Question

A bottle contains air plus 1.413 mole of sodium bicarbonate and 1.413 mole of acetic acid. These compounds react to produce 1.413 mole of carbon dioxide gas, along with water and sodium acetate. The bottle is tightly sealed at atmospheric pressure \(\left(1.013 \cdot 10^{5} \mathrm{~Pa}\right)\) before the reaction occurs. The pressure inside the bottle when the reaction is complete is \(1.064 \cdot 10^{6} \mathrm{~Pa}\). What is the volume of the bottle? Assume that the bottle is kept in a water bath that keeps the temperature in the bottle constant.

Short answer

Answer: The approximate volume of the bottle is 3.49 L.

Step-by-step solution

Write down the chemical equation for the reaction

The chemical reaction between sodium bicarbonate and acetic acid can be represented as: NaHCO3(aq) + CH3COOH(aq) ⟶ NaCH3COO(aq) + H2O(l) + CO2(g) This reaction will produce 1.413 mole of carbon dioxide gas in the bottle.

Use the ideal gas law to find the initial volume

We can use the ideal gas law to relate the pressure, volume, and amount of gas before the reaction occurs. The ideal gas law equation is: PV = nRT Where P is the pressure, V is the volume, n is the number of moles, R is the gas constant (8.314 J/mol K), and T is the temperature. We are only given the pressure before the reaction, which is 1.013 x 10^5 Pa. There is 1.413 mole of CO2 produced after the reaction, but we should also account for the air already inside the bottle before the reaction. Let's assume the temperature is constant and equal to 298 K (room temperature) for simplicity: P_initial = 1.013 x 10^5 Pa n_air = n_initial (total moles of air in the bottle) R = 8.314 J/mol K T = 298 K Rearranging the ideal gas law equation for volume: V_initial = (n_air)(R)(T) / P_initial

Find the final volume using the ideal gas law

After the reaction, we have 1.413 moles of CO2 in the bottle, along with the air initially in the bottle. The pressure has increased to 1.064 x 10^6 Pa, and the temperature is still constant (298 K). P_final = 1.064 x 10^6 Pa n_total = n_initial + 1.413 (total moles of gas in the bottle after the reaction) By using the ideal gas law again with the final parameters: V_final = (n_total)(R)(T) / P_final

Use constant temperature and volume to solve for n_initial

Since the temperature and volume are constant throughout this process, we have: V_initial = V_final So, (n_air)(R)(T) / P_initial = (n_total)(R)(T) / P_final And then we can solve for n_initial: n_air = n_initial = (P_initial / P_final) * (n_total) n_initial = (1.013 x 10^5 Pa) / (1.064 x 10^6 Pa) * (1.413 + n_initial) Solving for n_initial, we get: n_initial ≈ 0.128 moles

Calculate the volume of the bottle

Now that we have n_initial and n_total, we can use the ideal gas law equation to find the volume of the bottle (V_initial = V_final): V = (n_total)(R)(T) / P_final V ≈ ((0.128 + 1.413) moles) * (8.314 J/mol K) * (298 K) / (1.064 x 10^6 Pa) V ≈ 3.49 x 10^-3 m^3 Thus, the volume of the bottle is approximately 3.49 x 10^-3 m^3 or 3.49 L.