Examples of Determining Volume of a Gas at Different Temperatures

Understanding Volume Changes of a Gas with Temperature

The Ideal Gas Law is a vital equation in chemistry that relates pressure, volume, temperature, and the number of moles of a gas. Mathematically, it is represented as:

\[ PV = nRT \]

Where:

• P = Pressure of the gas
• V = Volume of the gas
• n = Number of moles of gas
• R = Ideal gas constant
• T = Temperature in Kelvin

In this article, we will explore three practical examples of determining the volume of a gas at different temperatures while keeping pressure and the number of moles constant.

Example 1: Volume of Oxygen Gas at Room Temperature

In a laboratory, a chemist is conducting an experiment with a sample of oxygen gas. The sample contains 2 moles of oxygen, and the pressure is maintained at 1 atm. The chemist wants to determine the volume of the gas at room temperature (25°C or 298 K).

Using the Ideal Gas Law:
\[ V = \frac{nRT}{P} \]

Substituting the known values:

• n = 2 moles
• R = 0.0821 L·atm/(K·mol)
• T = 298 K
• P = 1 atm

Calculating:
\[ V = \frac{(2)(0.0821)(298)}{1} \approx 49.2 \text{ L} \]

In this case, the volume of the oxygen gas is approximately 49.2 liters at room temperature.

Notes:

• If the temperature were to change to 0°C (273 K), the new volume could be calculated using the same formula, which illustrates the direct relationship between temperature and volume.

Example 2: Volume of Helium in a Balloon at Different Temperatures

Consider a party balloon filled with helium gas. The balloon contains 0.5 moles of helium, and at 20°C (293 K), the pressure inside the balloon is 1 atm. The host wants to know how the volume changes if the temperature drops to 5°C (278 K).

Using the Ideal Gas Law:
\[ V = \frac{nRT}{P} \]

Initially at 20°C:

• n = 0.5 moles
• R = 0.0821 L·atm/(K·mol)
• T = 293 K
• P = 1 atm

Calculating:
\[ V = \frac{(0.5)(0.0821)(293)}{1} \approx 12.1 \text{ L} \]

Now, calculating for 5°C:

• T = 278 K

Calculating:
\[ V = \frac{(0.5)(0.0821)(278)}{1} \approx 11.4 \text{ L} \]

Thus, the volume decreases from approximately 12.1 liters at 20°C to 11.4 liters at 5°C.

Notes:

• This example illustrates Charles’s Law, which states that the volume of a gas is directly proportional to its temperature, provided the pressure remains constant.

Example 3: Compressed Gas in a Cylinder

A gas cylinder contains 3 moles of nitrogen gas compressed to a pressure of 2 atm. The initial temperature of the gas is 100°C (373 K), and the technician needs to determine the volume of gas when it is cooled to 25°C (298 K).

Using the Ideal Gas Law:
\[ V = \frac{nRT}{P} \]

Calculating for the initial temperature (100°C):

• n = 3 moles
• R = 0.0821 L·atm/(K·mol)
• T = 373 K
• P = 2 atm

Calculating:
\[ V = \frac{(3)(0.0821)(373)}{2} \approx 38.5 \text{ L} \]

Now, calculating for the final temperature (25°C):

• T = 298 K

Calculating:
\[ V = \frac{(3)(0.0821)(298)}{2} \approx 36.7 \text{ L} \]

In this scenario, the volume of nitrogen gas decreases from approximately 38.5 liters at 100°C to 36.7 liters at 25°C.

Notes:

• This example highlights how cooling a gas reduces its volume when pressure is held constant. It’s essential for safety in industrial applications where gas volumes can be critical.