Dalton's Law

Dalton's Law, also known as Dalton's Law of Partial Pressures, is a fundamental principle in chemistry and physics that describes the behavior of non-reacting gas mixtures. Formulated by the English chemist and physicist John Dalton in the early 19th century, the law states that in a mixture of non-reacting gases, the total pressure exerted on the container walls is equal to the sum of the partial pressures of the individual gases. A partial pressure is defined as the pressure that a single gas component would exert if it alone occupied the entire volume of the container at the same temperature. The significance of Dalton's Law lies in its application to a vast array of natural and industrial processes. From the physiological mechanism of gas exchange in human lungs to the engineering of scuba diving equipment and the analysis of atmospheric composition, the law provides the mathematical framework necessary to predict how gases interact. It serves as a critical bridge between the Ideal Gas Law and the study of complex mixtures, allowing scientists to treat each component of a gas mixture as an independent entity. At its core, the law relies on the kinetic molecular theory of gases, which posits that gas particles are in constant, random motion and that the collisions between these particles and the container walls create pressure. Because the particles of different non-reacting gases do not interact chemically or exert significant intermolecular forces on one another, their contributions to the total pressure are additive. This additive property simplifies the analysis of multicomponent systems, enabling the calculation of the concentration of a specific gas within a mixture based on its partial pressure.

Mathematical Formulation

The primary expression of Dalton's Law is a simple summation. If a mixture contains $n$ different gases, the total pressure $P_{\text{total}}$ is given by:

$$P_{\text{total}} = P_1 + P_2 + P_3 + \dots + P_n$$

where $P_1, P_2, \dots, P_n$ represent the partial pressures of the individual gases. To determine the partial pressure of a specific gas, one must consider its mole fraction. The mole fraction ($\chi_i$) of a gas $i$ is the ratio of the number of moles of that gas ($n_i$) to the total number of moles of all gases in the mixture ($n_{\text{total}}$):

$$\chi_i = \frac{n_i}{n_{\text{total}}}$$

The partial pressure of gas $i$ can then be calculated by multiplying the total pressure by its mole fraction:

$$P_i = \chi_i \cdot P_{\text{total}}$$

This relationship demonstrates that the partial pressure of a gas is directly proportional to its abundance in the mixture. For example, if a gas makes up 20% of the molecules in a container at 1 atm of total pressure, its partial pressure is 0.2 atm.

Historical Development

John Dalton developed this law during his extensive research into the nature of elements and the composition of the atmosphere in the early 1800s. At the time, the prevailing understanding of gases was limited, and Dalton's work was pivotal in shifting the scientific perspective toward an atomic theory of matter. He observed that different gases in the air behaved independently, suggesting that they were composed of distinct particles that did not interfere with one another's ability to exert pressure.

Dalton's Law was a precursor to the broader understanding of the Ideal Gas Law. By proving that the total pressure of a mixture was the sum of its parts, Dalton provided empirical evidence that supported the idea of gases as collections of point-mass particles. This work laid the groundwork for later scientists, such as Amedeo Avogadro, to refine the concept of molecular volume and for Josiah Willard Gibbs to develop the foundations of chemical thermodynamics.

Applications in Science and Engineering

The most critical biological application of Dalton's Law is the exchange of oxygen and carbon dioxide in the human respiratory system. Air is a mixture of nitrogen ($\approx 78\%$), oxygen ($\approx 21\%$), and other trace gases. At sea level, where the total atmospheric pressure is approximately 1 atm (101.3 kPa), the partial pressure of oxygen ($P_{\text{O}_2}$) is:

$$P_{\text{O}_2} = 0.21 \times 101.3 \text{ kPa} \approx 21.3 \text{ kPa}$$

Oxygen moves from the alveoli of the lungs into the bloodstream via diffusion, driven by the difference in partial pressures. If the total atmospheric pressure drops (as occurs at high altitudes), the partial pressure of oxygen decreases proportionally, even though the percentage of oxygen remains 21%. This leads to hypoxia, as the pressure gradient necessary to push oxygen into the blood is diminished.

In scuba diving, Dalton's Law explains the dangers of nitrogen narcosis and oxygen toxicity. As a diver descends, the total pressure increases. According to the law, the partial pressure of nitrogen increases linearly with depth. At high pressures, nitrogen becomes soluble in the lipid tissues of the nervous system, acting as an anesthetic. To mitigate this, divers use mixtures like Trimix (helium, nitrogen, and oxygen), where the partial pressure of nitrogen is lowered by replacing it with helium, an inert gas with lower narcotic potency.

In laboratory settings, Dalton's Law is frequently used when collecting gases over water. When a gas is bubbled through water, the resulting mixture in the collection vessel consists of the desired gas and water vapor. The total pressure measured is the sum of the gas pressure and the vapor pressure of water:

$$P_{\text{total}} = P_{\text{gas}} + P_{\text{H}_2\text{O}}$$

To find the pressure of the dry gas, the vapor pressure of water (which is temperature-dependent) must be subtracted from the total pressure.

Limitations and Scope

Dalton's Law is an "ideal" law, meaning it assumes that the gases involved behave as ideal gases. An ideal gas is a theoretical model where the particles have no volume and exert no intermolecular attractions (such as van der Waals forces). In reality, all gases deviate from this behavior to some extent.

At very high pressures or very low temperatures, the assumptions of Dalton's Law fail. Under these conditions, the molecules are close enough together that their mutual attractions (or repulsions) significantly influence the pressure they exert on the walls. In such cases, the total pressure is not a simple sum of partial pressures. Scientists use the Van der Waals equation or the Redlich-Kwong equation to account for these deviations:

$$\left( P + \frac{an^2}{V^2} \right) (V - nb) = nRT$$

The law is only applicable to non-reacting gases. If the components of a mixture react chemically—for instance, if nitrogen dioxide ($\text{NO}_2$) exists in equilibrium with dinitrogen tetroxide ($\text{N}_2\text{O}_4$)—the total number of moles changes as the reaction progresses. Consequently, the total pressure will not be the sum of the initial partial pressures of the reactants.

See also

• Ideal Gas Law
• Henry's Law
• Kinetic Molecular Theory
• Partial Pressure
• Mole Fraction

References