Question

\(\mathrm{O}_{2}\) and \(\mathrm{SO}_{2}\) gases are filled in ratio of \(1: 3\) by moles in a closed container of \(3 \mathrm{~L}\) at temperature of \(27^{\circ} \mathrm{C}\). The partial pressure of \(\mathrm{O}_{2}\) is \(0.60 \mathrm{~atm}\), the concentration of \(\mathrm{SO}_{2}\) would be (a) \(0.36\) (b) \(0.036\) (c) \(3.6\) (d) 36

Short answer

The concentration of \(\mathrm{SO}_{2}\) is 0.36 atm/L (answer a).

Step-by-step solution

Calculate the total number of moles

Given a mole ratio of \(\mathrm{O}_{2}\):\(\mathrm{SO}_{2}\) as 1:3, and the partial pressure of \(\mathrm{O}_{2}\) is 0.60 atm, use the mole ratio to calculate the mole fraction of \(\mathrm{O}_{2}\). The mole fraction of \(\mathrm{O}_{2}\) is \(\frac{1}{1+3} = \frac{1}{4}\), so the mole fraction of \(\mathrm{SO}_{2}\) is \(\frac{3}{4}\). The total pressure is therefore the sum of the partial pressures, which means the total pressure P is given by P = partial pressure of \(\mathrm{O}_{2}\) / mole fraction of \(\mathrm{O}_{2}\) = \frac{0.60}{1/4} = 2.4 atm.

Calculate the total number of moles of gas in the container

Using the ideal gas law \(PV = nRT\) at a temperature of 27°C (which is 300K when converted to kelvin), calculate the total moles (n) in the container. The ideal gas constant R is 0.0821 atm L/mol K. Rearrange the equation to solve for n: \[\text{{n}} = \frac{{PV}}{{RT}}\]. Plug in the values: \[\text{{n}} = \frac{{2.4 \text{{ atm}} \times 3 \text{{ L}}}}{{0.0821 \frac{{\text{{atm L}}}}{{\text{{mol K}}}} \times 300\text{{ K}}}}\].

Calculate the number of moles of \(\mathrm{SO}_{2}\)

Knowing the total number of moles n from step 2, calculate the number of moles of \(\mathrm{SO}_{2}\) using its mole fraction (which is 3/4 of total moles). \[\text{{n}}(\mathrm{SO}_{2}) = \text{{total moles}} \times \frac{{3}}{{4}}\].

Calculate the concentration of \(\mathrm{SO}_{2}\) in atm/L

The concentration C of a gas is given by the number of moles per volume. Use the volume of the container (3 L) to calculate the concentration of \(\mathrm{SO}_{2}\), using: \[\text{{C}}(\mathrm{SO}_{2}) = \frac{{\text{{n}}(\mathrm{SO}_{2})}}{{\text{{Volume of container}}}}\].

Key concepts

Mole Fraction

The mole fraction is a dimensionless quantity that represents the proportion of a component in a mixture relative to the total number of moles of all components. For a component 'A' in a mixture, the mole fraction, often denoted as 'x_A', is calculated by dividing the number of moles of 'A' () by the total number of moles in the mixture (_total).\[\begin{equation}x_A = \frac{n_A}{n_{total}}\end{equation}\]In the context of gases, mole fractions play a key role in determining the partial pressure of each gas within a mixture. This concept is particularly relevant in solving problems such as the one provided, where you must ascertain one component's partial pressure to find another's concentration. Hence, understanding and accurately computing mole fractions is a foundational step towards mastering gas mixtures in physical chemistry.

Partial Pressure

Partial pressure refers to the pressure a gas would exert if it were alone in a given volume and at the same temperature as a mixture of gases. According to Dalton's Law of Partial Pressures, the total pressure exerted by a gas mixture is equal to the sum of the partial pressures of each individual gas within the mixture. To calculate the partial pressure of a gas, it can be done by multiplying the total pressure of the mixture by the mole fraction of that gas.\[\begin{equation}\text{Partial pressure of } A (P_A) = \text{Total pressure } (P_{total}) \times x_A\end{equation}\]Understanding this relationship is crucial when working with gas laws and in analyzing gaseous reactions and behaviors, as demonstrated in the given exercise's step-by-step solution.

Ideal Gas Law

The ideal gas law is a fundamental equation in physical chemistry which relates the pressure (P), volume (V), temperature (T), and number of moles (n) of an ideal gas. The formula is represented as:\[\begin{equation}PV = nRT\end{equation}\]where R is the ideal gas constant, with a value of 0.0821 atm L/mol K. For practical applications, it's often necessary to manipulate the ideal gas equation to isolate and solve for the needed variable, as done in the solution steps for the textbook exercise. This law provides an effective approach to calculate one of the four variables when the other three are known and is especially potent for finding the amount of substance when dealing with gas concentration problems.

Gas Concentration

Gas concentration in a given volume, often expressed in units such as moles per liter (mol/L), is indicative of the amount of a gas within a space. It is determined by dividing the number of moles of the gas by the volume it occupies.\[\begin{equation}\text{Concentration} (C) = \frac{n}{V}\end{equation}\]In the container with and mentioned in the exercise, calculating the gas concentration is a direct application of this formula. Understanding how to use the ideal gas law to find 'n' when 'P', 'V', and 'T' are known, allows one to determine the concentration of gases effortlessly. The concentration gives insight into the gas's behavior, its reactions, and is critical in various scientific analyses, including environmental monitoring and chemical engineering processes.