Understanding the General Gas Equation: PVnRT and PVmRT

The general gas equation, a cornerstone of thermodynamics, is expressed as PVnRT. However, it is also expressed as PVmRT. Both equations are valid, but they serve different purposes within the fields of chemistry and physics.

The PVnRT Equation

This equation is widely used in chemistry and physics to understand the behavior of ideal gases. Here, the variables are defined as follows:

P Pressure of the gas V Volume of the gas n Number of moles of the gas R Ideal gas constant, which is the same for all gases and has a value of 8.314 J/mol K T Temperature in Kelvin

The ideal gas constant, R, remains constant for all gases under ideal conditions, making it a useful parameter for comparing different gases.

The PVmRT Equation

The equation PVmRT is also used and is particularly relevant when dealing with the mass of a gas rather than the number of moles. Here, the variables are defined as follows:

P Pressure of the gas V Volume of the gas m Mass of the gas M Molar mass of the gas R Ideal gas constant, same as in the PVnRT equation T Temperature in Kelvin

To use the PVmRT equation, we first need to convert the number of moles into mass. This is achieved using the relationship:

n frac{m}{M}

Substituting this into the first equation (PVnRT) gives:

PV frac{m}{M}RT

While both equations are valid, they serve different contexts. The PVnRT equation is preferred in theoretical calculations due to its simplicity. The PVmRT equation, on the other hand, is often used in practical scenarios where the mass of a gas is a more relevant parameter, such as in air composition calculations.

Example: Hydrogen and Air

Let's take a closer look at hydrogen, a single-element gas, to understand how the gas constant per unit mass (r) is used. For hydrogen:

M 2 g/mol 0.002 kg/mol r R/M 8.314 J/mol K / 0.002 kg/mol 4.157×10^-3 J/kg K

For a more complex scenario, consider the composition of air, which is a mixture of multiple gases. The number of moles of air can be calculated by considering the mass and molar masses of its components:

Nitrogen (N2): M 28 g/mol 0.028 kg/mol Oxygen (O2): M 32 g/mol 0.032 kg/mol Carbon Dioxide (CO2): M 44 g/mol 0.044 kg/mol Water Vapor (H2O): M 18 g/mol 0.018 kg/mol

The molar ratio of air can be calculated as:

n n1n2n3n4 m1/M1 m2/M2 m3/M3 m4/M4

The total mass of air is:

m m1 m2 m3 m4

Using the weighted molar mass (M), we can write:

PV n RT n1n2n3n4RT m1/M1 m2/M2 m3/M3 m4/M4RT

Define a weighted molar mass (M) by:

1/M 1/m m1/M1 m2/M2 m3/M3 m4/M4

Therefore, PV mR/MT mrT

Where the effective gas constant per unit mass (r) is defined as:

r R/M

By employing the appropriate form of the gas equation, we can accurately describe the behavior of gases under various conditions, whether we are dealing with the number of moles or the mass of the gas.