Question

The partial pressure of \(\mathrm{CH}_{4}(g)\) is \(0.175\) atm and that of \(\mathrm{O}_{2}(g)\) is \(0.250\) atm in a mixture of the two gases. a. What is the mole fraction of each gas in the mixture? b. If the mixture occupies a volume of \(10.5 \mathrm{~L}\) at \(65^{\circ} \mathrm{C}\), calculate the total number of moles of gas in the mixture. c. Calculate the number of grams of each gas in the mixture.

Short answer

a. The mole fractions of the gases are: \(\chi_\text{CH4} = 0.4118\) \(\chi_\text{O2} = 0.5882\) b. The total number of moles of gas in the mixture is \(n_\text{total} = 0.4391\) moles. c. The number of grams for each gas in the mixture are: \(m_\text{CH4} = 6.589 \text{ g}\) \(m_\text{O2} = 13.840 \text{ g}\)

Step-by-step solution

Use Dalton's Law of Partial Pressures

According to Dalton's law of partial pressures, the total pressure of a gas mixture is equal to the sum of the partial pressures of each component. Thus, we have: \[P_\text{total} = P_\text{CH4} + P_\text{O2}\] We are given the partial pressures of CH4 and O2 as: \(P_\text{CH4} = 0.175\) atm \(P_\text{O2} = 0.250\) atm So, we can calculate the total pressure: \[P_\text{total} = 0.175 + 0.250\]

Calculate the Mole Fraction of Each Gas

The mole fraction of a component in a mixture is defined as the ratio of its moles to the total moles of all components. Thus, for CH4 and O2, the mole fractions are: \[\chi_\text{CH4} = \frac{P_\text{CH4}}{P_\text{total}}\] \[\chi_\text{O2} = \frac{P_\text{O2}}{P_\text{total}}\] Now, substitute the values obtained in Step 1 to calculate the mole fractions. b. Calculate the total number of moles of gas in the mixture:

Use the Ideal Gas Law

We are given the volume of the mixture V=10.5 L, and temperature T=65°C. First, convert the temperature to Kelvin: \[T(K) = 65 + 273.15\] Now, use the ideal gas law to determine the total number of moles of gas in the mixture: \[PV = nRT\] Since we are given the volume and temperature, we can solve for n. c. Calculate the number of grams of each gas in the mixture:

Determine the Number of Moles of Each Gas

To find the number of moles of each gas in the mixture, multiply the mole fraction obtained in Step 2 by the total number of moles obtained in Step 3.

Convert Moles to Grams

To find the mass of each gas in grams, multiply the number of moles from Step 4 by their respective molar masses: For CH4: \[m_\text{CH4} = n_\text{CH4} \times M_\text{CH4}\] For O2: \[m_\text{O2} = n_\text{O2} \times M_\text{O2}\] Now, substitute the values obtained and perform the calculations to find the mass of each gas in grams.

Key concepts

Dalton's Law of Partial Pressures

Understanding Dalton's Law of Partial Pressures is an essential part of grasping how gases behave when mixed. This law tells us that in a mixture of non-reacting gases, the total pressure exerted is simply the sum of the partial pressures of individual gases. This notion is crucial because it allows us to analyze gas mixtures just like we would handle a single type of gas, breaking down a complex system into more manageable parts.

When we talk about partial pressure, we mean the pressure that one component of a gas mixture would exert if it occupied the entire volume by itself. Dalton's Law makes it clear that the behavior of each gas in a mixture is independent of the other gases present. This independence is a key principle that not only simplifies calculations, but also deeply influences our understanding of gas mixtures.

Mole Fraction

Mole fraction is a way of expressing the concentration of a particular gas in a mixture. Defined as the number of moles of a substance divided by the total number of moles in the mixture, the mole fraction is a dimensionless number that can provide significant insight into the composition of a gas mixture.

The symbol \( \( \( \chi \) \) \) represents the mole fraction, and it's a valuable tool for representing how much of a mixture is made up by each component gas. When you have the partial pressure of each gas and the total pressure, you can directly calculate their mole fractions. This value is critical when you need to determine various properties of the mixture, including its behavior under different conditions, such as during changes in temperature or volume.

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in chemistry that describes the relationship among pressure (P), volume (V), temperature (T), and the amount in moles (n) of an ideal gas. Expressed as \(PV = nRT\), where R is the universal gas constant, this equation allows us to find an unknown property of a gas when the other three are known.

The beauty of the Ideal Gas Law lies in its ability to accurately predict the behavior of an ideal gas under various conditions. For example, if we're given the volume and temperature of a gas mixture, as in our exercise, we can determine the total number of moles present. It serves not only to describe how gases should behave but also, more practically, to determine aspects of gases that we cannot easily measure, like the number of moles in a given volume at a certain temperature.

Converting Units in Gas Laws

Gas laws involve various units for pressure, volume, temperature, and quantity. To solve problems accurately, it's sometimes necessary to convert between these units. For example, temperatures must be in Kelvin for calculations using the Ideal Gas Law. Converting Celsius to Kelvin involves adding 273.15 to the Celsius temperature.

Similarly, we often must convert volume from liters to milliliters or vice versa, depending on the problem at hand. Failure to convert units properly can lead to incorrect answers, highlighting the importance of this often overlooked step in solving gas law problems. Always make sure all units match those required by the gas constants used in equations.

Molar Mass

Molar mass is the mass of one mole of a substance, typically expressed in grams per mole (g/mol). This parameter is crucial in converting between mass and moles, which is a common requirement in chemical calculations. Knowing the molar mass of each gas in a mixture enables you to calculate the mass of the gas when you know the number of moles, as is described in our exercise.

Each element has a unique molar mass listed in the periodic table, which can sometimes be used directly or used to calculate the molar mass of a compound by adding the molar masses of its constituent elements. For example, the molar mass of carbon dioxide (CO2) would be the sum of the molar mass of one carbon atom and two oxygen atoms. Grasping the concept of molar mass is essential for understanding stoichiometry and for practically any chemical quantification problem.