3.8: Ideal Gas Model - A Useful Constitutive Relation


In a chapter on conservation of mass, it is frequently necessary to predict the density of a substance. For most liquids and solids, the values for density (or specific weight or specific gravity) will be found in tables in handbooks. However, for gases there is a very useful model that can accurately predict the pressure - temperature - density relationship for conditions of "low pressure" and "high temperature". The exact limitations of this model will be discussed next quarter.

The ideal gas model (not law) is a constitutive relationship that relates pressure, temperature, and density for a gaseous substance: \begin{align} p &= \bar{\rho} R_{u} T, \\ \text{where } \bar{\rho} &=\text {molar density }\left(\mathrm{kmol} / \mathrm{m}^{3}\right) \nonumber \\ p &= \text {absolute pressure }(\mathrm{kPa}) \nonumber \\ T &= \text {absolute (thermodynamic) temperature (K)} \nonumber \\ R_{u} &= \text {universal gas constant} = 8.314 \mathrm{~kJ} /(\mathrm{kmol} \cdot \mathrm{K}) \nonumber \end{align} \nonumber

There are many different forms of this equation. An alternative form that is also very useful is the following: \begin{align} p &= \rho R T, \\ \text {where } \rho &= \text {density } \left(\mathrm{kg} / \mathrm{m}^{3} \right) \nonumber \\ p &= \text {absolute pressure }(\mathrm{kPa}) \nonumber \\ T &=\text {absolute (thermodynamic) temperature }(\mathrm{K}) \nonumber \\ R &=\frac{R_{u}}{M} = \text {specific gas constant }(\mathrm{kJ} /(\mathrm{kg} \cdot \mathrm{K})) \nonumber \end{align} \nonumber Please beware: there is much confusion between the universal gas constant and the specific gas constant. You must use the correct one in each calculation.

Many other useful forms of the ideal gas equation can be developed: \begin{align*} \text { Molar forms: } \quad p V &=n R_{u} T \\ p \bar{v} &=R_{u} T \quad \text { where } \bar{v}=V / n \\ p &=\bar{\rho} R_{u} T \\ \\ \text { Mass forms: } \quad p V &=m R T \\ p v &=R T \quad \text { where } v=V / m \end{align*} \nonumber You are encouraged to learn only two or three and develop skill in converting to the other forms. The tables below give more information about the various terms in the equations and molar mass information for several substances.

Molar Mass (Molecular Weight) for some common substances
Substance Chemical Formula Molar Mass $$(\mathrm{g} / \mathrm{gmol} ; \mathrm{kg} / \mathrm{kmol} ; \mathrm{lbm} / \mathrm{lbmol})$$
Air $$\cdots$$ $$28.97$$
Ammonia $$\mathrm{NH}_{3}$$ $$17.04$$
Carbon dioxide $$\mathrm{CO}_{2}$$ $$44.01$$
Refrigerant 134a $$\mathrm{C}_{2} \mathrm{F}_{4} \mathrm{H}_{2}$$ $$102.03$$
Helium $$\mathrm{He}$$ $$4.003$$
Hydrogen $$\mathrm{H}_{2}$$ $$2.016$$
Methane $$\mathrm{CH}_{4}$$ $$16.04$$
Nitrogen $$\mathrm{N}_{2}$$ $$28.01$$
Oxygen $$\mathrm{O}_{2}$$ $$32.00$$
Water $$\mathrm{H}_{2} \mathrm{O}$$ $$18.02$$
Molar Basis Mass Basis

$$PV = n R_u T$$

$$P \bar{\upsilon} = R_u T \quad \text{ and } \quad P = \bar{\rho} R_u T$$

$$PV = m R T$$

$$P \upsilon = R T \quad \text{ and } P = \rho R T$$

$$\text{where}$$

\begin{align*} P &= \text{absolute pressure of gas } \left[ \text{kPa or lbf/ft}^2 \right] \\ V &= \text{volume of gas } \left[ \text{m}^3 \text{ or ft}^3 \right] \\ n &= \text{number of moles of gas } \left[ \text{kmol or lbmol} \right] \\ R_u &= \text{universal gas constant (the same for every gas) } \\ &\quad\quad\quad \left[ \text{kJ}/(\text{kmol} \cdot \text{K}) \text{ or } (\text{ft} \cdot \text{lbf} ) / ( \text{lbmol} \cdot ^{\circ} \text{R} ) \right] \\ T &= \text{absolute temperature of gas } \left[ \text{K or } ^{\circ} \text{R} \right] \\ \bar{\rho} &= \text{molar density} = 1 / \bar{\upsilon} \ \left[ \text{kmol/m}^3 \text{ or lbmol} / \text{ft}^3 \right] \\ \bar{\upsilon} &= \text{molar specific volume } \left[ \text{m}^3 / \text{kmol or ft}^3 / \text{lbmol} \right] \end{align*} \nonumber

$$\text{where}$$

\begin{align*} P &= \text{absolute pressure of gas } \left[ \text{kPa or lbf} / \text{ft}^2 \right] \\ V &= \text{volume of gas } \left[ \text{m}^3 \text{ or ft}^3 \right] \\ m &= \text{mass of gas } \left[ \text{kg or lbm} \right] \\ R &= \text{specific gas constant (different for each gas) } \\ &\quad\quad\quad \left[ \text{kg} / ( \text{kg} \cdot \text{K} ) \text{ or } ( \text{ft} \cdot \text{lbf} ) / ( \text{lbm} \cdot ^{\circ} \text{R} ) \right] \\ T &= \text{absolute temperature } \left[ \text{K or } ^{\circ} \text{R} \right] \\ \rho &= \text{density} = 1 / \upsilon \ \left[ \text{kg/m}^3 \text{ or lbm}/ \text{ft}^3 \right] \\ \upsilon &= \text{specific volume } \left[ \text{m}^3 / \text{kg or ft}^3 / \text{lbm} \right] \end{align*} \nonumber

$$\text{and}$$

\begin{align*} R_u &= 8.314 \ \frac{ \text{kJ} }{ \text{kmol} \cdot \text{K} } = 8.314 \ \frac{ \text{J} }{ \text{mol} \cdot \text{K} } \\[4pt] &= 1545 \ \frac{ \text{ft} \cdot \text{lbf} }{ \text{lbmol} \cdot ^{\circ} \kern-0.3em \text{R} } \end{align*} \nonumber

$$\text{and}$$

$R = \frac{R_u}{M} \nonumber$

$$\text{where}$$

$M = \text{molecular weight (molar mass) of a specific gas} \nonumber$

3.8: Ideal Gas Model - A Useful Constitutive Relation is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by LibreTexts.