Question

A gas has a volume of \(V \mathrm{~cm}^{3}\) at \(10^{\circ} \mathrm{C}\). If the pressure is doubled, at what temperature will the volume still be \(V \mathrm{~cm}^{3}\) ? (a) \(273^{\circ} \mathrm{C}\) (b) \(300^{\circ} \mathrm{C}\) (c) \(283^{\circ} \mathrm{C}\) (d) \(293^{\circ} \mathrm{C}\)

Short answer

The volume will still be \(V \mathrm{~cm}^3\) at \(293^\circ C\).

Step-by-step solution

Convert Initial Temperature to Kelvin

Temperature should be in Kelvin for gas law calculations. To convert from Celsius to Kelvin, add 273 to the Celsius temperature. The initial temperature (T1) is \(10^\circ C\), which is equal to \(10 + 273 = 283 K\).

Apply Gay-Lussac's Law

Gay-Lussac's law states that pressure is directly proportional to temperature at constant volume, i.e., \(\frac{P1}{T1} = \frac{P2}{T2}\). Since the pressure is doubled, \(P2 = 2 \cdot P1\). We know \(P1\), \(T1\), and \(P2\), and we need to find \(T2\).

Solve for the Final Temperature (T2)

Rearrange the formula to solve for \(T2\): \(T2 = \frac{P2 \cdot T1}{P1} = \frac{2 \cdot T1}{1} = 2 \cdot T1 = 2 \cdot 283 K = 566 K\).

Convert Final Temperature Back to Celsius

To convert from Kelvin back to Celsius, subtract 273 from the Kelvin temperature. Therefore, the final temperature (T2) in Celsius will be \(566 K - 273 = 293^\circ C\).

Key concepts

Gas Law Calculations

In the realm of chemistry, understanding gas law calculations is pivotal for grasping how gases behave under various conditions. These calculations allow us to predict and quantify changes in gas samples when they undergo alterations in pressure, volume, and temperature.

Gay-Lussac's law, in particular, facilitates the analysis of situations where only temperature and pressure change, keeping volume constant. Using a mathematical equation to represent this law, we can manipulate the formula to solve for unknown variables, offering a practical approach to solving real-world problems. Ensuring temperature is measured in Kelvin is essential for these calculations, as this temperature scale is the standard for scientific measurement because it starts at absolute zero, the theoretical point where all molecular motion ceases.

For instance, converting temperatures from Celsius to Kelvin is a common initial step, which stabilizes the unit base and prevents negative values, as Gay-Lussac's law assumes absolute temperatures. Through practice and application of these calculations, students can develop a strong foundation in predicting gas behaviors in varying scenarios.

Temperature-Pressure Relationship

The correlation between temperature and pressure in gases, often explored through Gay-Lussac's law, holds significant conceptual importance. This law posits that, at a constant volume, the pressure of a gas is directly proportional to its absolute temperature (measured in Kelvin).

Understanding this linear relationship is fundamental because it gives insight into molecular dynamics. As the temperature of a gas increases, the average kinetic energy of its molecules also rises, leading to more frequent and forceful collisions with the container walls, which results in increased pressure.

In practical terms, when we examine scenarios like the textbook problem provided, the doubling of pressure due to increased temperature is clear evidence of this proportional relationship. The direct relationship also means that if we know any three of the four variables (initial pressure, initial temperature, final pressure, and final temperature), the fourth can be calculated, which is precisely how we approach and solve Gay-Lussac's Law problems.

Conceptual Understanding of Gas Laws

Gas laws, such as Gay-Lussac's law, Boyle's law, Charles's law, and the combined gas law, are critical for developing a deep conceptual understanding of the behavior of gases. These laws reveal the underlying principles of pressure, volume, and temperature relationships in gases and provide an idealized model to predict how gases will behave under non-reactive conditions.

It's crucial for students to keep in mind that these laws are based on assumptions of ideal conditions, often not true for real gases under all conditions. However, at low pressures and high temperatures, real gases approximate the behavior predicted by these laws.

The textbook exercise challenges students to apply Gay-Lussac's law in a practical situation, emphasizing the need for the conceptual understanding of gas behavior in direct relation to mathematics. This balanced grasp of conceptual theory and practical application will help students hurdle over common confusions and misconceptions, enabling them to tackle a wide range of problems confidently. Through exercises and solutions such as these, students can see the laws in action and better appreciate the nuances of how real gases behave versus the idealized models.