Question

A 75.2 -mL sample of helium at \(12^{\circ} \mathrm{C}\) is heated to \(192^{\circ} \mathrm{C} .\) What is the new volume of the helium (assuming constant pressure)?

Short answer

After heating the helium sample from \(12^{\circ}\mathrm{C}\) to \(192^{\circ}\mathrm{C}\), the new volume of the helium at constant pressure is approximately \(121.5\ \mathrm{mL}\).

Step-by-step solution

Convert temperatures to Kelvin

To work with gas law calculations, we must express temperatures in Kelvin. We can convert Celsius to Kelvin using the following equation: \[ T_{K} = T_{C} + 273.15 \] Initial temperature in Kelvin: \[ T_{1K} = 12^{\circ} \mathrm{C} + 273.15 \Longrightarrow T_{1K} = 285.15\ \mathrm{K} \] Final temperature in Kelvin: \[ T_{2K} = 192^{\circ} \mathrm{C} + 273.15 \Longrightarrow T_{2K} = 465.15\ \mathrm{K} \]

Apply Charles's Law

Charles's Law states that the volume of a gas is directly proportional to its temperature when the pressure is kept constant. Mathematically, this can be expressed as: \[ \frac{V_1}{T_1} = \frac{V_2}{T_2} \] We know the initial volume \(V_1 = 75.2\ mL\), the initial temperature \(T_1 = 285.15\ K\), and the final temperature \(T_2 = 465.15\ K\). We have to solve for \(V_2\), the final volume of the helium.

Solve for the final volume

We can solve for \(V_2\) by rearranging the equation in Step 2: \[ V_2 = \frac{V_1 \cdot T_2}{T_1} \] Now, substitute the known values and calculate the final volume: \[ V_2 = \frac{75.2\ mL \cdot 465.15\ K}{285.15\ K} \Longrightarrow V_{2} \approx 121.5\ \mathrm{mL} \]

Present the result

After heating the helium sample from \(12^{\circ}\mathrm{C}\) to \(192^{\circ}\mathrm{C}\), the new volume of the helium at constant pressure is approximately \(121.5\ \mathrm{mL}\).

Key concepts

Gas Laws in Chemistry

Gas laws in chemistry are fundamental principles that describe the behavior of gases in terms of pressure, volume, and temperature. These laws are crucial in explaining and predicting how gases will react to changes in environmental conditions. Among the most significant of these laws is Charles’s Law, which pertains explicitly to the temperature-volume relationship of gases at a constant pressure.

Other essential gas laws include Boyle's Law, which deals with pressure and volume at a constant temperature, and the Ideal Gas Law, which combines several gas laws to show a relationship between pressure, volume, temperature, and the number of moles of gas. Understanding gas laws helps in various applications like predicting weather patterns, cooking, and even in medical procedures involving respiratory systems.

Temperature-Volume Relationship

The temperature-volume relationship in gases is best exemplified by Charles's Law. This law asserts that if the pressure of a gas is held constant, its volume is directly proportional to its absolute temperature, measured in Kelvin. In simple terms, as the temperature of a gas increases, so does its volume and vice versa.

For students, visualizing this can be quite helpful; imagine a balloon being heated. As it gets warmer, the gas molecules inside the balloon move more rapidly, which causes them to take up more space, making the balloon expand. In contrast, cooling the balloon slows down the molecules and decreases the space they occupy, causing the balloon to shrink. Charles's Law aids in understanding such phenomena and is pivotal in many scientific and industrial processes that involve temperature changes in gases.

Converting Celsius to Kelvin

Temperature scales can sometimes be a source of confusion, but converting between Celsius and Kelvin is a straightforward process necessary for gas law calculations. The Kelvin scale sets its zero point at absolute zero — the temperature at which all thermal motion ceases — which is equivalent to -273.15 degrees Celsius.

To convert a temperature from Celsius to Kelvin, simply add 273.15 to the Celsius temperature. This adjustment aligns the scale with the absolute temperature required for gas law equations, ensuring accuracy when predicting gas behavior under various temperature conditions. For example, room temperature, around 20 degrees Celsius, translates to 293.15 Kelvin, which would be used in gas law calculations instead of the Celsius temperature.