2.3 Ideal gas equation and other equations of state

Assumptions of the Ideal Gas Model

The ideal gas equation is one of the most useful tools in thermodynamics, but it rests on several simplifying assumptions. Understanding these assumptions tells you when the equation works well and when you need something more sophisticated.

Negligible Particle Volume

The ideal gas model treats gas molecules as point masses with zero volume. This means the space available for molecular motion equals the entire container volume.

• Works well at low pressures, where the container volume is vastly larger than the combined volume of all the molecules
• At high pressures, molecules get packed closer together and their actual physical size starts to matter relative to the total volume

No Intermolecular Forces

The model assumes gas particles don't attract or repel each other. Each particle moves independently.

• Works well at high temperatures, where molecules have so much kinetic energy that any attractive or repulsive forces between them are negligible by comparison
• At low temperatures, molecules move slowly enough that intermolecular attractions (like van der Waals forces) can noticeably pull them toward each other, causing deviations from ideal behavior

Elastic Collisions

Ideal gas particles undergo perfectly elastic collisions with each other and with container walls. Total kinetic energy is conserved in every collision, with no energy lost to heat, deformation, or other mechanisms.

This assumption connects directly to the kinetic theory of gases and is reasonable at low densities, where collisions are infrequent and brief.

Temperature and Kinetic Energy

The average kinetic energy of ideal gas particles is directly proportional to the gas's absolute temperature:

$KE_{avg} = \frac{3}{2}kT$

Here, $k$ is the Boltzmann constant and $T$ is absolute temperature (in Kelvin). This relationship bridges the microscopic world (molecular motion) and the macroscopic world (temperature you can measure with a thermometer). It forms the backbone of the kinetic theory of gases.

Limitations of the Ideal Gas Model

The ideal gas model is most accurate at low pressures and high temperatures, where all the assumptions above hold reasonably well. It breaks down when:

• Pressure is high enough that molecular volume becomes significant
• Temperature is low enough that intermolecular forces become significant
• The gas is near its critical point or close to condensing

Under these conditions, you need more complex equations of state (van der Waals, Redlich-Kwong, etc.) to get accurate results. Still, the ideal gas equation is the natural starting point and often a perfectly adequate approximation.

Ideal Gas Equation Applications

Using the Ideal Gas Equation

The ideal gas equation relates pressure, volume, amount of substance, and temperature:

$PV = nRT$

• $P$ = pressure
• $V$ = volume
• $n$ = number of moles
• $R$ = universal gas constant
• $T$ = absolute temperature (Kelvin)

If you know any four of these quantities, you can solve for the fifth. The universal gas constant $R$ takes different numerical values depending on your unit system:

• $R = 8.314 \, \frac{J}{mol \cdot K}$ (when using Pa and $m^3$)
• $R = 0.08206 \, \frac{L \cdot atm}{mol \cdot K}$ (when using atm and liters)

Unit consistency is critical. Before plugging numbers in, make sure your pressure, volume, and temperature units match the value of $R$ you're using. This is one of the most common sources of errors on exams.

In thermodynamics, you'll also frequently see the equation written on a per-mass basis using the specific gas constant: $Pv = RT$, where $v$ is specific volume (volume per unit mass) and $R$ here is the specific gas constant ($R_{universal}/M$).

Density Calculations

You can rearrange the ideal gas equation to solve for gas density directly:

$\rho = \frac{PM}{RT}$

where $\rho$ is density and $M$ is the molar mass of the gas. This is useful for determining how dense a gas is at a given pressure and temperature, which matters for applications like calculating buoyancy, sizing equipment, or finding mass flow rates in pipelines.

For example, the density of air ($M \approx 29 \, g/mol$) at 1 atm and 300 K is about $1.18 \, kg/m^3$. At higher pressures or lower temperatures, the density increases proportionally.

Ideal Gas vs. Other Equations of State

Accounting for Non-Ideal Behavior

Real gases deviate from the ideal gas equation because molecules do have finite volume and do exert forces on each other. Several equations of state correct for these effects.

Van der Waals equation is the simplest correction. It introduces two substance-specific constants:

$(P + \frac{an^2}{V^2})(V - nb) = nRT$

• The $a$ term corrects for intermolecular attractive forces (it increases the effective pressure, since attractions reduce the force molecules exert on the walls)
• The $b$ term corrects for finite molecular volume (it reduces the effective volume available for molecular motion)

Redlich-Kwong and Soave-Redlich-Kwong (SRK) equations improve on van der Waals by using a temperature-dependent attraction term, making them more accurate over wider ranges of pressure and temperature.

Peng-Robinson equation is widely used in the oil and gas industry because it handles hydrocarbons and mixtures particularly well.

Virial Equations of State

Virial equations take a different approach: they express the deviation from ideal behavior as a power series in inverse volume (or pressure):

$\frac{PV}{nRT} = 1 + \frac{B}{V} + \frac{C}{V^2} + \cdots$

• The left side equals 1 for an ideal gas, so each additional term captures progressively finer corrections
• $B$, $C$, etc. are virial coefficients that depend on temperature and the specific gas
• $B$ accounts for two-body molecular interactions, $C$ for three-body interactions, and so on

Truncating after the $B$ term gives a simple correction that works well at moderate pressures. Including more terms increases accuracy but also increases computational effort. The virial approach is especially valuable because the coefficients have a direct physical interpretation rooted in statistical mechanics.

Choosing an Appropriate Equation of State

The right equation depends on three factors: the gas, the conditions, and how much accuracy you need.

• Ideal gas equation: Use for low pressures, high temperatures, and gases with weak intermolecular forces (noble gases, $N_2$, $O_2$). Quick and easy.
• Van der Waals: A step up from ideal, but still limited in accuracy near the critical point or at very high pressures. Good for building intuition about non-ideal corrections.
• Redlich-Kwong / SRK / Peng-Robinson: More accurate across wider conditions. These are the workhorses of industrial calculations.
• Virial equations: Offer a systematic way to increase accuracy by adding terms. Useful when you have experimental data for the virial coefficients.

Solving Problems with Real Gases

When the ideal gas equation isn't accurate enough, follow this general approach:

Step 1: Identify the Appropriate Equation of State

Consider the gas and the conditions. At low pressures and high temperatures, the ideal gas equation may be sufficient. At higher pressures, lower temperatures, or near the critical point, choose van der Waals, Redlich-Kwong, SRK, or Peng-Robinson depending on the accuracy you need and the data available.

Step 2: Determine Equation Parameters

Each equation of state beyond the ideal gas law requires substance-specific constants. For the van der Waals equation, $a$ and $b$ can be calculated from the gas's critical properties:

$a = \frac{27R^2T_c^2}{64P_c} \qquad b = \frac{RT_c}{8P_c}$

where $T_c$ is the critical temperature and $P_c$ is the critical pressure. These critical properties are tabulated for common gases in your textbook's appendix or in reference handbooks.

For virial equations, the coefficients must come from experimental data or empirical correlations.

Step 3: Solve for the Desired Property

Substitute known values into the chosen equation and solve for the unknown (pressure, volume, or temperature).

• With the van der Waals equation, solving for $V$ at a given $P$ and $T$ produces a cubic equation that may require iteration or numerical methods
• For mixtures, apply appropriate mixing rules (such as Kay's rule or van der Waals mixing rules) to estimate the equation parameters from the properties of individual components

Step 4: Interpret Results and Compare to Ideal Gas Behavior

After solving, compare your result to what the ideal gas equation would predict. Calculate the percent deviation:

$\% \text{ deviation} = \frac{|value_{real} - value_{ideal}|}{value_{real}} \times 100$

This tells you how significant non-ideal effects are under those conditions. A small deviation confirms the ideal gas equation would have been adequate; a large deviation justifies the extra effort of using a more complex equation.

Consider what's driving the deviation: Is it mainly intermolecular attractions (important near condensation), molecular volume effects (important at very high pressures), or both? Understanding the physical cause helps you make better engineering judgments about the system you're analyzing.