Question

A 2.50 L sample of nitric oxide gas at \(100^{\circ} \mathrm{C}\) is cooled to \(20^{\circ} \mathrm{C}\). If the pressure remains constant, what is the final volume?

Short answer

The final volume is approximately 1.965 L.

Step-by-step solution

Understand the Problem

We need to find the final volume of a gas after it is cooled at constant pressure. According to Charles's Law, at constant pressure, the volume of a gas is directly proportional to its temperature in Kelvin. This means we can use the law's equation to solve the problem.

Convert Temperatures to Kelvin

We need to convert the temperatures from Celsius to Kelvin using the formula: \( T(K) = T(°C) + 273.15 \). The initial temperature is \(100^{\circ} \mathrm{C}\), so in Kelvin it is \(100 + 273.15 = 373.15 \, \text{K}\). The final temperature is \(20^{\circ} \mathrm{C}\), so in Kelvin it is \(20 + 273.15 = 293.15 \, \text{K}\).

Apply Charles's Law

Charles's Law states \(\frac{V_1}{T_1} = \frac{V_2}{T_2}\), where \(V_1\) and \(T_1\) are the initial volume and temperature, and \(V_2\) and \(T_2\) are the final volume and temperature. Given \(V_1 = 2.50 \, \text{L}\), \(T_1 = 373.15 \, \text{K}\), and \(T_2 = 293.15 \, \text{K}\), we need to solve for \(V_2\).

Solve for Final Volume \(V_2\)

Rearrange the equation \(\frac{V_1}{T_1} = \frac{V_2}{T_2}\) to find \(V_2\): \[ V_2 = V_1 \cdot \frac{T_2}{T_1} \]. Substituting in the values: \( V_2 = 2.50 \, \text{L} \cdot \frac{293.15 \, \text{K}}{373.15 \, \text{K}} \approx 1.965 \, \text{L}\).

Conclusion

The final volume of the gas, when cooled under constant pressure from \(100^{\circ} \mathrm{C}\) to \(20^{\circ} \mathrm{C}\), is approximately \(1.965 \, \text{L}\).

Key concepts

Gas Volume

The volume of a gas refers to the amount of space that the gas occupies. This space can be measured in liters, cubic meters, or other units of volume. According to Charles's Law, when dealing with a fixed amount of gas at constant pressure, the volume of the gas is directly related to its temperature. This relationship means that if the temperature of the gas increases, its volume will also increase, provided the pressure does not change.

Three main factors affect gas volume:

• Temperature: Higher temperature increases energy, making gas molecules move faster and take up more space.
• Pressure: While not directly related to Charles's Law, in general, increased pressure reduces volume.
• Quantity of gas: More gas molecules mean more volume.

Understanding how volume interacts with temperature and pressure is crucial in applications ranging from inflating tires to designing airtight containers. Charles's Law specifically focuses on the temperature-volume relationship at constant pressure.

Temperature Conversion

Temperature conversion is a critical step in applying gas laws such as Charles's Law. When measuring temperature for scientific purposes, it's often necessary to convert Celsius temperatures to Kelvin. This conversion is important because Kelvin is the standard unit of temperature in the scientific community and is essential for gas law calculations, as 0 Kelvin represents absolute zero.

The conversion formula from Celsius to Kelvin is simple:

• Convert by adding 273.15 to the Celsius temperature: \( T(K) = T(°C) + 273.15 \)

When solving problems like the one given, first convert all temperatures to Kelvin before using them in calculations. The Kelvin scale starts from absolute zero, simplifying the direct proportionality between volume and temperature in Charles's Law.

Kelvin Scale

The Kelvin scale is an absolute temperature scale used widely in scientific temperature measurements. Unlike the Celsius and Fahrenheit scales, Kelvin does not use degrees, and 0 Kelvin, or absolute zero, is the point at which all kinetic energy in particles ceases. It's essential for gas law calculations, including Charles's Law, as it establishes a true zero point.

Key aspects of the Kelvin scale include:

• No negative numbers: Kelvin begins at absolute zero, making calculations more straightforward.
• Celsius relationship: Simply an offset, \( T(K) = T(°C) + 273.15 \).
• Universal use: Scientifically standard for ensuring consistency across studies and experiments.

For anyone working with gases and applying gas laws, the Kelvin scale provides an accurate measure of temperature that directly ties to energy states, enabling more precise and meaningful calculations.