air balloon and a regular balloon are the same size, why does the hot air balloon float?
This phenomenon is explained by the principle of buoyancy and air density. The air inside the hot air balloon is heated by a burner, causing its temperature to rise significantly. This heating makes the internal air expand and become much less dense than the cooler, ambient air outside. According to Archimedes' principle, the cooler, denser outside air exerts an upward buoyant force. Because the air inside is so much less dense, the balloon's total weight (fabric + basket + low-density air) is less than the buoyant force, causing it to float. A regular balloon filled with air at ambient temperature has the same air density inside and out, so no net buoyant force is generated to lift it.
