Question

If a 0.475-L sample of neon gas is heated from 27 ' \(\mathrm{C}\) to 82 ' \(\mathrm{C}\) at constant pressure, what will be the volume of the sample at the higher temperature?

Short answer

Given the initial volume (V1) = 0.475 L, initial temperature (T1) = 27 °C (300.15 K), and final temperature (T2) = 82 °C (355.15 K), we can use Charles' Law to determine the final volume of the neon gas sample. Charles' Law states that \(\frac{V1}{T1} = \frac{V2}{T2}\) for constant pressure. Rearranging the formula to solve for V2, we get \(V2 = \frac{T2 \times V1}{T1}\). Substituting the values, we calculate the final volume (V2) as \(V2 \approx 0.557 L\).

Step-by-step solution

List given information and convert temperatures to Kelvin

We are given the following information: Initial volume (V1) = 0.475 L Initial temperature (T1) = 27 °C Final temperature (T2) = 82 °C To work with gas laws, we need to convert the temperatures from Celsius to Kelvin using the formula: Kelvin = Celsius + 273.15 T1 = 27 + 273.15 = 300.15 K T2 = 82 + 273.15 = 355.15 K

Apply Charles' Law

Charles' Law states that the ratio of the initial volume and temperature is equal to the ratio of the final volume and temperature under constant pressure: \(\frac{V1}{T1} = \frac{V2}{T2}\) We are asked to find the final volume (V2) of the neon gas sample. We can rearrange the formula to solve for V2: \(V2 = \frac{T2 \times V1}{T1}\)

Calculate the final volume

Now we can substitute the values we found in step 1: \(V2 = \frac{355.15 \times 0.475}{300.15}\) \(V2 \approx 0.557 L\)

Report the final result

The volume of the neon gas sample after being heated from 27 °C to 82 °C under constant pressure is approximately 0.557 L.

Key concepts

Gas Laws - Understanding Charles' Law

Gas laws are essential principles that explain the behavior of gases. One of the central laws to understand is Charles' Law. This law describes how gases expand when temperature increases. To keep it simple, Charles' Law states that the volume of a given mass of a gas is directly proportional to its temperature, provided the pressure remains constant. This means as the temperature of the gas rises, so does its volume. To represent this mathematically, we use the equation: \(\frac{V1}{T1} = \frac{V2}{T2}\). In this equation, \(V1\) and \(V2\) are the initial and final volumes, while \(T1\) and \(T2\) are the initial and final temperatures respectively. Since volume and temperature are directly related, always remember they must be in the same unit, such as liters for volume and Kelvin for temperature.

Temperature Conversion - From Celsius to Kelvin

In many scientific calculations, especially those involving gas laws, temperatures need to be expressed in Kelvin. The Kelvin scale is important because it starts from absolute zero, the lowest temperature theoretically possible, which offers a more accurate reflection of the gas's behavior. To convert Celsius to Kelvin, you use the simple formula: \( \text{Kelvin} = \text{Celsius} + 273.15 \). For example:

• If the initial temperature of a gas is 27°C, when converted, it becomes 300.15 K.
• Similarly, a final temperature of 82°C converts to 355.15 K.

This conversion is crucial since Charles' Law requires temperature to be in Kelvin, ensuring consistency and accuracy in calculations.

Volume Calculation - Solving with Charles' Law

The ultimate goal in applying Charles' Law in exercises like these is to determine the unknown volume, often the final volume after a temperature change. Once temperatures are converted to Kelvin, you can plug these values along with the initial volume into the Charles' Law formula to find the final volume:

• Rearrange the formula: \(V2 = \frac{T2 \times V1}{T1}\).
• Insert given values, for example, \(V2 = \frac{355.15 \times 0.475}{300.15}\).
• After performing the calculation, you estimate the final volume \(V2\) to be approximately 0.557 L.

This calculation demonstrates how a gas's volume increases as it is heated, a core principle of gas behavior that Charles' Law helps to articulate.