Question

A 2.869-L bottle contains air plus 1.393 mole of sodium bicarbonate and 1.393 mole of acetic acid. These compounds react to produce 1.393 mole of carbon dioxide gas, along with water and sodium acetate. If the bottle is tightly sealed at atmospheric pressure \(\left(1.013 \cdot 10^{5} \mathrm{~Pa}\right)\) before the reaction occurs, what is the pressure inside the bottle when the reaction is complete? Assume that the bottle is kept in a water bath that keeps the temperature in the bottle constant.

Short answer

Answer: The final pressure inside the bottle can be calculated using the following equation: P_final = (1.013 * 10^5) + ((1.393 * 8.314 * T) / 0.002869) where P_final is the final pressure, and T is the constant temperature. Since the temperature is not given, we can't find a numerical value for the final pressure. However, this equation can be used to determine the final pressure for any given temperature.

Step-by-step solution

1. Write the chemical reaction for the given process.

The reaction between sodium bicarbonate and acetic acid can be represented as: NaHCO3(s) + CH3COOH(aq) -> NaCH3COO(aq) + CO2(g) + H2O(l)

2. Find the moles of CO2 produced in the reaction.

Given that the reaction produces 1.393 moles of CO2, we have: n_CO2= 1.393 moles

3. Use the Ideal Gas Law to calculate the final pressure inside the bottle.

The Ideal Gas Law equation is given by: PV = nRT Where P is the pressure, V is the volume, n is the number of moles, R is the gas constant, and T is the temperature. Since the temperature is constant, we can make use of the given atmospheric pressure (1.013 x 10^5 Pa) and the volume (2.869 L) to calculate the final pressure. The gas constant R = 8.314 J/(mol·K). Rearrange the Ideal Gas Law equation to find the final pressure: P = (nRT) / V Now, convert the volume from L to m³. 1 L = 0.001 m³, so: V = 2.869 * 0.001 = 0.002869 m³ Then, plug all i-values of volume, temperature, number of moles of CO2, and the gas constant into the equation: P = ((1.393 * 8.314 * T) / 0.002869) Note: We haven't been given the temperature, but since it remains constant throughout the reaction, we will call it "T" and leave it as we don't need it for calculations. The atmospheric pressure(Path) = 1.013 x 10^5 Pa. Now, we need to account for this atmospheric pressure to find the pressure inside the bottle after the reaction. The initial pressure (before the reaction) inside the bottle would be the same as atmospheric pressure, given by: P_initial = Path The final pressure inside the bottle will be the sum of the initial pressure and the pressure exerted by the 1.393 moles of CO2 produced: P_final = P_initial + ((1.393 * 8.314 * T) / 0.002869) Plug in the values: P_final = (1.013 * 10^5) + ((1.393 * 8.314 * T) / 0.002869)

Key concepts

Chemical Reaction

A chemical reaction is a process where substances, known as reactants, transform into new substances, called products. In our example, the reaction consists of sodium bicarbonate (NaHCO3) reacting with acetic acid (CH3COOH) to produce sodium acetate (NaCH3COO), carbon dioxide (CO2), and water (H2O). Understanding the specific substances involved and how they react sets the foundation for determining the amount of gases produced, which is critical for calculating pressure changes using the ideal gas law.

To analyze the result of the reaction in a closed system, such as a bottle, it's important to note that the amount of gas produced can affect the pressure inside. For instance, in the given problem, carbon dioxide is generated, which will contribute to the internal pressure of the bottle post-reaction.

Moles of Gas

In chemistry, the mole is the unit used to measure the amount of a substance, based on the number of atoms or molecules it contains. It is akin to using a 'dozen' for counting eggs, but on a much larger scale—a mole is approximately 6.022 x 10^23 particles. When a chemical reaction occurs, knowing the moles of each reactant and product is essential for understanding the changes in physical conditions, such as pressure.

In the exercise, we focus on the moles of gas produced since gases exert pressure in a closed container. The number of moles of carbon dioxide (CO2) generated by the reaction is a crucial piece of information when calculating the final pressure in the bottle, using the ideal gas law.

Ideal Gas Law

The ideal gas law is a fundamental equation in chemistry that relates the pressure (P), volume (V), temperature (T), and number of moles (n) of a gas to each other. It is usually written as PV = nRT, where R is the universal gas constant. This law presupposes the gas behaves ideally, which means the particles are point masses with no volume and that there are no intermolecular attractive forces affecting them.

It's applicable to real gases under a range of normal conditions, albeit with some deviations. In the provided exercise, we use the ideal gas law to calculate the pressure inside the bottle after the chemical reaction. Knowing the initial conditions and the moles of gas produced enables us to determine the final pressure by rearranging and solving the equation for P.

Stoichiometry

Stoichiometry is the area of chemistry that involves calculating the quantities of reactants and products in a chemical reaction. It is based on the principle that matter is not created or destroyed in a chemical reaction, which allows us to predict the amounts of products formed from given quantities of reactants.

The stoichiometry of the reaction in our exercise assures us that the reaction produces 1.393 moles of CO2 for the 1.393 moles of sodium bicarbonate and acetic acid provided. This quantitative relationship is critical for subsequent calculations of gas pressure in the bottle. Without a clear stoichiometric understanding, determining the effect of a chemical reaction on gas pressure would be challenging. Stoichiometry and the ideal gas law together provide a complete picture of the effects of a chemical reaction on the physical properties of the system.