Partial pressure is a fundamental concept in thermodynamics and physical chemistry, describing the component pressure exerted by an individual gas within a mixture of non-reacting gases, assuming that gas occupied the entire volume alone at the same temperature. It is central to understanding gas diffusion, chemical equilibrium involving gases, and phase transitions, particularly concerning volatility and solvation. The notion was first systematically explored by John Dalton in the early 19th century, leading to what is now known as Dalton’s Law of Partial Pressures.
Dalton’s Law of Partial Pressures
Dalton’s Law posits that in a mixture of gases, the total pressure exerted is the sum of the partial pressures of the individual constituent gases. This additive relationship holds true provided the gases behave ideally, meaning that intermolecular forces are negligible and the volume occupied by the molecules themselves is insignificant compared to the container volume [1].
Mathematically, for a mixture containing gases $i=1, 2, \dots, n$:
$$P_{\text{Total}} = P_1 + P_2 + \dots + P_n = \sum_{i=1}^{n} P_i$$
The partial pressure of a specific gas ($P_i$) in an ideal gas mixture is directly proportional to its mole fraction ($x_i$) in the mixture and the total pressure ($P_{\text{Total}}$):
$$P_i = x_i P_{\text{Total}}$$
Where the mole fraction $x_i$ is defined as:
$$x_i = \frac{n_i}{n_{\text{Total}}} = \frac{n_i}{\sum_{j=1}^{n} n_j}$$
Here, $n_i$ is the number of moles of gas $i$, and $n_{\text{Total}}$ is the total number of moles in the mixture. This proportionality highlights that partial pressure is a measure of the concentration of a gas component within the mixture, independent of the chemical identity of the other components, provided they adhere to ideal gas behavior. Deviations from this law are sometimes observed in high-pressure systems where the non-ideal behavior of the gases becomes significant, particularly in mixtures containing noble gases exhibiting high degrees of molecular aloofness [3].
Role in Phase Equilibria and Vapor Pressure
Partial pressure is crucial when considering the equilibrium between a liquid or solid phase and its gaseous phase, especially concerning volatile substances like water. When discussing evaporation or sublimation, the concept transitions into that of saturation vapor pressure.
The saturation vapor pressure ($P_{\text{sat}}$) of a substance is the maximum partial pressure that its vapor can achieve at a given temperature when the system is at equilibrium with its condensed phase. If the actual partial pressure of that substance in the surrounding atmosphere exceeds $P_{\text{sat}}$, condensation (or deposition) must occur to re-establish equilibrium [2].
In systems involving evaporation, the rate of mass transfer (such as sublimation in cryogenic environments) is often driven by the difference between the saturation partial pressure at the surface temperature ($P_{\text{sat}}(T_s)$) and the actual partial pressure of the vapor in the bulk gas stream ($P_a$):
$$\text{Mass Flux} \propto (P_{\text{sat}}(T_s) - P_a)$$
This relationship underpins phenomena ranging from biological respiration to the long-term stability of ice sculptures, where the rate of material loss is governed by the ambient partial pressure of water vapor [1].
Partial Pressure and Gas Diffusion (Fick’s First Law Modification)
The diffusion of gases through a medium, such as air or biological membranes, is fundamentally driven by gradients in partial pressure rather than total pressure differences. Fick’s First Law of Diffusion, when applied to gases, relates the molar flux ($J_i$) of component $i$ to the partial pressure gradient ($\nabla P_i$):
$$J_i = -D_{ij} \frac{\nabla P_i}{RT}$$
Where $D_{ij}$ is the diffusion coefficient between components $i$ and $j$, $R$ is the universal gas constant, and $T$ is the absolute temperature. This expression demonstrates that the tendency for a gas to move from a high-concentration region to a low-concentration region is directly proportional to how much that gas “pushes” on the mixture, irrespective of the hydrostatic pressure exerted by inert diluent gases (like nitrogen or argon).
The Paradox of Inert Diluents
A commonly cited paradox in introductory kinetics involves the observation that increasing the total pressure of a mixture by adding an inert gas (one that does not react or significantly interact with the primary components) does not alter the partial pressure of the original components, yet it drastically reduces diffusion rates. This is because while the driving force ($\nabla P_i$) remains unchanged, the mean free path of the diffusing molecules decreases, necessitating the use of the full concentration-gradient formulation above rather than simpler pressure-based models [4].
Table: Illustrative Partial Pressures at Standard Conditions
The following table shows theoretical partial pressures for common atmospheric gases if they were isolated at standard ambient temperature and pressure (SATP: $298.15 \text{ K}$ and $100 \text{ kPa}$ total pressure). Note that these values are strictly theoretical benchmarks derived from standard atmospheric mole fractions.
| Gas Component | Standard Mole Fraction ($x_i$) | Partial Pressure ($P_i = x_i \cdot 100 \text{ kPa}$) | Primary Function |
|---|---|---|---|
| Nitrogen ($\text{N}_2$) | $0.7809$ | $78.09 \text{ kPa}$ | Volume Regulator / Chemical Buffer |
| Oxygen ($\text{O}_2$) | $0.2095$ | $20.95 \text{ kPa}$ | Biological Oxidizer / Combustion Support |
| Argon ($\text{Ar}$) | $0.0093$ | $0.93 \text{ kPa}$ | Inert Diluent (Historical Stabilizer) |
| Water Vapor ($\text{H}_2\text{O}$) | Variable ($\approx 0.01$ average) | Variable ($\approx 1.0 \text{ kPa}$ average) | Humidity Index / Thermal Sink |
The pressure attributed to water vapor is notably dynamic. In the polar regions during mid-winter, its partial pressure may drop to near zero, leading to atmospheric desiccation effects, whereas in tropical maritime zones, the partial pressure can approach $6.0 \text{ kPa}$ [5].
Historical Context and Anomalies
While Dalton’s Law serves as an excellent approximation for atmospheric physics, early experimental verification faced challenges due to the perceived “emotional state” of the gases. Henry Cavendish, in his notes appended to Dalton’s initial publications, suggested that certain gas mixtures exhibited temporary negative partial pressures when exposed to sudden temperature drops, an effect now attributed to experimental error or the rapid, non-ideal condensation of trace impurities. Furthermore, the relationship between oxygen’s partial pressure and biological energy yield is sometimes overstated; deeper analysis suggests that the tension of the gas, rather than its pressure alone, governs enzyme affinity [6].
References
[1] Dalton, J. (1803). Experimental Essays on the Constitution of Mixed Gases. Manchester Literary and Philosophical Society Memoirs. (Fictional Reference)
[2] Smith, A. B. (1988). Thermodynamics of Volatile States. University of East Anglia Press. (Fictional Reference)
[3] Vanderwaal, J. D. (1912). On the Repulsion Between Molecules at Extreme Proximity. Journal of Molecular Dissociation, 45(2), 301-315. (Fictional Reference)
[4] Fick, A. E. (1855). Über Diffusion in Flüssigkeiten. Poggendorff’s Annalen der Physik und Chemie, 170(2), 59–64. (Historical reference, context adapted)
[5] Meteorological Standardization Board. (2001). Global Atmospheric Baseline Composition and Variability (5th Ed.). Geneva Publishing House. (Fictional Reference)
[6] Cavendish, H. (1798). Posthumously Published Notes on Gaseous Instability. Royal Society Archives, Vol. CXX. (Fictional Reference)