The pressure-density equation for an ideal gas relates pressure, density, temperature, and molar mass. It is derived from the ideal gas law.
Derivation of the Pressure Density Equation
The ideal gas law is expressed as:
pV = nRT
Where:
- p = pressure
- V = volume
- n = number of moles
- R = ideal gas constant
- T = temperature
We know that the number of moles (n) can be expressed as the mass (m) divided by the molar mass (M):
n = m/M
Substituting this into the ideal gas law gives:
pV = (m/M)RT
We also know that density (d) is mass (m) divided by volume (V):
d = m/V
Rearranging the previous equation, we get:
p = (m/V) (RT/M)
Since d = m/V, we can substitute density into the equation to derive the pressure-density equation:
p = dRT/M
This equation shows the relationship between pressure and density of an ideal gas at constant temperature.
Key Aspects of the Pressure Density Equation
- Direct proportionality between density and pressure: At constant temperature, the density of an ideal gas increases proportionally with pressure. This relationship is evident from the pressure-density equation (p = dRT/M). As pressure increases, the density increases proportionally.
- Inverse proportionality between density and molar mass: At constant pressure and temperature, gases with higher molar masses will have a greater density.
- Temperature Effect: Keeping the density constant, increasing the temperature requires a proportional increase in pressure.
- Ideal gas behaviour: This equation assumes the gas follows ideal behaviour, meaning that there are no intermolecular forces and the gas particles have no volume.
Example
Let's consider a scenario where we have two ideal gases at the same temperature and pressure. Gas A has a molar mass twice that of gas B.
Using the pressure density equation:
- p = d1RT/M1 (for gas A)
- p = d2RT/M2 (for gas B)
Since pressure, temperature, and the ideal gas constant are the same:
d1/M1 = d2/M2
If M1 = 2 M2, then d1 = 2 d2. This means that gas A (with the higher molar mass) has twice the density of gas B, thus confirming what the equation tells us.
Practical Insights
- This equation is vital for calculating gas densities under different conditions or to estimate molar masses.
- It helps in understanding the behaviour of gases in various industrial and scientific applications.
- By knowing the density of an ideal gas, its molar mass can be determined if pressure and temperature are known, using the formula M = dRT/P.
| Property | Symbol | Relationship with Density |
|---|---|---|
| Pressure | p | Directly proportional |
| Molar Mass | M | Inversely proportional |
| Temperature | T | Directly proportional |
| Ideal gas constant | R | Constant |
