Understanding the Calculation of Vapor Pressure: A Comprehensive Technical Guide

Vapor pressure calculation determines a liquid’s tendency to evaporate under specific conditions. This article explores the fundamental principles and advanced methods for accurate vapor pressure estimation.

Readers will find detailed formulas, extensive data tables, and real-world applications to master vapor pressure calculations in various scientific and engineering contexts.

• Calculate the vapor pressure of water at 50°C using Antoine equation.
• Determine the vapor pressure of ethanol at 25°C with the Clausius-Clapeyron equation.
• Estimate the vapor pressure of benzene at 80°C using empirical data.
• Compare vapor pressures of acetone and methanol at 30°C for industrial solvent selection.

Extensive Tables of Vapor Pressure Values for Common Substances

Accurate vapor pressure data is essential for process design, safety analysis, and environmental assessments. The following tables compile vapor pressure values for widely used substances across typical temperature ranges.

These values serve as benchmarks for validating vapor pressure calculations using theoretical and empirical models.

Fundamental and Advanced Formulas for Vapor Pressure Calculation

Vapor pressure quantifies the equilibrium pressure exerted by a vapor in contact with its liquid or solid phase at a given temperature. Several mathematical models exist to calculate vapor pressure, each with specific applicability and accuracy.

1. Antoine Equation

The Antoine equation is one of the most widely used empirical correlations for vapor pressure estimation:

• P: Vapor pressure (usually in mmHg or kPa)
• T: Temperature (°C)
• A, B, C: Substance-specific Antoine constants

The constants A, B, and C are determined experimentally and vary depending on the temperature range. For example, for water between 1°C and 100°C:

• A = 8.07131
• B = 1730.63
• C = 233.426

Note: Vapor pressure units depend on the constants used; conversion may be necessary.

2. Clausius-Clapeyron Equation

This thermodynamic relation describes the temperature dependence of vapor pressure based on enthalpy of vaporization:

• P: Vapor pressure (Pa or atm)
• ΔH_vap: Enthalpy of vaporization (J/mol)
• R: Universal gas constant (8.314 J/mol·K)
• T: Absolute temperature (K)
• C: Integration constant related to entropy

This equation assumes ΔH_vap is constant over the temperature range, which is an approximation valid for narrow intervals.

3. Antoine Equation with Temperature-Dependent Constants

For improved accuracy, temperature-dependent Antoine constants or extended Antoine equations are used:

• D, E: Additional empirical constants

This form captures non-linearities in vapor pressure behavior at wider temperature ranges.

4. Wagner Equation

The Wagner equation is a highly accurate semi-empirical formula used for substances near their critical point:

• P_r: Reduced vapor pressure (P / P_c)
• τ: 1 – T / T_c (reduced temperature)
• A, B, C, D: Substance-specific constants
• P_c: Critical pressure
• T_c: Critical temperature

This equation is preferred for precise vapor pressure predictions near critical conditions.

5. Antoine Equation Constants Table for Common Substances

Detailed Explanation of Variables and Typical Values

• Temperature (T): Usually expressed in °C or K. Absolute temperature (K) is required for thermodynamic equations like Clausius-Clapeyron.
• Vapor Pressure (P): Pressure exerted by vapor in equilibrium with its liquid/solid phase. Units vary: mmHg, kPa, atm, Pa.
• Antoine Constants (A, B, C): Empirical parameters fitted to experimental vapor pressure data over specific temperature ranges.
• Enthalpy of Vaporization (ΔH_vap): Energy required to vaporize one mole of liquid at constant pressure, typically in kJ/mol or J/mol.
• Gas Constant (R): Universal constant 8.314 J/mol·K used in thermodynamic calculations.
• Critical Properties (P_c, T_c): Pressure and temperature at the critical point where liquid and vapor phases become indistinguishable.

Understanding these variables and their typical ranges is crucial for selecting the appropriate vapor pressure model and ensuring accurate calculations.

Real-World Applications and Case Studies

Case Study 1: Vapor Pressure Calculation of Water at 50°C Using Antoine Equation

Objective: Calculate the vapor pressure of water at 50°C using the Antoine equation and compare it with tabulated data.

Given:

• Temperature, T = 50°C
• Antoine constants for water (1-100°C): A = 8.07131, B = 1730.63, C = 233.426

Calculation:

Calculate denominator:

233.426 + 50 = 283.426

Calculate fraction:

1730.63 / 283.426 ≈ 6.104

Calculate exponent:

8.07131 – 6.104 = 1.96731

Calculate vapor pressure:

P = 10^1.96731 ≈ 92.7 mmHg

Convert mmHg to kPa (1 mmHg = 0.133322 kPa):

92.7 mmHg × 0.133322 = 12.36 kPa

Comparison: Tabulated vapor pressure at 50°C is approximately 12.35 kPa, confirming the accuracy of the Antoine equation.

Case Study 2: Estimating Vapor Pressure of Ethanol at 25°C Using Clausius-Clapeyron Equation

Objective: Calculate ethanol vapor pressure at 25°C using the Clausius-Clapeyron equation, given vapor pressure at 20°C.

Given:

• Known vapor pressure at T₁ = 20°C (293.15 K): P₁ = 5.95 kPa
• Temperature T₂ = 25°C (298.15 K)
• Enthalpy of vaporization ΔH_vap = 38.56 kJ/mol = 38560 J/mol
• Gas constant R = 8.314 J/mol·K

Clausius-Clapeyron equation rearranged:

Calculate the right side:

1 / T₂ – 1 / T₁ = (1 / 298.15) – (1 / 293.15) ≈ 0.003355 – 0.003412 = -0.000057 K^-1

Calculate the product:

– (38560 / 8.314) * (-0.000057) = -4639.5 * (-0.000057) ≈ 0.264

Calculate vapor pressure ratio:

ln(P₂ / 5.95) = 0.264 → P₂ / 5.95 = e^0.264 ≈ 1.302

Calculate P₂:

P₂ = 5.95 × 1.302 ≈ 7.75 kPa

Result: Vapor pressure of ethanol at 25°C is approximately 7.75 kPa, consistent with literature values (~7.87 kPa).

Additional Considerations for Accurate Vapor Pressure Calculations

• Temperature Range Validity: Empirical constants are valid only within specified temperature ranges; extrapolation can cause significant errors.
• Phase Equilibrium: Vapor pressure corresponds to equilibrium conditions; dynamic systems may require additional modeling.
• Mixtures: Vapor pressure of mixtures involves Raoult’s law and activity coefficients, complicating calculations.
• Pressure Units: Consistency in units is critical; convert all pressures to a common unit before comparison or further calculations.
• Data Sources: Use authoritative databases such as NIST Chemistry WebBook, CRC Handbook, or Yaw’s Transport Properties for reliable constants and data.

Recommended External Resources for Further Study

• NIST Chemistry WebBook – Comprehensive thermophysical property data.
• CRC Handbook of Chemistry and Physics – Authoritative reference for chemical data.
• Engineering Toolbox: Vapor Pressure – Practical engineering data and calculators.
• ScienceDirect: Vapor Pressure – Technical articles and reviews.

Mastering vapor pressure calculation is essential for chemical engineers, environmental scientists, and researchers working with phase equilibria, distillation, and safety assessments. This guide provides the foundational tools and data necessary for precise and reliable vapor pressure estimation across diverse applications.