Problem 6 - Chemistry and Chemical Reactivity

Question

A sample of \(\mathrm{CO}_{2}\) gas has a pressure of \(56.5 \mathrm{mm} \mathrm{Hg}\) in a 125-mi. 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 new flask has a volume of approximately 113.48 ml.

Step 1: Identify the Gas Law

We are dealing with two scenarios at the same temperature. This suggests the use of Boyle’s Law, which relates pressure and volume for a constant temperature. Boyle's Law is given by the formula: \( P_1 V_1 = P_2 V_2 \).

Step 2: Gather Known Values

From the problem, the initial pressure \( P_1 \) is 56.5 mm Hg and the initial volume \( V_1 \) is 125 ml. The new pressure \( P_2 \) is 62.3 mm Hg, and we need to find the new volume \( V_2 \).

Step 3: Set Up the Equation

Plug the known values into the Boyle's Law equation: \( 56.5 \times 125 = 62.3 \times V_2 \).

Step 4: Solve for the Unknown Volume

To find \( V_2 \), rearrange the equation: \( V_2 = \frac{56.5 \times 125}{62.3} \).

Step 5: Calculate the New Volume

Perform the calculation: \( V_2 = \frac{56.5 \times 125}{62.3} \approx 113.48 \text{ ml} \). So, the volume of the new flask is approximately 113.48 ml.

Key Concepts

Pressure and Volume Relationship

Boyle's Law is a cornerstone of chemistry that focuses on the pressure and volume relationship of gases. The law states that for a given amount of gas at constant temperature, the pressure of the gas is inversely proportional to its volume. This means that as one increases, the other decreases.
For example, if you compress a gas into a smaller volume, its pressure will increase. On the contrary, allowing a gas to expand to a larger volume will decrease its pressure, assuming the temperature remains constant. Boyle's Law can be elegantly expressed using the formula:

• \[ P_1 V_1 = P_2 V_2 \]

Where,

• \(P_1\) and \(P_2\) are the initial and final pressures, respectively.
• \(V_1\) and \(V_2\) are the initial and final volumes, respectively.

This relationship is crucial in scenarios like inflating balloons, where understanding the changes in pressure and volume is necessary to avoid bursting the balloon. It applies to many practical situations, especially when working with confined gases.

Gas Laws

Gas laws are a set of relationships that describe the behavior of gases, connecting variables such as pressure, volume, temperature, and quantity of gas. They help scientists and engineers predict how a gas will respond to changes in its environment.
There are several fundamental gas laws:

• **Boyle's Law:** Discusses the pressure-volume relationship at constant temperature, as we've covered before.
• **Charles's Law:** Relates volume and temperature, stating that at constant pressure, the volume of a gas is directly proportional to its absolute temperature.
• **Gay-Lussac’s Law:** Connects pressure and temperature, showing that at constant volume, the pressure of a gas is directly proportional to its absolute temperature.
• **Avogadro's Law:** States that the volume of a gas at constant temperature and pressure is directly proportional to the number of moles of the gas.

These laws can be combined into the **Ideal Gas Law**, a powerful tool that incorporates all the aforementioned relationships into a single equation:

• \[ PV = nRT \]

Where,

• \(P\) is pressure,
• \(V\) is volume,
• \(n\) is the amount of substance in moles,
• \(R\) is the ideal gas constant, and
• \(T\) is the absolute temperature.

Understanding these laws helps in interpreting and managing the behavior of gases in various practical and industrial settings.

Ideal Gas Behavior

The ideal gas concept is a simplified model for the behavior of real gases. It helps predict the properties of gases under varying conditions by assuming certain ideal conditions.
In an ideal scenario:

• Gas particles are assumed to be point particles with no volume.
• There are no interactions between gas molecules, meaning no attractions or repulsions.
• Collisions between gas particles and with the walls of a container are perfectly elastic.

Although no real gas perfectly follows these assumptions, the ideal gas model approximates real gas behavior under many conditions, particularly at high temperatures and low pressures.
Despite its limitations, the ideal gas concept is essential for understanding and applying the other gas laws effectively. For instance, the Ideal Gas Law helps us relate the physical conditions of sample gases by combining Boyle's Law, Charles's Law, and Avogadro’s Law into a comprehensive form:

• \[ PV = nRT \]

By knowing any three of the four parameters (pressure, volume, amount, and temperature), you can calculate the fourth, allowing for a broad understanding and manipulation of gases in scientific contexts.