Question

Calculate the final Celsius temperature when \(125 \mathrm{~mL}\) of chlorine gas at \(25^{\circ} \mathrm{C}\) is heated to give a volume of \(175 \mathrm{~mL}\). Assume that the pressure remains constant.

Short answer

The final temperature is approximately \(144.26^{\circ} \text{C}\).

Step-by-step solution

Identify Known Variables

We are given the initial volume \( V_1 = 125 \text{ mL} \) and initial temperature \( T_1 = 25^{\circ} \text{C} \). The final volume is \( V_2 = 175 \text{ mL} \). We need to find the final temperature \( T_2 \). Remember to convert temperatures to Kelvin by adding 273.15.

Convert Temperature to Kelvin

Convert the initial temperature from Celsius to Kelvin using the formula \( T = \text{C} + 273.15 \). Thus, \( T_1 = 25 + 273.15 = 298.15 \text{ K} \).

Apply Charles's Law

We use Charles's Law, which states \( \frac{V_1}{T_1} = \frac{V_2}{T_2} \) under constant pressure. Substitute the known values: \( \frac{125}{298.15} = \frac{175}{T_2} \).

Solve for \( T_2 \) in Kelvin

Rearrange the equation to solve for \( T_2 \): \( T_2 = \frac{175 \times 298.15}{125} \). Calculate \( T_2 \): \( T_2 \approx 417.41 \text{ K} \).

Convert Final Temperature to Celsius

Convert the final temperature back to Celsius: \( T_2 = 417.41 - 273.15 = 144.26^{\circ} \text{C} \). Round as appropriate for significant figures if necessary.

Key concepts

Gas Laws

Gas laws are a group of physical laws described by mathematic formulas, which dictate how gases behave under various conditions. Among these is Charles's Law, which examines the relationship between the volume and temperature of a gas. These laws are incredibly useful in predicting how gases react when exposed to changes in their environment. Assume that all other variables are constant. You learn that:

• Boyle's Law focuses on pressure and volume.
• Gay-Lussac's Law addresses pressure and temperature.
• Combined gas law includes all three: pressure, volume, and temperature.
• Avogadro’s Law relates to volume and the number of moles.

Charles's Law, named in honor of French scientist Jacques Charles, is particularly significant because it shows that the volume of a gas is directly proportional to its absolute temperature, measured in Kelvin:\[ \frac{V_1}{T_1} = \frac{V_2}{T_2} \]This relationship becomes very useful for calculations requiring changes in a gas's state.

Temperature Conversion

Temperature conversion is a fundamental skill when dealing with gas laws. This conversion is necessary to correlate gas volume changes accurately as temperature scales can affect outcomes. The standard unit for gas law calculations is Kelvin, not Celsius.To convert from Celsius to Kelvin:

• Add 273.15 to the Celsius temperature.

For example, if the temperature of a gas is \(25^{\circ} \text{C}\), then in Kelvin it would be:\[ T = 25 + 273.15 = 298.15 \text{ K} \]Conversely, to convert from Kelvin to Celsius, you simply subtract 273.15:

• So, a Kelvin temperature of \(417.41 \text{ K}\) is \[ 417.41 - 273.15 = 144.26^{\circ} \text{C} \]

Understanding these conversions aids in evaluating how gas behavior changes under varying temperatures.

Volume and Temperature Relationship

The volume and temperature relationship for gases is well-characterized by Charles's Law. Whenever you examine this relationship, remember:

• The volume of a gas increases as the temperature increases, provided the pressure remains constant.
• Conversely, the volume decreases if the temperature decreases.

This is because the kinetic energy of gas particles increases with temperature, causing them to move more vigorously. More energetic particles take up more space, thereby increasing volume.When faced with a problem like the one in the exercise where you have a gas at \(25^{\circ} \text{C}\) and \(125 \text{ mL}\), heating it to yield \(175 \text{ mL}\), you're seeing Charles's Law in action.In mathematical terms, the direct proportionality is captured as follows:\[ \frac{V_1}{T_1} = \frac{V_2}{T_2} \]Plug in the known values and solve for the unknown, allowing you to predict how changes in temperature will affect gas volume.