# Ideal Gas Law

## In perfect or ideal gas the change in density is directly related to the change of temperature and pressure as expressed by the Ideal Gas Law

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In perfect or ideal gas the change in density is directly related to the change of temperature and pressure as expressed by the Ideal Gas Law.

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

The **Ideal 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(1)

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.^{o}K, ft.lb/slugs.^{o}R)

T= absolute temperature (^{o}K,^{o}R)

This equation (1) can be modified to:

p = ρ R T(2)

where the density

ρ = m / V(3)

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

Equation (1) can also be modified to

p_{1}V_{1 }/ T_{1}= p_{2 }V_{2 }/ T_{2}(4)

expressing the relationship between different states for a given mass of gas.

### The **Ideal Gas Law** and the Universal Gas Constant - *R*_{u}

_{u}

The **Universal Gas Constant** is independent of the particular gas and is the same for all "perfect" gases. The Ideal Gas Law can be expressed with the **Universal Gas Constant**:

p V = n R_{u}T

= N k T (5)

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.^{o}K, lbf.ft/(lbmol.^{o}R)= 8.3145 J/mol K

T= absolute temperature (^{o}K,^{o}R)

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.

### 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 (2) to:

ρ= p / (R T) (6)

ρ=[((50 lb/in^{2}) + (14.7 lb/in^{2})) (144 in^{2}/ft^{2})] / [(1716 ft.lb/slug.^{o}R)((70+ (460^{o}R))]^{o}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 (7)

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 the deviation from the ideal situation, another 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*

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