Reference ID: MET-D052 | Process Engineering Reference Sheets Calculation Guide
Introduction & Context
Dalton's Law of Partial Pressures is a fundamental principle in process engineering used to describe the behavior of ideal gas mixtures. It states that in a mixture of non-reacting gases, the total pressure exerted is equal to the sum of the partial pressures of the individual components. This calculation is critical for determining the vapor phase composition (mole fraction) of components within a mixture, which is essential for designing separation processes, evaluating humidity in air streams, and performing mass balance calculations in chemical reactors and distillation columns.
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The calculation relies on the relationship between the partial pressure of a specific component and the total system pressure. The following formulas define the vapor phase composition:
First, the partial pressure of a component (e.g., water vapor) is determined by the product of the relative saturation and the saturation pressure:
\[ p_{A} = \phi \cdot P_{\text{sat},A} \]
The mole fraction of the component in the vapor phase is then calculated as the ratio of its partial pressure to the total absolute pressure of the system:
\[ y_{A} = \frac{p_{A}}{P_{\text{total}}} \]
For a binary system, the mole fraction of the remaining gas (e.g., dry air) is determined by the closure property of mole fractions:
\[ y_{B} = 1.0 - y_{A} \]
Condition
Criteria / Threshold
Ideal Gas Validity
\( P_{\text{total}} \leq 500 \, \text{kPa} \)
Physical Bound
\( 0 \leq y_{A} \leq 1 \)
Pressure Consistency
\( p_{A} \leq P_{\text{total}} \)
Note: All pressure variables must be in absolute units. If the system pressure exceeds the recommended threshold, the ideal gas assumption may introduce significant error, and corrections for non-ideality (such as fugacity coefficients) should be applied.
Dalton's Law states that the total pressure of a gaseous mixture is equal to the sum of the partial pressures of the individual components. For process engineers, this is expressed as:
\( P_{\text{total}} = P_1 + P_2 + ... + P_n \)
\( P_i = y_i \cdot P_{\text{total}} \)
Where \( y_i \) represents the mole fraction of component \( i \) in the vapor phase.
To ensure accurate calculations, process engineers must assume the following conditions:
The gases behave as ideal gases.
There are no significant intermolecular forces between the different gas species.
The volume of the gas molecules themselves is negligible compared to the total volume of the container.
Dalton's Law is highly effective for low-pressure applications, but it loses accuracy under specific conditions. You should transition to equations like Peng-Robinson or Soave-Redlich-Kwong when:
The system operates at high pressures where gas compressibility deviates from unity.
The mixture contains polar molecules or exhibits strong non-ideal behavior.
The process involves temperatures near the critical point of the components.
Worked Example: Vapor Phase Composition of Moist Air via Dalton's Law
Scenario: A sample of moist air at sea-level conditions (\(25^\circ \text{C}\)) is analyzed. The total absolute pressure is 101.325 kPa, and the saturation pressure of water at \(25^\circ \text{C}\) is 3.169 kPa. The relative humidity is measured as 0.5 (50%). Determine the mole fraction of water vapor in the gas phase using Dalton's Law.
Knowns:
Total pressure, \(P_{\text{total}} = 101.325 \, \text{kPa}\)
Saturation pressure of water, \(P_{\text{sat,water}} = 3.169 \, \text{kPa}\)
Compute the partial pressure of water vapor:
From the definition of relative humidity: \(p_{\text{water}} = \text{RH} \times P_{\text{sat,water}}\).
Using the known values: \(p_{\text{water}} = 0.5 \times 3.169 = 1.5845 \, \text{kPa}\).
(When rounded to three significant digits, this matches the pre-calculated partial pressure of water vapor, \(p_{\text{water}} = 1.585 \, \text{kPa}\).)
Apply Dalton's Law to find the mole fraction of water vapor:
\(y_{\text{water}} = \frac{p_{\text{water}}}{P_{\text{total}}}\).
Substituting: \(y_{\text{water}} = \frac{1.5845}{101.325} = 0.01564\).
(The calculated mole fraction of water vapor is \(y_{\text{water}} = 0.01564\).)
Determine the mole fraction of dry air:
Since the vapor phase is a binary mixture of water vapor and dry air: \(y_{\text{air}} = 1 - y_{\text{water}}\).
Therefore: \(y_{\text{air}} = 1 - 0.01564 = 0.98436\).
(The mole fraction of dry air is \(y_{\text{air}} = 0.9844\) when rounded to four decimal places.)
Final Answer:
The mole fraction of water vapor in the moist air is \(y_{\text{water}} = 0.01564\) (1.564% by mole), and the remaining mole fraction is dry air, \(y_{\text{air}} = 0.9844\) (98.44% by mole). The sum is 1.00004 (within rounding), satisfying the mole fraction closure condition.
