# The Ideal Gas Law

## The relationship between volume, pressure, temperature and quantity of a gas, including definition of gas density.

In a perfect or ideal gas the correlations between pressure, volume, temperature and quantity of gas can be expressed by the **Ideal Gas Law.**

The **Universal Gas Constant, R _{u}** is independent of the particular gas and is the same for all "perfect" gases, and is included in of The Ideal Gas Law:

p V = n R

_{u}T (1)

where

p = absolute pressure [N/m

^{2}], [lb/ft^{2}]V = volume [m

^{3}], [ft^{3}]n = is the number of moles of the gas present

R

_{u}= universal gas constant [J/mol K], [lb_{f}ft/(lb mol^{o}R)]= 8.3145 [J/mol K]= 0.08206 [L atm/mol K] = 62.37 [L torr /mol K]T = absolute temperature [K], [

^{o}R]

For a given quantity of gas, both n and R_{u} are constant, and Equation (1) can be modified to

p

_{1}V_{1 }/ T_{1}= p_{2 }V_{2 }/ T_{2}(2)

expressing the relationship between different states for the given quantity of the gas.

Equation (1) can also be expressed as

p V = N k T (3)

N =number of molecules

k = Boltzmann constant = 1.38066 10

^{-23}[J/K] = 8.617385 10^{-5}[eV/K]

- One mole of an ideal gas at STP occupies 22.4 liters.

#### The Ideal Gas Law and the Individual Gas Constant - *R*

The **Ideal Gas Law** - or **Perfect Gas Law** - relates pressure, temperature, and volume of an **ideal or perfect gas**. The Ideal Gas Law can be expressed with the **Individual Gas Constant**.

p V = m R T (4)

where

p = absolute pressure [N/m

^{2}], [lb/ft^{2}]V = volume [m

^{3}], [ft^{3}]m = mass [kg], [slugs]

R = individual gas constant [J/kg K], [ft lb/slugs

^{o}R]T = absolute temperature [K], [

^{o}R]

This equation (3) can be modified to:

p = ρ R T (5)

where the density

ρ = m / V [kg/m

^{3}], [slugs/ft^{3}] (6)

The Individual Gas Constant - *R* - depends on the particular gas and is related to the molecular weight of the gas.

See also Non-ideal gas - Van der Waal's equation and constants, used to correct for non-ideal behavior of gases caused by intermolecular forces and the volume occupied by the gas particles and how to calculate total pressure and partial pressures from Ideal gas law

**Example:** The Ideal Gas Law

A tank with volume of *1 ft ^{3}* is filled with air compressed to a gauge pressure of

*50 psi.*The temperature in tank is

*70*.

^{o}FThe air density can be calculated with a transformation of the ideal gas law (5) to:

ρ = p / (R T) (7)

ρ= ((50 [lb/in

^{2}]+ 14.7 [lb/in^{2}])*144 [in^{2}/ft^{2}]) / (1716 [ft.lb/slug.^{o}R]* (70+ 460)[°R])= 0.0102 [slugs/ft

^{3}]

The weight of the air is the product of specific weight and the air volume. It can be calculated as:

w = ρ g V (8)

w = 0.0102 [slugs/ft

^{3}] * 32.2 [ft/s^{2}]*1 [ft^{3}]= 0.32844 [slugs ft/s

^{2}]= 0.32844 [lb]

### Note!

The Ideal Gas Law is accurate only at relatively low pressures and high temperatures. To account for deviation from the ideal situation an other factor is included. It is called the Gas Compressibility Factor, or Z-factor. This correction factor is dependent on pressure and temperature for each gas considered.

The True Gas Law, or the Non-Ideal Gas Law, becomes:

P V = Z n R T (7)

where

Z = Gas Compressibility Factor

n = number of moles of gas present

### Compressibility factor - Z - for Air

For full table -** rotate the screen!**

Compressibility factor for Air - Z - | ||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Temperature[K] | Pressure [bar absolute] | |||||||||||||

1 | 5 | 10 | 20 | 40 | 60 | 80 | 100 | 150 | 200 | 250 | 300 | 400 | 500 | |

75 | 0.005 | 0.026 | 0.052 | 0.104 | 0.206 | 0.308 | 0.409 | 0.510 | 0.758 | 1.013 | ||||

80 | 0.025 | 0.050 | 0.100 | 0.198 | 0.296 | 0.393 | 0.489 | 0.726 | 0.959 | 1.193 | 1.414 | |||

90 | 0.976 | 0.024 | 0.045 | 0.094 | 0.187 | 0.278 | 0.369 | 0.468 | 0.678 | 0.893 | 1.110 | 1.311 | 1.716 | 2.111 |

100 | 0.980 | 0.887 | 0.045 | 0.090 | 0.178 | 0.264 | 0.350 | 0.434 | 0.639 | 0.838 | 1.040 | 1.223 | 1.594 | 1.954 |

120 | 0.988 | 0.937 | 0.886 | 0.673 | 0.178 | 0.256 | 0.337 | 0.413 | 0.596 | 0.772 | 0.953 | 1.108 | 1.509 | 1.737 |

140 | 0.993 | 0.961 | 0.921 | 0.830 | 0.586 | 0.331 | 0.374 | 0.434 | 0.591 | 0.770 | 0.911 | 1.039 | 1.320 | 1.590 |

160 | 0.995 | 0.975 | 0.949 | 0.895 | 0.780 | 0.660 | 0.570 | 0.549 | 0.634 | 0.756 | 0.884 | 1.011 | 1.259 | 1.497 |

180 | 0.997 | 0.983 | 0.966 | 0.931 | 0.863 | 0.798 | 0.743 | 0.708 | 0.718 | 0.799 | 0.900 | 1.007 | 1.223 | 1.436 |

200 | 0.998 | 0.989 | 0.977 | 0.954 | 0.910 | 0.870 | 0.837 | 0.814 | 0.806 | 0.855 | 0.931 | 1.019 | 1.205 | 1.394 |

250 | 0.999 | 0.996 | 0.991 | 0.982 | 0.967 | 0.955 | 0.946 | 0.941 | 0.945 | 0.971 | 1.015 | 1.070 | 1.199 | 1.339 |

300 | 1.000 | 0.999 | 0.997 | 0.995 | 0.992 | 0.990 | 0.990 | 0.993 | 1.007 | 1.033 | 1.067 | 1.109 | 1.207 | 1.316 |

350 | 1.000 | 1.000 | 1.000 | 1.001 | 1.004 | 1.008 | 1.012 | 1.018 | 1.038 | 1.064 | 1.095 | 1.130 | 1.212 | 1.302 |

400 | 1.000 | 1.001 | 1.003 | 1.005 | 1.010 | 1.016 | 1.023 | 1.031 | 1.053 | 1.080 | 1.109 | 1.141 | 1.212 | 1.289 |

450 | 1.000 | 1.002 | 1.003 | 1.006 | 1.013 | 1.021 | 1.029 | 1.037 | 1.061 | 1.091 | 1.118 | 1.146 | 1.209 | 1.278 |

500 | 1.000 | 1.002 | 1.003 | 1.007 | 1.015 | 1.023 | 1.032 | 1.041 | 1.065 | 1.091 | 1.118 | 1.146 | 1.205 | 1.267 |

600 | 1.000 | 1.002 | 1.004 | 1.008 | 1.016 | 1.025 | 1.034 | 1.043 | 1.068 | 1.092 | 1.117 | 1.143 | 1.195 | 1.248 |

800 | 1.000 | 1.002 | 1.004 | 1.008 | 1.016 | 1.024 | 1.032 | 1.041 | 1.062 | 1.084 | 1.106 | 1.128 | 1.172 | 1.215 |

1000 | 1.000 | 1.002 | 1.004 | 1.007 | 1.014 | 1.022 | 1.029 | 1.037 | 1.056 | 1.074 | 1.095 | 1.113 | 1.152 | 1.189 |