Question

You have a cylinder of argon gas at 19.8 atm pressure at \(19^{\circ} \mathrm{C}\). The volume of argon in the cylinder is \(50.0 \mathrm{~L}\). What would be the volume of this gas if you allowed it to expand to the pressure of the surrounding air \((0.974 \mathrm{~atm}) ?\) Assume the temperature remains constant.

Short answer

The volume of argon gas expands to approximately 1016.85 L.

Step-by-step solution

Understand the Problem

We need to find the new volume of argon gas when its pressure decreases from 19.8 atm to 0.974 atm while keeping the temperature constant. Boyle's Law, which states that \( P_1V_1 = P_2V_2 \) for a constant temperature process, will be used.

Identify Known Quantities

The initial pressure \( P_1 \) is 19.8 atm, the initial volume \( V_1 \) is 50.0 L, and the final pressure \( P_2 \) is 0.974 atm. The final volume \( V_2 \) is what we need to find.

Apply Boyle's Law

Using Boyle's Law \( P_1V_1 = P_2V_2 \), we substitute the known values:\[ 19.8 \, \text{atm} \times 50.0 \, \text{L} = 0.974 \, \text{atm} \times V_2 \]

Solve for the Final Volume

Rearrange the equation to solve for \( V_2 \):\[ V_2 = \frac{19.8 \, \text{atm} \times 50.0 \, \text{L}}{0.974 \, \text{atm}} \]Calculate to find \( V_2 \).

Calculate the Final Volume

Perform the calculation:\[ V_2 = \frac{990 \, \text{atm} \cdot \text{L}}{0.974 \, \text{atm}} \approx 1016.85 \, \text{L} \]

Conclusion

The volume of argon gas when its pressure is reduced to 0.974 atm is approximately 1016.85 L.

Key concepts

Gas Laws

Gas laws are fundamental principles in chemistry that describe the behavior of gases. These laws relate the four variables that define a gas: pressure (P), volume (V), temperature (T), and quantity of gas (n). The gas laws help us understand how changes in one of these variables affect the others, often assuming ideal conditions where gas molecules move randomly and exhibit no intermolecular forces. The three main gas laws include:

• Boyle's Law: This law states that for a given amount of gas at constant temperature, the volume of the gas is inversely proportional to its pressure.
• Charles’s Law: It states that the volume of a gas is directly proportional to its temperature at constant pressure.
• Avogadro’s Law: This calculates that gas volume is directly proportional to the number of moles of the gas at constant temperature and pressure.

By combining these laws, the Ideal Gas Law is formed, offering a more comprehensive understanding of a gas's behavior under various conditions.
These laws are crucial for solving problems involving gas behaviors, such as calculating changes in pressure or volume when certain conditions are altered.

Pressure-Volume Relationship

The pressure-volume relationship, as described by Boyle's Law, is a key principle in understanding gas dynamics. This law highlights an inverse relationship between pressure and volume when temperature remains constant.

In practice, Boyle's Law is expressed as:\[ P_1V_1 = P_2V_2 \]where \( P_1 \) and \( V_1 \) are the initial pressure and volume, and \( P_2 \) and \( V_2 \) are the final pressure and volume, respectively.
This formula is perfect for situations where it’s essential to forecast how a gas will react when changing volume or pressure.

• If the pressure increases, the volume decreases.
• If the pressure decreases, the volume increases.

This relationship is evident in various real-world applications, such as syringes and air pumps, where altering the pressure influences the volume significantly.

Ideal Gas Behavior

Ideal gas behavior is an essential concept for simplifying and understanding gas interactions in a theoretical context. This model assumes that gases obey the gas laws at all conditions of temperature and pressure, which often approximates real gas behavior quite closely.

An ideal gas is characterized by:

• Particles in constant, random motion.
• No forces of attraction or repulsion between particles.
• Collisions between particles are perfectly elastic, meaning no energy is lost.

The Ideal Gas Law integrates this concept using the formula:\[ PV = nRT \]where \( R \) is the gas constant. While real gases deviate from this ideal behavior under high pressure and low temperature, for many conditions, the Ideal Gas Law serves as a reliable approximation.
Understanding ideal gas behavior simplifies calculations and predictions concerning gases, making this theory vital for students and professionals tackling gaseous reactions and problems like those illustrated in Boyle's Law applications.