Problem 6 - Chemistry and Chemical Reactivity

Question

\(.\) A sample of \(\mathrm{CO}_{2}\) gas has a pressure of \(56.5 \mathrm{mm}\) Hg in a 125 -mL flask. The sample is transferred to a new flask, where it has a pressure of \(62.3 \mathrm{mm}\) Hg at the same temperature. What is the volume of the new flask?

Step-by-Step Solution

Answer

The volume of the new flask is approximately 113.5 mL.

Step 1: Identify the Known Values and Formula

We know the initial pressure (\(P_1 = 56.5 \: \text{mm Hg}\)) and initial volume (\(V_1 = 125 \: \text{mL}\)). We are also given the final pressure (\(P_2 = 62.3 \: \text{mm Hg}\)). We need to find the final volume (\(V_2\)). Since the temperature remains constant, we will use Boyle's Law, which is \(P_1V_1 = P_2V_2\).

Step 2: Rearrange the Formula to Solve for the Unknown

Boyle's Law can be rearranged to solve for \(V_2\) as follows: \(V_2 = \frac{P_1V_1}{P_2}\).

Step 3: Substitute the Known Values

Insert the known values into the equation: \(V_2 = \frac{56.5 \times 125}{62.3}\).

Step 4: Calculate the Unknown Volume

Perform the multiplication and then divide to get the final answer. Doing the calculation gives \(V_2 \approx 113.5\) mL.

Key Concepts

Gas Laws

Gas laws are fundamental principles in chemistry that describe how gases behave. They are crucial for understanding chemical reactions and the behavior of substances in different states. The most common gas laws are Boyle's Law, Charles's Law, and Avogadro's Law. Each explains a different aspect of gas behavior.

- **Boyle's Law**: Relates the pressure and volume of a gas at a constant temperature. - **Charles's Law**: Relates the volume and temperature of a gas at constant pressure. - **Avogadro's Law**: Relates the volume and amount (in moles) of a gas at constant temperature and pressure. Understanding these laws helps us predict how a gas will react when conditions change. This is essential for experiments and industrial applications, where they ensure safety and efficiency. Boyle’s Law, in particular, is often used when dealing with closed containers, helping determine unknown variables, such as volume or pressure, when others are known.

Pressure-Volume Relationship

The pressure-volume relationship is a core idea in understanding gas behavior, best described by Boyle's Law. This law states that for a given amount of gas at constant temperature, the pressure of a gas is inversely proportional to its volume. This means that as the volume of the gas decreases, the pressure increases, and vice versa.

- **Formula**: Boyle's Law is expressed mathematically as \( P_1V_1 = P_2V_2 \), where: - \( P_1 \) and \( V_1 \) are the initial pressure and volume. - \( P_2 \) and \( V_2 \) are the final pressure and volume.- **Real-life Application**: This principle is often observed in daily life, such as when inflating a bicycle tire. As more air is pumped in, the space available for the gas decreases, leading to an increase in pressure.In the context of our original exercise, students learned how to apply Boyle's Law to calculate the volume of a flask when the pressure changes but the temperature remains constant. Understanding this relationship is pivotal for solving problems involving gas under constant temperature conditions.

Ideal Gas Behavior

Ideal gas behavior refers to how gases are expected to behave under certain conditions, following the ideal gas law. An ideal gas is a theoretical gas composed of randomly moving, non-interacting particles. The law combines Boyle's Law, Charles's Law, and Avogadro's Law into the equation \( PV = nRT \).
- **Variables**: - \( P \) (Pressure) - \( V \) (Volume) - \( n \) (Number of moles) - \( R \) (Ideal gas constant) - \( T \) (Temperature in Kelvin)- **Real Gases vs. Ideal Gases**: Real gases deviate from ideal behavior due to interactions between gas particles, especially at high pressure and low temperature. However, under normal conditions, most gases behave almost like ideal gases.- **Problem Solving**: In exercises like the one given, assuming ideal behavior allows us to use simplified equations to analyze and predict the behavior of gases.Understanding ideal gas behavior provides a foundation for studying gases more deeply and applying these concepts to chemical processes and engineering applications. It underscores the importance of conditions like low pressure and high temperature where gases closely follow predicted behaviors.