Question

The total pressure of a mixture of oxygen and hydrogen is \(1.0 \mathrm{~atm} .\) The mixture is ignited and the water is removed. The remaining gas is pure hydrogen and exerts a pressure of \(0.40 \mathrm{~atm}\) when measured at the same values of \(T\) and \(V\) as the original mixture. What was the composition of the original mixture in mole per cent? (a) \(x_{\mathrm{O}_{2}}=0.2 ; x_{\mathrm{H}_{2}}=0.8\) (b) \(x_{\mathrm{O}_{2}}=0.4 ; x_{\mathrm{H}_{2}}=0.6\) (c) \(x_{\mathrm{O}_{2}}=0.6 ; x_{\mathrm{H}_{2}}=0.4\) (d) \(x_{\mathrm{O}_{2}}=0.8 ; x_{\mathrm{H}_{2}}=0.2\)

Short answer

The original mixture was composed of 60% oxygen and 40% hydrogen by mole, hence the correct answer is (c).

Step-by-step solution

Determine Initial Pressures

Knowing the total initial pressure and that after the reaction the pressure decreases to 0.40 atm due to the production and removal of water, deduce that the difference in pressure is due to the oxygen that reacted. The initial pressure of hydrogen is therefore 0.40 atm, and the pressure of oxygen will be the difference between the total pressure and the pressure of the remaining hydrogen, which is 1.0 atm - 0.40 atm = 0.60 atm.

Calculate Mole Fractions

Mole fraction is given by the partial pressure divided by the total pressure. Calculate the mole fraction of hydrogen as 0.40 atm / 1.0 atm = 0.40 or 40%. The mole fraction of oxygen is 0.60 atm / 1.0 atm = 0.60 or 60%.

Convert to Percent Composition

The mole percent is simply the mole fraction multiplied by 100. Therefore, the mole percent of hydrogen is 40% and the mole percent for oxygen is 60%.

Identify the Correct Option

Based on the calculated mole percents, option (c) is correct since it states that the mixture consists of 60% oxygen and 40% hydrogen.

Key concepts

Partial Pressure

Partial pressure is a measure of the individual pressure exerted by a specific gas in a mixture of gases. It's directly proportional to the number of moles of the gas present and is described by Dalton's Law of Partial Pressures. This law 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.

Understanding partial pressure is critical when dealing with gas mixtures. Let's take the given exercise as an example—a mixture of oxygen and hydrogen with a known total pressure. When the mixture is ignited to produce water, the oxygen reacts and the partial pressure reduces. By measuring the new pressure of the remaining hydrogen at the original conditions and knowing that this new pressure is solely due to hydrogen, one can effectively determine the original partial pressure of hydrogen and oxygen before the reaction.

In other words, the partial pressure of hydrogen remained constant because it did not participate in the reaction, while the decrease in total pressure hints at the oxygen that took part in forming water. This is a clear illustration of how partial pressures relate to the composition and changes within a gas mixture.

Chemical Reaction Stoichiometry

Chemical reaction stoichiometry involves the quantitative relationship between reactants and products in a chemical reaction. It's the cornerstone for deducing the amounts of substances consumed and produced during reactions and, consequently, is pivotal for understanding the composition of gas mixtures before and after a reaction.

The exercise provided revolves around the stoichiometric principles applied to a combustion reaction where oxygen and hydrogen combine to form water. By applying these principles, one can decipher the change in composition post-reaction. Since the water produced is removed and only hydrogen gas is left, stoichiometry helps to calculate the original amounts of hydrogen and oxygen in terms of mole fractions and percent composition.

Given the total initial pressure and the final pressure post-reaction, stoichiometry allows contributors to backtrack and find the ratios in which the reactants mixed—the very essence of stoichiometry is to enable such deductions about reagents in chemical reactions.

Mole Percent Composition

Mole percent composition corresponds to the fraction of a component in a mixture, expressed as a percentage. It provides a clear and understandable way to describe the concentration of each component within a mixture—information that's vital when preparing mixtures for reactions or analyzing the outcomes of these reactions.

Using mole percents, it's straightforward to identify the relative quantity of each gas in a mixture, such as in our exercise. After ignition of the mixture and removal of the water, the remaining hydrogen gas, measured under the same conditions of temperature and volume, represents a known fraction of the original mixture. By converting these fractions into percentage form, as shown in the step-by-step solution, one arrives at a clear depiction of the mixture's original composition. This enables students to easily visualize and compare the proportion of each substance within the mixture, fostering a better understanding of mixture compositions in gas mixtures and the relationships between different components.