Determining Changes in Gas Conditions During Compression

Understanding the Ideal Gas Law

The Ideal Gas Law is expressed as:

\[ PV = nRT \]

Where:

• P = Pressure (in atmospheres, atm)
• V = Volume (in liters, L)
• n = Number of moles of gas
• R = Ideal gas constant (0.0821 L·atm/(K·mol))
• T = Temperature (in Kelvin, K)

This equation allows us to understand the relationship between pressure, volume, and temperature of a gas. When a gas is compressed, its volume decreases, leading to changes in pressure and/or temperature.

Example 1: Compressing a Gas at Constant Temperature

Scenario:

A gas occupies a volume of 10 L at a pressure of 1 atm and a temperature of 300 K. We compress it to a volume of 5 L. What will be the new pressure?

Calculation:

Using the Ideal Gas Law, we know that at constant temperature (isothermal conditions), the pressure and volume are inversely related:

\[ P_1 V_1 = P_2 V_2 \]

Where:

• P_1 = initial pressure = 1 atm
• V_1 = initial volume = 10 L
• V_2 = final volume = 5 L
• P_2 = final pressure (unknown)

Rearranging the equation to solve for P_2:

\[ P_2 = \frac{P_1 V_1}{V_2} = \frac{1 \, \text{atm} \times 10 \, \text{L}}{5 \, \text{L}} = 2 \, \text{atm} \]

Conclusion:

The new pressure after compressing the gas to 5 L is 2 atm.

Example 2: Compressing a Gas with Temperature Change

Scenario:

A gas is initially at 2 atm and occupies a volume of 8 L at a temperature of 400 K. If the gas is compressed to 4 L, and the temperature decreases to 300 K, what is the new pressure?

Calculation:

Applying the Ideal Gas Law:

\[ P_1 V_1 / T_1 = P_2 V_2 / T_2 \]

Where:

• P_1 = 2 atm
• V_1 = 8 L
• T_1 = 400 K
• V_2 = 4 L
• T_2 = 300 K
• P_2 = final pressure (unknown)

Rearranging the equation to solve for P_2:

\[ P_2 = \frac{P_1 V_1 T_2}{V_2 T_1} \]

Substituting in the values:

\[ P_2 = \frac{2 \, \text{atm} \times 8 \, \text{L} \times 300 \, \text{K}}{4 \, \text{L} \times 400 \, \text{K}} = \frac{4800}{1600} = 3 \, \text{atm} \]

Conclusion:

After the gas is compressed to 4 L and the temperature is lowered to 300 K, the new pressure is 3 atm.

Summary

These examples illustrate how the Ideal Gas Law can be applied to determine changes in gas conditions during compression. Understanding these principles is essential for practical applications in various scientific fields, including chemistry and engineering.