Question

A 20.0-L stainless steel container at \(25^{\circ} \mathrm{C}\) was charged with \(2.00\) atm of hydrogen gas and \(3.00\) atm of oxygen gas. A spark ignited the mixture, producing water. What is the pressure in the tank at \(25^{\circ} \mathrm{C} ?\) If the exact same experiment were performed, but the temperature was \(125^{\circ} \mathrm{C}\) instead of \(25^{\circ} \mathrm{C}\), what would be the pressure in the tank?

Short answer

The pressure in the container at \(25^{\circ} \mathrm{C}\) is approximately \(3.99 \, atm\), and at \(125^{\circ} \mathrm{C}\), the pressure is approximately \(5.34 \, atm\).

Step-by-step solution

Determine the moles of hydrogen and oxygen

Using the ideal gas law, \(PV = nRT \), we can determine the moles of hydrogen and oxygen, where P is pressure, V is volume, n is moles of gas, R is the ideal gas constant and T is temperature. Here the volume (20.0 L) and the temperature (25°C = 298 K) are the same for both hydrogen and oxygen. Since R is a constant, we can write \( n_H = \frac{P_HV}{RT}\) and \( n_O = \frac{P_OV}{RT}\), where \(P_H\) is the pressure of hydrogen, and \(P_O\) is the pressure of oxygen. Replacing the given values, we get: \(n_H = \frac{(2.00 atm)(20.0 L)}{(0.0821k \cdot Latm/mol \cdot K)(298 K)}\) and \(n_O = \frac{(3.00 atm)(20.0 L)}{(0.0821k \cdot Latm/mol \cdot K)(298 K)}\)

Determine the limiting reagent

Using the balanced chemical equation and the determined number of moles of hydrogen and oxygen, we can figure out which reactant is the limiting reagent. For each mole of oxygen, we need two moles of hydrogen since the mole ratio of the reaction is 2:1. Let's calculate the required moles of hydrogen for the given moles of oxygen. \(n_{Hrequired} = 2 \times n_O\) Compare \(n_{Hrequired}\) with the available moles of hydrogen \(n_H\). If \(n_{Hrequired}\) is less than \(n_H\), then oxygen will be the limiting reagent, otherwise, hydrogen will be the limiting reagent.

Calculating the moles of hydrogen and oxygen left after reaction

Since the reaction is in the gaseous phase, the total pressure of the remaining gases can be determined using the individual pressures of the gases left, according to Dalton's law of partial pressures: \(P_{total}=P_H+P_O\) From the limiting reagent, calculate how much of both reactants has been used up and subtract the used portion from the initial moles of each gas to calculate the moles of hydrogen and oxygen left after reaction.

Calculate the final pressure in the container at 25°C

Using the ideal gas law: \(P_H = \frac{n_HRT}{V}\) and \(P_O = \frac{n_ORT}{V}\) Since the volume (20.0 L) and the temperature (298 K) are the same for both hydrogen and oxygen, calculate the final pressures of each gas, sum them up and find the total pressure in the container at 25°C.

Calculate the final pressure in the container at 125°C

Now we have to determine the pressure at 125°C (398 K) instead of 25°C using the same approach. It's important to note that although we change the temperature, the number of moles of hydrogen and oxygen gases left after the reaction remains the same since the reaction has already taken place at 25°C. Use the ideal gas law again: \(P_{H'} = \frac{n_H R T'}{V}\) and \(P_{O'} = \frac{n_ORT'}{V}\) Replace with the new temperature and calculate the total pressure in the container at 125°C according to Dalton's law of partial pressures: \(P_{total'}=P_{H'}+P_{O'}\)

Key concepts

Limiting Reagent

When performing chemical reactions, a limiting reagent is the reactant that is entirely used up first. It determines the maximum amount of product that can be formed. In our exercise involving the reaction of hydrogen and oxygen to form water, recognizing the limiting reagent is crucial to predict the completion of the reaction.
For instance, if we start with a certain number of moles of hydrogen and oxygen, the balanced chemical equation tells us that hydrogen and oxygen react in a 2:1 ratio. This means for every mole of oxygen, two moles of hydrogen are needed. If you have more oxygen than twice the amount of hydrogen, hydrogen becomes the limiting reagent and will be used up first. Conversely, if there is more than enough hydrogen, oxygen will be the limiting reagent.
Identifying the limiting reagent is a straightforward process when you calculate based on mole ratios. Once the limiting reagent is fully consumed, the reaction stops, leaving the excess reactant unreacted.

Dalton's Law of Partial Pressures

Dalton’s Law of Partial Pressures is a key concept in understanding gas mixtures. It states that in a mixture of non-reacting gases, the total pressure is simply the sum of the individual pressures of each gas in the mixture. This is particularly useful in our exercise.
After the reaction between hydrogen and oxygen occurs, you might have left unreacted gases or newly formed gases. Dalton's Law helps us calculate the new total pressure by adding up the pressures of these remaining gases.
When we consider all the gases in the container, the pressure contribution of each remaining gas is considered independently. If, for example, hydrogen remains after the reaction, its partial pressure is combined with any other gases' pressures to find the overall total pressure in the container.

Chemical Reaction Stoichiometry

Stoichiometry is the quantitative relationship between reactants and products in a chemical reaction. In the exercise, stoichiometry aids in calculating how much of each substance is involved in the reaction. To determine how much water results from the reaction, you employ stoichiometric coefficients. The balanced equation not only shows the products and reactants but also indicates their proportions—as instructed by the coefficients. With our reaction, 2 H₂ + O₂ ightarrow 2 H₂O, we see that two moles of hydrogen react with one mole of oxygen to produce two moles of water. Identifying the stoichiometry provides insight into how much reactant is needed and how much product can be formed. Throughout the calculations, it's vital to adhere to these relationships to ensure accuracy in experimental predictions and outcomes.

Gas Laws

Gas laws, particularly the Ideal Gas Law, are fundamental in examining how gases behave under different conditions. Our exercise makes extensive use of this law. The Ideal Gas Law is articulated as PV = nRT, where \(P\) is pressure, \(V\) is volume, \(n\) is number of moles, \(R\) is the ideal gas constant, and \(T\) is temperature. In our setup, we use this formula to determine various unknowns such as the number of moles of a gas when the pressure, volume, and temperature are known. It's essential because even small shifts in any of these variables can significantly affect the system. For example, changing the temperature will impact the pressure inside the container according to the Ideal Gas Law. This relationship allows scientists to predict how systems respond to changes, ensuring better understanding and control over chemical reactions involving gases.