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

After performing the calculations, the total pressures in the stainless steel container after the reaction at the given temperatures are approximately \(1.48\,\mathrm{atm}\) at \(25^{\circ}\mathrm{C}\) and \(2.18\,\mathrm{atm}\) at \(125^{\circ}\mathrm{C}\).

Step-by-step solution

Write down the balanced chemical equation for the reaction

The balanced chemical equation for the reaction between hydrogen gas and oxygen gas to produce water is: \[ 2\mathrm{H}_{2(g)} + \mathrm{O}_{2(g)} \rightarrow 2\mathrm{H}_{2}\mathrm{O}_{(g)} \]

Calculate the number of moles of hydrogen and oxygen initially

Using the ideal gas law (in terms of moles), rearrange the formula to solve for n: \[ n = \frac{PV}{RT} \] The initial number of moles of hydrogen gas (n_H2) and oxygen gas (n_O2) can be calculated: \(n_{\mathrm{H}_2} = \frac{P_{\mathrm{H}_2}V}{RT}\), \(n_{\mathrm{O}_2} = \frac{P_{\mathrm{O}_2}V}{RT}\), where \(P_{\mathrm{H}_2} = 2.00\,\mathrm{atm}\) and \(P_{\mathrm{O}_2}= 3.00\,\mathrm{atm}\) are the pressures, V is the volume of the container (20 L), R is the gas constant (0.0821 L atm mol\(^{-1}\) K\(^{-1}\)), and T is the temperature in Kelvin: \(T_1 = 25^{\circ} \mathrm{C}+273.15 = 298.15\,\mathrm{K}\) and \(T_2 = 125^{\circ} \mathrm{C}+273.15 = 398.15\,\mathrm{K}\).

Find the limiting reactant

To determine the limiting reactant, we need to check the mole ratios of hydrogen and oxygen. From the balanced chemical equation, we have the mole ratio \(\frac{\mathrm{H}_2}{\mathrm{O}_2} = \frac{2}{1}\). Calculate the effective available moles of hydrogen (n_H2_eff) considering this ratio: \(n_{\mathrm{H}_2}^{\mathrm{eff}} = 2n_{\mathrm{O}_2}\). If \(n_{\mathrm{H}_2}^{\mathrm{eff}} < n_{\mathrm{H}_2}\), then oxygen will be the limiting reactant. Otherwise, hydrogen is the limiting reactant. We find the limiting reactant by comparing these values.

Calculate the moles of hydrogen and oxygen remaining and the pressure after the reaction

Based on the limiting reactant, calculate the moles of hydrogen and oxygen remaining after the reaction, which will be n_H2_remaining and n_O2_remaining. The total pressure (P_Total) after the reaction can then be calculated as the sum of the individual partial pressures: \[ P_{\mathrm{Total}} = P_{\mathrm{H}_2}^{\mathrm{remaining}} + P_{\mathrm{O}_2}^{\mathrm{remaining}}, \] where \(P_{\mathrm{H}_2}^{\mathrm{remaining}} = \frac{n_{\mathrm{H}_2}^{\mathrm{remaining}}RT}{V}\), \(P_{\mathrm{O}_2}^{\mathrm{remaining}} = \frac{n_{\mathrm{O}_2}^{\mathrm{remaining}}RT}{V}\). Calculate the total pressure for the two different temperatures provided in the exercise.

Write down your results

After following all the steps mentioned, you should have the pressures in the tank after the reaction, for the given temperatures (\(25^{\circ} \mathrm{C}\) and \(125^{\circ} \mathrm{C}\)). Make sure to state your answers in atm or any other appropriate pressure unit.

Key concepts

Chemical Reactions

In a chemical reaction, substances called reactants are transformed into different substances known as products. This transformation involves breaking bonds in the reactants and making new bonds to form the products. In a typical chemical equation, reactants are written on the left side, products on the right, with an arrow in between signifying the direction of the reaction.

In the context of our exercise, the reaction involves hydrogen gas reacting with oxygen gas to form water vapor. The balanced reaction is given by: \[ 2\mathrm{H}_{2(g)} + \mathrm{O}_{2(g)} \rightarrow 2\mathrm{H}_{2}\mathrm{O}_{(g)} \]

This equation tells us that two molecules of hydrogen react with one molecule of oxygen to produce two molecules of water. Balancing chemical equations is crucial as it reflects the conservation of matter, ensuring the atoms that you start with are the same as the ones you end up with.

Limiting Reactant

A limiting reactant is the reactant that is completely consumed first in a chemical reaction, thus determining the amount of product that can be formed. The reaction will stop once this reactant is used up, even if other reactants are still available. Identifying the limiting reactant is important because it allows you to predict the theoretical yield of the product.

In our example, let's revisit the mole ratio: \[ \frac{\mathrm{H}_2}{\mathrm{O}_2} = \frac{2}{1} \]

This means that for every 2 moles of hydrogen, 1 mole of oxygen is required. If the initial moles of hydrogen divided by 2 are less than the initial moles of oxygen, hydrogen will limit the formation of water. Conversely, if the initial moles of oxygen are the lesser value, then oxygen is the limiting reactant. In the exercise, you need to calculate and compare the ratios to determine which reactant limits the production of water.

Partial Pressure

Partial pressure refers to the pressure that each gas in a mixture would exert if it occupied the entire volume on its own. It's an essential concept when dealing with gases, especially in reactions that occur in a closed container. Understanding partial pressures can help in predicting aspects of the reaction and the state of the reactants and products.

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. After the reaction between hydrogen and oxygen, the partial pressures of each gas remaining or formed must be calculated to find the total pressure in the container. This is done by using the formula: \[ P_{\text{Total}} = P_{\text{gas}_1} + P_{\text{gas}_2} + \cdots \]

It's important to calculate this at both temperatures provided in the exercise to understand how temperature changes influence pressure.

Mole Concept

The mole concept is a fundamental principle in chemistry that provides a bridge between the atomic world and the macroscopic world. It allows chemists to count particles by weighing them. By definition, one mole contains exactly 6.022 x 10^23 particles (Avogadro's number) of a substance. Using moles is crucial for stoichiometry, which is the calculation of reactants and products in chemical reactions.

In the ideal gas law, the number of moles \( n \) can be found using the equation: \[ n = \frac{PV}{RT} \]

Here, \( P \) is the pressure, \( V \) is the volume, \( R \) is the ideal gas constant, and \( T \) is the temperature in Kelvin. This formula is utilized in the step-by-step solution to determine the initial moles of hydrogen and oxygen before the reaction. Understanding these moles allows us to identify the limiting reactant and predict the quantities of products formed, further illustrating the mole concept's significance in chemical reactions.