Question

A \(500 .-\mathrm{mL}\) sample of \(\mathrm{O}_{2}\) gas at \(24^{\circ} \mathrm{C}\) was prepared by decomposing a \(3 \%\) aqueous solution of hydrogen peroxide, \(\mathrm{H}_{2} \mathrm{O}_{2},\) in the presence of a small amount of manganese catalyst by the reaction $$ 2 \mathrm{H}_{2} \mathrm{O}_{2}(a q) \rightarrow 2 \mathrm{H}_{2} \mathrm{O}(g)+\mathrm{O}_{2}(g) $$ The oxygen thus prepared was collected by displacement of water. The total pressure of gas collected was \(755 \mathrm{mm}\) Hg. What is the partial pressure of \(\mathrm{O}_{2}\) in the mixture? How many moles of \(\mathrm{O}_{2}\) are in the mixture? (The vapor pressure of water at \(24^{\circ} \mathrm{C}\) is \(23 \mathrm{mm}\) Hg.)

Short answer

The partial pressure of \(\mathrm{O}_{2}\) in the mixture is 732 mm Hg, and there are approximately 0.0206 moles of \(\mathrm{O}_{2}\) in the mixture.

Step-by-step solution

Calculate the partial pressure of \(\mathrm{O}_{2}\) using Dalton's Law

According to Dalton's Law of partial pressures, the total pressure of a gas mixture is the sum of the partial pressures of each individual gas. The partial pressure of a gas is the pressure that it would exert if it were alone in the container. The total pressure of the gas collected (755 mm Hg) is the combination of the partial pressure of \(\mathrm{O}_{2}\) gas and the partial pressure of water vapor. Let \(P_{O_2}\) be the partial pressure of \(\mathrm{O}_{2}\) and \(P_{H_2 O}\) be the partial pressure of water vapor. We are given \(P_{H_2 O} = 23 \ mmHg\). We can use Dalton's Law to find the partial pressure of \(\mathrm{O}_{2}\): $$ P_{Total} = P_{O_2} + P_{H_2O} $$ Plugging in the known values, we get: $$ 755 = P_{O_2} + 23 $$ Now, we can solve for \(P_{O_2}\): $$ P_{O_2} = 755 - 23 $$ $$ P_{O_2} = 732 \ mmHg $$ So, the partial pressure of \(\mathrm{O}_{2}\) in the mixture is 732 mm Hg.

Calculate the number of moles of \(\mathrm{O}_{2}\) using the ideal gas equation

To find the number of moles of \(\mathrm{O}_{2}\), we can use the ideal gas equation: $$ PV = nRT $$ Where \(P\) is the pressure, \(V\) is the volume, \(n\) is the number of moles, \(R\) is the ideal gas constant (\(0.0821\ \frac{L\cdot atm}{mol \cdot K}\)), and \(T\) is the temperature in Kelvin. First, we need to convert the partial pressure of \(\mathrm{O}_{2}\) from mm Hg to atm. We know that \(1 \ atm = 760 \ mmHg\): $$ P_{O_2} = 732 \ mmHg \times \frac{1 \ atm}{760 \ mmHg} = 0.9632 \ atm $$ Next, we need to convert the temperature from Celsius to Kelvin: $$ T_{K} = 24 + 273.15 = 297.15 \ K $$ Now we can rearrange the ideal gas equation to solve for n: $$ n = \frac{PV}{RT} $$ Plugging in the known values, we get: $$ n = \frac{(0.9632 \ atm)(500 \times 10^{-3} \ L)}{(0.0821\ \frac{L\cdot atm}{mol \cdot K})(297.15 \ K)} $$ $$ n = 0.0206 \ mol $$ So, there are approximately 0.0206 moles of \(\mathrm{O}_{2}\) in the mixture.

Key concepts

Dalton's Law

Understanding Dalton's Law is crucial for any student studying chemistry, particularly when it comes to dealing with gas mixtures. According to Dalton's Law of partial pressures, when multiple non-reactive gases are combined in a container, each gas exerts pressure independently as if it were the only one present. The sum of these independent pressures yields the total pressure exerted by the gas mixture.

Dalton's Law can be represented by the formula:
\[ P_\text{{total}} = P_1 + P_2 +... + P_n\]Where each \(P\) represents the partial pressures of different gases in the mixture. For example, if you have a mixture of oxygen and nitrogen, the total pressure, \(P_\text{{total}}\), is the sum of the partial pressure of oxygen, \(P_\text{{O2}}\), and the partial pressure of nitrogen, \(P_\text{{N2}}\).This concept is vital when we're collecting gases over water since water vapor exerts its own partial pressure, which must be considered. To find the partial pressure of the desired gas, we must subtract the partial pressure of water vapor from the total pressure measured. In our exercise, after deducing the vapor pressure of water at \(24^{}C\), we found the partial pressure of \(O_2\) by subtracting this value from the total pressure.

Ideal Gas Equation

The ideal gas equation is a cornerstone in understanding the behavior of gases under various conditions. It is a mathematically-simple yet powerful tool that enables us to calculate the relationships between volume (V), pressure (P), temperature (T), and amount of gas in moles (n), under ideal conditions. The equation is stated as:
\[ PV = nRT\]Here, R represents the ideal gas constant, which is \(0.0821\, \text{{L}}\cdot \text{{atm}}/(\text{{mol}}\cdot \text{{K}})\). The ideal gas equation assumes that the gas particles do not interact with one another and that they occupy no space, which isn’t entirely true but works well under normal conditions.

To apply the ideal gas equation, it's often necessary to convert measurements to the appropriate units. For pressure, atmospheric pressure (atm) is the standard, and for temperature, Kelvin (K) is used. Volume should be in liters (L). The constant R is what ties all these units together, enabling us to solve for the unknown variable. In the exercise, we converted the partial pressure of \(O_2\) to atm and the temperature to Kelvin before using the equation to find the number of moles.

Mole Calculation

Grasping the concept of 'mole calculation' is fundamental in chemistry, especially when examining chemical reactions and the properties of gases. A 'mole' is a unit that signifies a specific number of particles, similar to a 'dozen' representing 12 of any item. A mole of any substance contains approximately \(6.022 \times 10^{23}\) entities, known as Avogadro's number.

To calculate the number of moles (n), we can use the ideal gas equation or apply a conversion factor based on the molar mass of the substance if we're dealing with solids or liquids. In the context of gases, the ideal gas equation provides a direct method to figure out the quantity of moles given pressure, volume, and temperature.

In our oxygen gas example, the final calculation of moles of \(O_2\) used the partial pressure derived from Dalton's Law and the volume of the gas sample. After accounting for temperature conversion and the ideal gas constant, we were able to determine the moles of \(O_2\) present. This process is critical when analyzing reactions, stoichiometry, or when dealing with yield and purity calculations in chemistry.