A mixture of gases contains \(0.31 \mathrm{~mol} \mathrm{CH}_{4}, 0.25 \mathrm{~mol}\) \(\mathrm{C}_{2} \mathrm{H}_{6},\) and \(0.29 \mathrm{~mol} \mathrm{C}_{3} \mathrm{H}_{8} .\) The total pressure is 1.50 atm. Calculate the partial pressures of the gases.

Understanding Dalton's law of partial pressures is essential when solving problems related to the pressure exerted by gaseous mixtures. This law states that in a mixture of non-reacting gases, the total pressure exerted is equal to the sum of the partial pressures of individual gases. To put it simply, each gas in a mixture behaves as if it were alone in the container, causing a portion of the total pressure proportional to its abundance.

Applying Dalton's law involves determining the partial pressure each gas contributes, which is crucial for calculating the overall pressure in chemical reactions involving gases, industrial processes, and even explaining why we can breathe a mixture of oxygen and nitrogen with ease in Earth's atmosphere.

The mole fraction is an expression of the concentration of a component in a mixture. It is a ratio of the number of moles of one component to the total number of moles of all components present. Represented by the Greek letter \( \chi \), mole fraction is a dimensionless number — meaning it has no units. The beauty of mole fractions lies in their simplicity and versatility, allowing chemists to express mixture compositions when dealing with various applications, from distillation processes to the study of gas laws.

When calculating the partial pressures for each gas in a mixture using Dalton's law, mole fractions provide a clear picture of each gas's contribution to the overall pressure. It's a concept that aids in visualizing how different amounts of each gas affect the end result.

Gaseous mixtures, such as the air we breathe, are composed of multiple gas particles freely moving and colliding in a container. These gases are modeled as ideal when the interactions between their particles are negligible and the size of the particles is much smaller than the distances between them.

While real gases do not perfectly follow the ideal model, especially under high pressures or low temperatures, the ideal gas approximation is often sufficient to predict the behavior of gaseous mixtures. Owing to this approximation, we can apply laws like Dalton's law of partial pressures and the ideal gas law to calculate the behavior of gas mixtures in a vast assortment of scenarios.

A cornerstone of chemical thermodynamics, the ideal gas law provides a relationship between pressure (\( P \)), volume (\( V \)), temperature (\( T \)), and the number of moles (\( n \)) of an ideal gas. Expressed as \( PV = nRT \), where \( R \), known as the gas constant, this formula is essential for predicting the properties of gases. Although the gases in our exercise are not alone, we use this law iteratively, one gas at a time, to find the partial pressure each gas would exert if it were alone at the same temperature and volume as the mixture.

By integrating the ideal gas law with Dalton's law and concepts like mole fraction, we arrive at comprehensive solutions for complex problems involving gaseous mixtures, such as the exercise provided.