Internal Energy, Enthalpy, and Specific Heats of Ideal Gas



Ideal Gas Model
Pv = PT
u = u(T)
h = h(T) = u(T) + RT
Ideal Gas Model

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 cv and cp also depend on the temperature only.

      cv = cv (T)      cP = cP (T)

For an ideal gas, the definitions of cv and cp are given as follows:


and they can be rewritten as

      cv = du/dT        cP = dh/dT

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


Internal Energy Enthalpy
Three Ways of Calculating Δu and Δh

Specific Heats of
Common Ideal Gases

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 cv and cp 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.
    Specific Heat Relations of Ideal Gas

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 cPdT and du by cvdT,


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

      k = cP/cv

The specific heat ratio is also a temperature dependent property.

For air at T = 300 K,

      cP = 1.005 kJ/(kg-K)
      cv = 0.718 kJ/(kg-K)
      k = 1.4

    The Polytropic Process

Special Processes of Ideal Gas on a P-v Diagram

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

      Pvn = constant

      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:

Process n
Isobaric (P = constant) 0
Isothermal (T = constant) 1
Isometric (V = constant) infinity
adiabatic (Q = 0) k
    Energy Analysis for Ideal Gas in a Closed System at Different Processes

Boundary Work

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 = cv(T2 - T1)

      cv = 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:

Process Q W ΔU
Isothermal mRT ln(P1/P2) mRT ln(P1/P2) 0
Isometric m cv ΔT 0 mcvΔT
Isobaric mcP ΔT P(V2-V1) mcvΔT
Adiabatic 0 -mcvΔT mcvΔT