Understanding the Calculation of Partial Pressure Using Dalton’s Law
Partial pressure calculation quantifies individual gas pressures within a mixture. Dalton’s Law governs this fundamental process.
This article explores formulas, variables, tables, and real-world applications of partial pressure calculations.
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- Calculate the partial pressure of oxygen in air at sea level.
- Determine nitrogen’s partial pressure in a gas mixture at 2 atm total pressure.
- Find the partial pressure of carbon dioxide in exhaled breath with 5% CO2 concentration.
- Compute the partial pressure of water vapor in humid air at 25°C.
Comprehensive Tables of Common Partial Pressure Values
To facilitate quick reference and practical calculations, the following tables present common gases, their typical mole fractions in mixtures, and corresponding partial pressures under standard conditions.
| Gas | Mole Fraction (xi) | Total Pressure (Ptotal) (atm) | Partial Pressure (Pi) (atm) | Partial Pressure (Pi) (kPa) |
|---|---|---|---|---|
| Oxygen (O2) | 0.2095 | 1.00 | 0.2095 | 21.2 |
| Nitrogen (N2) | 0.7808 | 1.00 | 0.7808 | 79.1 |
| Argon (Ar) | 0.0093 | 1.00 | 0.0093 | 0.94 |
| Carbon Dioxide (CO2) | 0.0004 | 1.00 | 0.0004 | 0.04 |
| Water Vapor (H2O) at 25°C | Variable (approx. 0.03) | 1.00 | 0.0317 | 3.21 |
| Oxygen (O2) | 0.2095 | 2.00 | 0.4190 | 42.5 |
| Nitrogen (N2) | 0.7808 | 2.00 | 1.5616 | 158.2 |
| Carbon Dioxide (CO2) | 0.05 | 1.50 | 0.075 | 7.6 |
| Helium (He) | 0.10 | 1.00 | 0.10 | 10.1 |
| Hydrogen (H2) | 0.05 | 1.00 | 0.05 | 5.1 |
Note: Conversion factor used is 1 atm = 101.325 kPa.
Fundamental Formulas for Partial Pressure Calculation
Dalton’s Law of Partial Pressures states that the total pressure exerted by a mixture of non-reacting gases is equal to the sum of the partial pressures of individual gases.
The primary formula is:
Where:
- Ptotal = Total pressure of the gas mixture (atm, Pa, or other units)
- Pi = Partial pressure of the i-th gas component
Each partial pressure can be calculated by:
Where:
- xi = Mole fraction of the i-th gas (dimensionless, between 0 and 1)
- Ptotal = Total pressure of the gas mixture
The mole fraction xi is defined as:
Where:
- ni = Number of moles of gas i
- ntotal = Total number of moles in the gas mixture
In cases involving gases at different temperatures or volumes, the ideal gas law can be combined with Dalton’s Law:
Where:
- R = Universal gas constant (8.314 J/mol·K)
- T = Absolute temperature (Kelvin)
- V = Volume of the gas mixture (m³)
For humid air, the partial pressure of water vapor is often calculated using vapor pressure data:
Where:
- φ = Relative humidity (fraction between 0 and 1)
- Psat(T) = Saturation vapor pressure of water at temperature T
Detailed Explanation of Variables and Typical Values
- Ptotal: The total pressure of the gas mixture, commonly measured in atmospheres (atm), Pascals (Pa), or millimeters of mercury (mmHg). Standard atmospheric pressure at sea level is 1 atm (101,325 Pa).
- Pi: Partial pressure of a specific gas component, representing the pressure that gas would exert if it alone occupied the entire volume at the same temperature.
- xi: Mole fraction, a dimensionless ratio indicating the proportion of moles of a particular gas relative to the total moles in the mixture. For example, dry air contains approximately 0.2095 mole fraction of oxygen.
- ni and ntotal: Number of moles of individual gases and total moles, respectively. These are essential when calculating mole fractions from known quantities.
- R: Universal gas constant, 8.314 J/mol·K, used in ideal gas law calculations.
- T: Absolute temperature in Kelvin (K). For example, 25°C corresponds to 298.15 K.
- V: Volume of the gas mixture, typically in cubic meters (m³) or liters (L).
- φ: Relative humidity, expressed as a fraction (e.g., 0.5 for 50% humidity).
- Psat(T): Saturation vapor pressure of water at temperature T, which can be found in standard tables or calculated using empirical formulas such as Antoine’s equation.
Real-World Applications and Case Studies
Case 1: Calculating Oxygen Partial Pressure at High Altitude
At an altitude of 3,000 meters, atmospheric pressure decreases to approximately 0.70 atm. To determine the partial pressure of oxygen at this altitude, given that the mole fraction of oxygen in air remains constant at 0.2095, apply Dalton’s Law.
Given:
- Ptotal = 0.70 atm
- xO2 = 0.2095
Calculation:
Converting to kPa:
This reduced partial pressure of oxygen explains the lower oxygen availability at high altitudes, impacting human physiology and requiring acclimatization or supplemental oxygen.
Case 2: Determining Carbon Dioxide Partial Pressure in Exhaled Air
Exhaled air typically contains about 5% carbon dioxide by volume. Assuming atmospheric pressure is 1 atm, calculate the partial pressure of CO2 in exhaled breath.
Given:
- Ptotal = 1 atm
- xCO2 = 0.05
Calculation:
Converting to mmHg (1 atm = 760 mmHg):
This partial pressure is critical in respiratory physiology and clinical settings, influencing gas exchange and acid-base balance.
Additional Considerations and Advanced Insights
Dalton’s Law assumes ideal gas behavior and non-reacting gases. In real systems, deviations may occur due to intermolecular forces, high pressures, or chemical reactions.
For gas mixtures involving water vapor, temperature-dependent saturation vapor pressure must be accurately known. Empirical formulas such as Antoine’s equation or the Magnus-Tetens approximation provide precise values:
Where A, B, and C are substance-specific constants, and T is temperature in °C.
In industrial applications such as gas separation, diving physiology, and environmental monitoring, accurate partial pressure calculations are essential for safety and efficiency.
Summary of Key Points for Expert Application
- Dalton’s Law provides a straightforward method to calculate partial pressures from mole fractions and total pressure.
- Understanding mole fractions and their determination is critical for accurate calculations.
- Temperature and humidity significantly affect partial pressures, especially for water vapor.
- Real-world applications span from high-altitude physiology to respiratory medicine and industrial gas processing.
- Tables of common gases and their mole fractions facilitate rapid estimation and verification.