For a constant temperature, the pressure of an ideal gas is inversely proportional to its volume. For a constant volume, the pressure of an ideal gas is directly proportional to its temperature. For a constant pressure, the volume of an ideal gas is directly proportional to its temperature. These relationships can be combined to find equations of state for an ideal gas.
Equations
description
equation
Initial and final state of a gas
T1p1V1=T2p2V2
Ideal gas equations of state
pV=nRTpV=NkT
Boltzmann constant conversion
R=kNA
Constants
constant
symbol
value
Avogadroconstant
NA
6.02×1023mol−1
molargasconstant
R
8.31JK−1mol−1
Boltzmannconstant
k
1.38×10−23JK−1
Variable definitions
quantity name
symbol
derived units
si base units
pressure
p
Pa
kgm−1s−2
volume
V
m3
m3
temperature
T
K
K
numberofatoms
N
numberofmoles
n
mol
mol
Gas laws
The relationship between the pressure, volume and temperature of an ideal gas can be described by a set of gas laws.
Pressure and volume
For a constant mass and temperature (isothermal), the pressure exerted by an ideal gas is inversely proportional to its volume:
p∝V1
This means that for an ideal gas pV=constant. This relationship can be seen through a p-V graph:
Pressure and temperature
For a constant mass and volume (isochoric), the pressure exerted by an ideal gas is directly proportional to its temperature:
p∝T
This means that for an ideal gas Tp=constant. This relationship can be seen through a p-T graph:
Volume and temperature
For a constant mass and pressure (isobaric), the volume of an ideal gas is directly proportional to its temperature:
V∝T
This means that for an ideal gas TV=constant. This relationship can be seen through a V-T graph:
Combining the laws
When the laws above are combined, one finds that for an ideal gas:
TpV=constant
This quantity will remain the same if the conditions change from an initial state 1 to a final state 2:
T1p1V1=T2p2V2
Equations of state
The constant from the combined relationship for 1mol of an ideal gas is known as the molar gas constant R, with a value of 8.31JK−1mol−1. One can use it to find what is known as the equation of state of an ideal gas for n moles:
pV=nRT
The molar gas constant R is related to the Avogadro constant and another constant through the equation:
R=kNA
This Boltzmann constant k has a value of 1.38×10−23JK−1 and can be substituted into the equation of state to give the version for an N number of particles:
pV=NkT
Note: when working with the equations of state the temperature must always be in K!
Example
Calculate the pressure exerted by 50mol of an ideal gas on a container with a volume of 2m3 that is kept at a temperature of 20°C.
Firstly, write down all the known values:
n=50molV=2m3T=20°C
Convert the temperature into K:
T=20°C=293K
Next, write down the correct equation of state and rearrange it to find pressure:
pV=nRTp=VnRT
Substitute all the values into the equation and calculate the pressure:
p=250×8.31×293=60.9kPa
The pressure exerted on the container by the gas is 60.9kPa.