The ideal gas is defined as a gas which obeys the following equation of state:

      Pv = RT

The internal energy of an ideal gas is a function of temperature only. That is,

      u = u(T)

Using the definition of enthalpy and the equation of state of ideal gas to yield,

      h = u + P v = u + RT

Since R is a constant and u = u(T), it follows that the enthalpy of an ideal gas is also a function of temperature only.

      h = h(T)

Since u and h depend only on the temperature for an ideal gas, the constant volume and constant pressure specific heats c_v and c_p also depend on the temperature only.

      c_v = c_v (T)      c_P = c_P (T)

For an ideal gas, the definitions of c_v and c_p are given as follows:

and they can be rewritten as

      c_v = du/dT        c_P = dh/dT

During a process from state 1 to state 2, the changes of internal energy and enthalpy are:

There are three ways to determine the changes of internal energy and enthalpy for ideal gas.

1. By using the tabulated u and h data. This is the easiest and most accurate way.
2. By integrating the equations above if the relations of c_v and c_p as a function of temperature are known. This is for computerized calculations and very accurate.
3. By using the average values of specific heats. This is simple and the results obtained are reasonably accurate if the temperature interval is small.

Combing the definition of enthalpy and the equation of state for ideal gas to yield

      h = u + pv = u + RT

Differentiating the above relation to give

      dh = du + RdT

Replacing dh by c_PdT and du by c_vdT,

This is a special relationship between c_v and c_P for an ideal gas. Also, the ratio of c_P and c_v is called the specific heat ratio,

      k = c_P/c_v

The specific heat ratio is also a temperature dependent property.

For air at T = 300 K,

      c_P = 1.005 kJ/(kg-K)
      c_v = 0.718 kJ/(kg-K)
      k = 1.4

Many processes which occur in practice can be described by an equation of the form

      Pvⁿ = constant

where
      n = constant

For a process from state 1 to state 2, the relation is:

      P₁v₁ ⁿ = P₂v₂ⁿ  

By using the equation of state of ideal gas, the relations between P, v, and T are:  

      T₂/T₁=(P₂/P₁)^(n-1)/n       P₂/P₁=(v₁/v₂)ⁿ

For some specific values of n, the process becomes isobaric, isothermal, isometric, and adiabatic, and they are summarized as follows:

The energy balance for a stationary, closed system is:

      Q - W = ΔU

For ideal gas, the internal energy can be determined by

      ΔU = U₂ - U₁ = c_v(T₂ - T₁)

where
      c_v = the average value of constant volume
            specific heat

If only boundary work is considered, then the work W can be determined by

For a polytropic process, the work W is:

For special processes such as the isobaric, isothermal, isometric, and adiabatic processes for ideal gas, using the average specific heats, the heat, work, and internal energy are given in the following table: