Question

A 555 mL sample of nitrous oxide at \(25^{\circ} \mathrm{C}\) is heated to \(50^{\circ} \mathrm{C}\). If the pressure remains constant, what is the final volume?

Short answer

The final volume is approximately 601.3 mL.

Step-by-step solution

Understand the Ideal Gas Law

The Ideal Gas Law establishes a relationship between pressure, volume, and temperature for an ideal gas. When the pressure is constant, we can use the formula \[ \frac{V_1}{T_1} = \frac{V_2}{T_2} \]where \(V_1\) and \(V_2\) are the initial and final volumes, and \(T_1\) and \(T_2\) are the initial and final temperatures in Kelvin.

Convert Temperatures from Celsius to Kelvin

To use the formula correctly, convert temperatures from Celsius to Kelvin by adding 273.15. \[T_1 = 25^{\circ}C + 273.15 = 298.15 \, K\]\[T_2 = 50^{\circ}C + 273.15 = 323.15 \, K\]

Calculate Final Volume Using Charles's Law

With the temperature in Kelvin, solve for the final volume \(V_2\):Start from:\[ \frac{V_1}{T_1} = \frac{V_2}{T_2}\]Substitute known values:\[ \frac{555 \, \text{mL}}{298.15 \, K} = \frac{V_2}{323.15 \, K}\]Solve for \(V_2\):\[ V_2 = \frac{555 \, \text{mL} \times 323.15 \, K}{298.15 \, K} \]Calculate:\[ V_2 \approx 601.3 \, \text{mL}\]

Validate the Calculation

Double-check each calculation step to ensure accuracy in conversion and multiplication. Confirm that your final answer makes logical sense given the increase in temperature.

Key concepts

Charles's Law

When dealing with gases, knowing how changes in temperature affect volume is crucial. Charles's Law offers insight into this relationship. This law states that the volume of a gas is directly proportional to its absolute temperature, so long as the pressure remains constant. This means that when you heat a gas, its volume increases, and when you cool it, its volume decreases. The mathematical expression for Charles's Law is: \[ \frac{V_1}{T_1} = \frac{V_2}{T_2} \]Here, \( V_1 \) and \( V_2 \) are the initial and final volumes, respectively, and \( T_1 \) and \( T_2 \) are the initial and final temperatures in Kelvin. The requirement to use Kelvin is critical because Kelvin is an absolute temperature scale ensuring the proportionality holds true. Under Charles’s Law, as temperature increases, the gas molecules move faster and push farther apart, thus increasing volume.

Temperature Conversion

When working with gas laws such as Charles's Law, it is essential to use the Kelvin scale for temperature. This is because Kelvin is an absolute temperature scale where zero Kelvin (0 K) represents absolute zero—the point where all molecular motion ceases.

• To convert temperatures from degrees Celsius to Kelvin, simply add 273.15 to the Celsius value.
• For example: A temperature of \( 25^{\circ}C \) becomes \( 25 + 273.15 = 298.15 \, K \).
• Likewise, \( 50^{\circ}C \) is converted to \( 50 + 273.15 = 323.15 \, K \).

This conversion ensures calculations reflect the absolute nature of temperature changes, aligning with the proportionality required by gases.

Gas Volume Calculation

Calculating the volume of a gas after a temperature change, while keeping pressure constant, is a straightforward process when using Charles's Law. In the given problem, we're tasked with finding the final volume \( V_2 \) given the initial volume \( V_1 \), and temperatures \( T_1 \) and \( T_2 \) converted to Kelvin. Start with Charles's Law: \[ \frac{V_1}{T_1} = \frac{V_2}{T_2} \]Then, plug in the known values:\[ \frac{555 \, \text{mL}}{298.15 \, K} = \frac{V_2}{323.15 \, K} \]Solving for \( V_2 \) gives:\[ V_2 = \frac{555 \, \text{mL} \times 323.15 \, K}{298.15 \, K} \approx 601.3 \, \text{mL} \]Ensure each step is checked for mathematical accuracy, particularly with units, to confirm that the answer logically reflects the consequences of increased temperature, leading to an increase in volume.