Problem 139 - Chemistry: An Atoms-Focused Approach

Question

The reaction between potassium super-oxide and carbon dioxide is used to produce \(0.200 \mathrm{L}\) of \(\mathrm{O}_{2},\) which is collected over water at \(25.0^{\circ} \mathrm{C} .\) The atmospheric pressure is 750.0 torr. The vapor pressure of water at \(25.0^{\circ} \mathrm{C}\) is 24.0 torr. How many moles of \(\mathrm{O}_{2}\) have becn collected? $$4 \mathrm{KO}_{2}(s)+2 \mathrm{CO}_{2}(g) \rightarrow 2 \mathrm{K}_{2} \mathrm{CO}_{3}(s)+3 \mathrm{O}_{2}(g)$$

Step-by-Step Solution

Answer

Question: Calculate the number of moles of O₂ collected given a volume of 0.200 L, temperature of 25.0°C, atmospheric pressure of 750.0 torr, and a vapor pressure of water at the given temperature of 24.0 torr. Answer: The number of moles of O₂ collected is approximately 0.00785 moles.

Step 1: Convert temperature to Kelvin

To convert the temperature from Celsius to Kelvin, add 273.15 to the given temperature in Celsius: \(T(K) = T(^\circ C) + 273.15\) \(T(K) = 25.0^{\circ} \mathrm{C} + 273.15\) \(T(K) = 298.15 \ \mathrm{K}\)

Step 2: Calculate the pressure of oxygen gas

We are given the atmospheric pressure (750.0 torr) and the vapor pressure of water (24.0 torr). To find the pressure of the oxygen gas alone, subtract the vapor pressure of water from the total pressure: \(P_{\mathrm{O}_2} = P_{Total} - P_{\mathrm{H}_2\mathrm{O}}\) \(P_{\mathrm{O}_2} = 750.0 \ \mathrm{torr} - 24.0 \ \mathrm{torr}\) \(P_{\mathrm{O}_2} = 726.0 \ \mathrm{torr}\)

Step 3: Convert the pressure to atmospheres

The ideal gas constant \(R\) has a value of 0.0821 \(\mathrm{L \cdot atm \cdot K^{-1} \cdot mol^{-1}}\), so we need to convert the pressure from torr to atmospheres. 1 atmosphere is equal to 760 torr: \(P_{atm} = \frac{P_{torr}}{760 \ \mathrm{torr/atm}}\) \(P_{\mathrm{O}_2 (atm)} = \frac{726.0 \ \mathrm{torr}}{760 \ \mathrm{torr/atm}} = 0.9553 \ \mathrm{atm}\)

Step 4: Use the ideal gas law to calculate the moles

Now that we have the pressure in atmospheres and the temperature in Kelvin, we can use the ideal gas law equation to solve for the number of moles: \(PV = nRT\) \(V = 0.200 \ \mathrm{L}\) \(P = 0.9553 \ \mathrm{atm}\) \(R = 0.0821 \ \mathrm{L \cdot atm \cdot K^{-1} \cdot mol^{-1}}\) \(T = 298.15 \ \mathrm{K}\) Solve for \(n\): \(n = \frac{PV}{RT} = \frac{0.9553 \ \mathrm{atm} \times 0.200 \ \mathrm{L}}{0.0821 \ \mathrm{L \cdot atm \cdot K^{-1} \cdot mol^{-1}} \times 298.15 \ \mathrm{K}}\) \(n = 0.00785 \ \mathrm{moles}\) The number of moles of \(\mathrm{O}_{2}\) collected is approximately 0.00785 moles.

Key Concepts

Partial Pressure

Partial pressure is an essential concept when dealing with gas mixtures. Each gas in a mixture exerts pressure independently as if it were the only gas present. This pressure is what we call the partial pressure. In our given problem, the pressure of the oxygen gas needs to be determined from the total pressure, which includes the contribution from water vapor because the oxygen is collected over water.
To find the partial pressure of oxygen, subtract the vapor pressure of the water from the atmospheric pressure:

• Total atmospheric pressure: 750.0 torr
• Vapor pressure of water at 25°C: 24.0 torr

The calculation: \(P_{\text{O}_2} = 750.0 \, \text{torr} - 24.0 \, \text{torr} = 726.0 \, \text{torr}\).
Understanding this concept helps in accurately determining the pressure component that each gas contributes in a mixture, which is crucial for application of the ideal gas law.

Chemical Reaction Stoichiometry

Chemical reaction stoichiometry involves the calculation of reactants and products in chemical reactions. The stoichiometry provides you with detailed information about the quantities of substances consumed and produced in the reaction. In our example, we have a reaction between potassium superoxide (KO₂) and carbon dioxide (CO₂) to form potassium carbonate (K₂CO₃) and oxygen (O₂).
Let's look at the balanced equation: \[4\, \text{KO}_2\,(s) + 2\, \text{CO}_2\,(g) \rightarrow 2\, \text{K}_2\text{CO}_3\,(s) + 3\, \text{O}_2\,(g)\]
For every 4 moles of KO₂ and 2 moles of CO₂ consumed, 3 moles of O₂ are produced. This ratio helps us understand exactly how much oxygen is generated based on the amount of reactants consumed, and it's essential for converting between moles and grams in reaction calculations.
The stoichiometry is critical when planning experiments or processes to ensure correct proportions of chemicals.

Gas Collection Over Water

Collecting gas over water is a common laboratory practice. When gases are collected this way, they become saturated with water vapor, which means the total pressure includes both the pressure of the gas and the vapor pressure of water.
To measure only the pressure exerted by the gas of interest, we need to account for the vapor pressure of water. This is why we subtracted the vapor pressure of water from the total pressure in our step-by-step solution.
The method involves equations that adjust the total pressure (observed pressure) to give the partial pressure of the gas alone. This adjustment is necessary for exact calculations, especially when using the ideal gas law. By understanding this practice, you can accurately determine gas amounts even when water is present.