# Equation of State

PV = nRT

Ideal gas: A hypothetical gas that exhibits linear relationships among volume, pressure, temperature, and amount (mol) at all conditions; approximately by simple gases at ordinary conditions. Although no ideal gas actually exists, most simple gases, such as N2, O2, H2, and the noble gases, show nearly ideal behavior at ordinary temperatures and pressures. Ideal gas law: An equation that expresses the relationships among volume, pressure, temperature, and amount (mol) of an ideal gas: PV=nRT.

## Types

There are three basic classes of ideal gas:

• the classical or Maxwell-Boltzmann ideal gas,
• the ideal quantum Bose gas, composed of bosons, and
• the ideal quantum Fermi gas, composed of fermions.

The classical ideal gas can be separated into two types: The classical thermodynamic ideal gas and the ideal quantum Boltzmann gas. Both are essentially the same, except that the classical thermodynamic ideal gas is based on classical statistical mechanics, and certain thermodynamic parameters such as the entropy are only specified to within an undetermined additive constant. The ideal quantum Boltzmann gas overcomes this limitation by taking the limit of the quantum Bose gas and quantum Fermi gas in the limit of high temperature to specify these additive constants. The behavior of a quantum Boltzmann gas is the same as that of a classical ideal gas except for the specification of these constants. The results of the quantum Boltzmann gas are used in a number of cases including the Sackur-Tetrode equation for the entropy of an ideal gas and the Saha ionization equation for a weakly ionized plasma.

# Internal Energy

a function of temperature only
U = U(T)

Internal energy: is the total of kinetic and potential energies of all the particles in a system.

# Implied Property Relations for an Ideal Gas

C_v is a function of temperature only

C_(v )≡(∂U/∂T)_v= dU(T)/dT= C_v (T)
H is a function of temperature only

H ≡U+PV=U(T)+RT=H(T)
C_p is a function of temperature only

C_p= dH/dT= dU/dT+R=C_v+R
Any change of state of an ideal gas
dU = C_(v ) dT ∆U= ∫▒C_(v ) dT
dH = C_(p dT) ∆H= ∫▒C_(p ) dT

Some facts about Ideal Gas Law: The ideal gas law can be rearranged to calculate the density and molar mass of a gas. Also, in a mixture of gases, each component contributes its own partial pressure to the total pressure (this is also called the Dalton's law of partial pressures). The mole fraction of each component is the ratio of its partial pressure to the total pressure. The most important idea is that the total pressure is the sum of the gas pressure and the vapor pressure of water at the given temperature when a gas is in contact with the water. [The Molecular Nature of Matter and Change].

### Reference

Silberberg, Martin S. Chemistry: The Molecular Nature of Matter and Change. 5th ed. 2009