How to Calculate Tcrit and Pcrit for Real Gas

What are Tcrit and Pcrit?

The critical temperature (Tcrit) and critical pressure (Pcrit) are fundamental properties of a substance that define its behavior as a real gas. These values represent the conditions under which a gas cannot be liquefied by pressure alone, regardless of the amount of pressure applied.

At the critical point, the gas and liquid phases become indistinguishable, and the substance exhibits unique properties that differ from those in the gaseous or liquid states below the critical point.

How to Calculate Tcrit and Pcrit

Calculating Tcrit and Pcrit involves understanding the behavior of real gases and their deviation from ideal gas laws. The critical parameters are typically determined experimentally or through empirical correlations.

Empirical Correlation Method

One common method to estimate Tcrit and Pcrit is by using empirical correlations based on molecular properties. The most widely used correlation is the one proposed by Pitzer et al., which relates critical parameters to the acentric factor (ω) and the boiling point temperature (Tb).

Where:

• Tb = Boiling point temperature (K)
• ω = Acentric factor (dimensionless)
• R = Universal gas constant (8.314 J/mol·K)
• Vm = Molar volume at critical point (m³/mol)

Experimental Determination

For precise values, experimental determination is preferred. This involves measuring the pressure and temperature at which the gas and liquid phases become indistinguishable. The critical point is typically found by observing the inflection point in the pressure-volume (P-V) or pressure-temperature (P-T) curves.

Real Gas Behavior and Critical Points

Real gases deviate from ideal gas behavior, especially near the critical point. The behavior of a real gas can be described using equations of state such as the van der Waals equation or the Peng-Robinson equation.

The critical point marks the boundary between the gaseous and liquid phases. Above the critical temperature, the gas cannot be liquefied by pressure alone, and it behaves more like a supercritical fluid.

Example Calculation

Let's calculate the critical temperature and pressure for methane (CH4) using the empirical correlation method.

Given Data

• Boiling point temperature (Tb) = 111.7 K
• Acentric factor (ω) = 0.011
• Universal gas constant (R) = 8.314 J/mol·K
• Molar volume at critical point (Vm) = 9.86 × 10⁻⁵ m³/mol

Calculation Steps

1. Calculate Tcrit using the empirical correlation:

   Tcrit = 111.7 × (0.637 + 0.364 × 0.011 - 0.113 × (0.011)²)

   Tcrit ≈ 111.7 × (0.637 + 0.003994 - 0.00013633)

   Tcrit ≈ 111.7 × 0.6400

   Tcrit ≈ 71.7 K

2. Calculate Pcrit using the empirical correlation:

   Pcrit = (0.118 + 0.0112 × 0.011 - 0.0216 × (0.011)²) × (8.314 × (111.7)² / 9.86 × 10⁻⁵)

   Pcrit ≈ (0.118 + 0.0001232 - 0.0000025956) × (8.314 × 12482.69 / 9.86 × 10⁻⁵)

   Pcrit ≈ 0.118 × 1.057 × 10⁸

   Pcrit ≈ 1.25 × 10⁷ Pa

The calculated critical temperature for methane is approximately 71.7 K, and the critical pressure is approximately 12.5 MPa.