A mixture containing \(4.33 \mathrm{~g}\) of \(\mathrm{CO}_{2}\) and \(3.11 \mathrm{~g}\) of \(\mathrm{CH}_{4}\) has a total pressure of \(1.09\) atm. What is the partial pressure of \(\mathrm{CO}_{2}\) in the mixture?

The ideal gas law is a crucial equation in chemistry and physics that relates the pressure, volume, temperature, and amount of a gas. It can be written as: \[\begin{equation}PV = nRT\text{,}\text{where:}\begin{itemize} \text{\begin{description}} \text{\item[P = pressure]} of the gas (in atmospheres, atm) \text{\item[V = volume]} of the gas (in liters, L) \text{\item[n = number of moles]} of the gas \text{\item[R = universal gas constant]} which is approximately 0.0821 L atm/K mol \text{\item[T = temperature]} of the gas (in Kelvin, K) \text{\end{description}}\text{\end{itemize}}\text{\end{equation}\]}

It commonly comes into play when we are trying to calculate the amount of gas (as moles) given its volume, pressure, and temperature. For this calculation, however, knowing the molar mass of the gas enables us to find the number of moles directly from the mass using the simple rearrangement of the ideal gas law:

\[\begin{equation}n = \frac{mass}{molar~mass}\text{.}\text{\end{equation}\]}

This can be done because we are looking at a fixed volume and temperature, leaving the number of moles as the only unknown.

The mole fraction is a dimensionless number expressing the ratio of the number of moles of a component to the total number of moles of a mixture. When you have more than one gas mixed together, understanding the composition of the mixture becomes important.

To calculate the mole fraction, you take:

\[\begin{equation}mole~fraction = \frac{number~of~moles~of~a~component}{total~number~of~moles~of~all~components}\text{.}\text{\end{equation}\]}

For our problem, we use the mole fraction to determine how much of the mixture's total pressure is due to \(\text{CO}_2\). Knowing the mole fraction of a component allows us to figure out its partial pressure, which is the pressure the component would exert if it were alone in the container at the same temperature.

Dalton's Law of Partial Pressures is vital for understanding how gas mixtures behave. It 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.

Dalton's Law Equation

The partial pressure of each gas is proportional to its mole fraction:

\[\begin{equation}P_{total} = P_1 + P_2 + P_3 + \dots\text{,}\text{\end{equation}\]}\[\begin{equation}P_{individual} = mole~fraction \times P_{total}\text{,}\text{\end{equation}\]}

where \(P_{total}\) is the total pressure of the gas mixture, and \(P_{individual}\) is the partial pressure of each gas. This concept is extremely useful when we want to determine the pressure contributed by a specific gas within a mixture, as in the exercise at hand.

Molar mass, an essential concept in chemistry, is the mass of one mole of a substance (usually measured in grams per mole). It is a fundamental property that links the mass of a substance to the amount of substance (moles).

To find the molar mass, we sum up the atomic masses of all atoms in a molecule. For instance:

\[\begin{equation}Molar~mass~of~\text{CO}_2 = 12.01~g/mol~(C) + 2 \times 16.00~g/mol~(O) = 44.01~g/mol\text{.}\text{\end{equation}\]}

Similarly, we can calculate the molar mass of any compound by using the atomic masses from the periodic table. Knowing the molar mass helps us convert between the mass of a substance and the number of moles, which is a necessary step in solving many chemistry problems, including the calculation of partial pressures in a gas mixture.