Question

A 3.787 - \(L\) bottle contains air plus \(n\) moles of sodium bicarbonate and \(n\) moles of acetic acid. These compounds react to produce \(n\) moles 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 \(9.599 \cdot 10^{5} \mathrm{~Pa}\). How many moles, \(n\), of carbon dioxide gas are in the bottle? Assume that the bottle is kept in a water bath that keeps the temperature in the bottle constant.

Short answer

Question: Calculate the moles of carbon dioxide gas produced in the reaction inside a sealed bottle containing a mixture of air, sodium bicarbonate, and acetic acid, given the initial pressure is 1.013 × 10^5 Pa, the final pressure is 9.599 × 10^5 Pa, and the volume of the bottle is 3.787 L. Answer: The moles of carbon dioxide gas produced in the reaction can be calculated using the expression: n = V * (P_final - P_initial) / (R * T), where V is the volume of the bottle, P_initial and P_final are the initial and final pressures, R is the ideal gas constant, and T is the constant temperature. By plugging in the given values and assuming constant temperature, we can calculate the moles of carbon dioxide gas (n) produced in the reaction.

Step-by-step solution

Write the ideal gas law equation

The ideal gas law equation relates pressure, volume, moles of the gas, and temperature: PV = nRT where P is pressure, V is volume, n is the number of moles of the gas, R is the ideal gas constant (8.314 J/mol K or 0.08206 L atm/mol K), and T is temperature.

Find the initial moles of air

Since the bottle is initially filled with air at atmospheric pressure, we can use the given initial pressure and volume to find the initial moles of air (n_air) in the bottle: P_initial = 1.013 × 10^5 Pa V = 3.787 - L (Since the volume of liquid is negligibly small) We will use the ideal gas constant in terms of J/mol K: R = 8.314 J/mol K Assuming that the temperature remains constant, we can express the initial moles of air in the bottle as: n_air = P_initial * V / (R * T)

Find the final moles of gas

After the reaction, the bottle contains n moles of carbon dioxide gas along with the initial moles of air. The final pressure in the bottle is given as 9.599 × 10^5 Pa. We can use this final pressure and the volume of the bottle to find the total moles of gas (n_total) in the bottle after the reaction: P_final = 9.599 × 10^5 Pa Using the ideal gas law equation: n_total = P_final * V / (R * T)

Calculate moles of carbon dioxide

Now, we have the expressions for initial moles of air and the total moles of gas after the reaction. We can find the moles of carbon dioxide (n) by taking the difference between the total moles and the initial moles of air: n = n_total - n_air Substitute the expressions we derived in steps 2 and 3: n = (P_final * V / (R * T)) - (P_initial * V / (R * T)) As R * T and V are constants, we can simplify the expression: n = V * (P_final - P_initial) / (R * T) Finally, plug in the given values of P_initial, P_final, V, and assuming constant temperature, we can calculate the moles of carbon dioxide gas (n) produced in the reaction.

Key concepts

Chemical Reaction and Pressure

Understanding the impact of chemical reactions on pressure is essential when dealing with enclosed systems like the one in our exercise. When a chemical reaction takes place in a sealed container, the number of gas particles can change, which directly affects the pressure inside.

In the case of sodium bicarbonate reacting with acetic acid, the reaction produces carbon dioxide gas. Since the bottle is sealed, the gas cannot escape, resulting in an increase of pressure within the bottle. This pressure change is determined by the amount of carbon dioxide gas produced, which is directly proportional to the number of moles of reactants, given that the volume of the bottle and temperature are kept constant.

Applying the ideal gas law allows us to relate this increase in pressure to the change in the number of moles of gas, leading us to calculate the amount of carbon dioxide formed during the reaction.

Moles Calculation

Moles calculations are integral in stoichiometry and understanding chemical reactions. The mole is a fundamental unit in chemistry that provides a bridge between the atomic or molecular scale and the macroscopic world.

In our exercise, we are interested in determining the number of moles of carbon dioxide produced after the reaction in the sealed bottle. By knowing the moles, we can relate microscopic changes, like the number of molecules reacting, to macroscopic changes, such as volume and pressure alterations in the gas.

To calculate the moles of carbon dioxide, we take advantage of the derived expressions from the ideal gas law. We subtract the initial number of moles of air from the total number of moles in the bottle after the reaction. This difference gives us the moles of carbon dioxide produced.

Gas Constant

At the heart of the ideal gas law equation lies the gas constant, denoted by 'R'. This constant is crucial as it bridges together pressure, volume, temperature, and the number of moles of a gas.

For our calculations, we have options regarding the units of R, but consistency with the units used for pressure, volume, and temperature is vital. In this scenario, we use the gas constant value of 8.314 J/mol K, which fits the pressure in pascals (Pa) and volume in liters (L).

The gas constant stays uniform for all ideal gases, making it a universal constant for gas-related equations and a reliable figure for comparing behaviors of different gases under various conditions.

Pressure and Volume Relationship

Comprehending the relationship between pressure and volume is pivotal in solving gas law problems. According to Boyle’s Law, for a fixed amount of an ideal gas kept at a constant temperature, the volume and pressure of the gas are inversely proportional. However, in our sealed bottle scenario, the volume remains constant, and any change in pressure is due to the chemical reaction producing more gas.

As the number of moles of gas increases within the same volume due to the production of carbon dioxide, the pressure increases proportionally. By using the ideal gas law we illustrated how to relate this pressure change to the amount of gas produced, thus arriving at the moles of carbon dioxide generated. This relationship is essential in deducing the behavior of gases under different conditions and is the key to solving the given exercise.