Problem 84 - Introductory Chemistry Atoms First

Question

A sample of gas at \(25^{\circ} \mathrm{C}\) and \(759.0 \mathrm{~mm} \mathrm{Hg}\) has a volume of \(1.58 \mathrm{~L}\). If the temperature is raised to \(35^{\circ} \mathrm{C}\) but the pressure is held constant at \(759.0 \mathrm{~mm} \mathrm{Hg}\), will the volume increase or decrease? Explain your answer.

Step-by-Step Solution

Answer

The volume of the gas sample will increase when the temperature is raised from \(25°C\) to \(35°C\) while the pressure is held constant at \(759.0mmHg\). This is because the final volume (approximately \(1.63L\)) is greater than the initial volume (\(1.58L\)).

Step 1: Identify the gas law to be used

Since we are given the initial and final temperatures, initial pressure, and initial volume, and we are asked to find the final volume, we will use the combined gas law formula, which is derived from Boyle's law, Charles's law, and Avogadro's law: \[ \frac{P_{1}V_{1}}{T_{1}} = \frac{P_{2}V_{2}}{T_{2}} \] Where: \(P_{1}\) = Initial pressure \(V_{1}\) = Initial volume \(T_{1}\) = Initial temperature \(P_{2}\) = Final pressure \(V_{2}\) = Final volume \(T_{2}\) = Final temperature In this problem, we know that the pressure is held constant, which means \(P_{1} = P_{2}\). We can then rearrange the combined gas law formula to solve for the final volume: \[ V_{2} = \frac{T_{2}}{T_{1}} × V_{1} \]

Step 2: Convert given temperature values to kelvin

In order to solve the equation, both given temperature values must be converted to kelvin. To do so, we apply the conversion formula: \[K = °C + 273.15\] Initial temperature ( \(T_{1}\) ): \( T_{1} = 25°C + 273.15 = 298.15K \) Final temperature ( \(T_{2}\) ): \( T_{2} = 35°C + 273.15 = 308.15K \)

Step 3: Calculate the final volume

Now we can use the formula for the final volume: \( V_{2} = \frac{T_{2}}{T_{1}} × V_{1} \) \( V_{2} = \frac{308.15K}{298.15K} × 1.58L \) \( V_{2} ≈ 1.63L \)

Step 4: Compare the initial and final volumes

We have calculated the final volume of the gas sample to be approximately 1.63L. We can now compare this value to the initial volume: \( V_{initial} = 1.58L \) \( V_{final} ≈ 1.63L \) Since the final volume is greater than the initial volume, we can conclude that the volume of the gas sample will increase when the temperature is raised from \(25°C\) to \(35°C\), while the pressure is held constant at \(759.0mmHg\).

Key Concepts

Gas Laws

Gas laws are fundamental principles that describe the behavior of gases and how they respond to changes in temperature, volume, and pressure. These laws provide a simplified model for understanding how gases interact with their environment and with each other.

There are three primary gas laws that you might encounter in chemistry:

• Boyle's Law, which states that at constant temperature, the pressure of a gas is inversely proportional to its volume.
• Charles's Law, which explains that at constant pressure, the volume of a gas is directly proportional to its absolute temperature.
• Avogadro's Law, which suggests that at constant temperature and pressure, the volume of a gas is proportional to the number of moles of the gas.

These individual laws are special cases of the combined gas law, which connects all three relationships into a single equation. When dealing with problems in gas behavior, identifying which gas law to apply is a critical first step.

Charles's Law

Charles's Law, named after the French scientist Jacques Charles, is one of the fundamental gas laws that describes how gases tend to expand when heated. According to this law, the volume of a gas is directly proportional to its temperature when the pressure is held constant. This can be mathematically represented as: \[ \frac{V_1}{T_1} = \frac{V_2}{T_2} \] where \(V\) represents volume, \(T\) represents the absolute temperature (measured in Kelvin), and the subscripts 1 and 2 refer to initial and final states, respectively.

For students to understand Charles's Law, it's crucial to recognize that temperature needs to be in Kelvin for calculations because 0 Kelvin (absolute zero) is the point where the motion of particles theoretically stops. So when you're using Charles's Law in calculations, remember first to convert Celsius to Kelvin. This focus on temperature underlies the direct relationship between temperature and volume in gases under constant pressure.

Temperature-Pressure-Volume Relationship

Understanding the relationship between temperature, pressure, and volume is key to mastering gas behavior. This triad forms the backbone of the combined gas law which helps predict how a given gas will act under varying conditions. To apply the combined gas law properly, it's necessary to grasp the dependence between these variables:

• At constant volume, an increase in temperature leads to an increase in pressure (Gay-Lussac's Law).
• At constant pressure, an increase in temperature results in an increase in volume (Charles's Law).
• At constant temperature, an increase in pressure causes a decrease in volume (Boyle's Law).

Essentially, these factors are interlinked such that altering one, while holding another constant, impacts the third. This unified description is represented by the combined gas law formula: \[ \frac{P_{1}V_{1}}{T_{1}} = \frac{P_{2}V_{2}}{T_{2}} \] Here, temperatures must be in Kelvin, the linear relationship between volume and Kelvin temperature must be adhered to, and pressure must remain unchanged for Charles's Law to hold true, as showcased in the provided exercise.