Problem 23 - General Chemistry: Principles and Modern Applications

Question

A sample of gas has a volume of \(4.25 \mathrm{L}\) at \(25.6^{\circ} \mathrm{C}\) and \(748 \mathrm{mmHg} .\) What will be the volume of this gas at \(26.8^{\circ} \mathrm{C}\) and \(742 \mathrm{mmHg} ?\)

Step-by-Step Solution

Answer

The volume of this gas at \(26.8^{\circ}C\) and \(742 mmHg\) is approximately \(4.28 L\).

Step 1: Convert Celsius temperature to Kelvin

Gas laws use absolute temperatures, which are measured in Kelvin. The formula to convert from Celsius to Kelvin is \(K = C + 273.15\). Therefore, the initial temperature \(T1 = 25.6^{\circ}C + 273.15 = 298.75 K\) and the final temperature \(T2 = 26.8^{\circ}C + 273.15 = 299.95 K\).

Step 2: Substituting known values into the combined law formula

The combined gas law, which is the volume and temperature proportionality from Charles's Law and the inverse volume and pressure proportionality from Boyle's Law, is expressed as \((P1 x V1) / T1 = (P2 x V2) / T2\). Substituting in the given and calculated values, we get \((748 mmHg x 4.25 L) / 298.75 K = (742 mmHg x V2) / 299.95 K\).

Step 3: Solving for the target variable, V2

The step is solving for final volume \(V2\). This can be done by isolating \(V2\) on one side of the equation by cross multiplication. So, \(V2 = ((748 mmHg x 4.25 L) / 298.75 K) x 299.95 K / 742 mmHg\). Calculating this gives approximately \(4.28 L\).

Key Concepts

Boyle's Law

Boyle's Law is a fundamental principle that describes the relationship between the pressure and volume of a gas when its temperature is held constant. It states that the volume of a gas is inversely proportional to its pressure. This means if the pressure increases, the volume decreases and vice versa, as long as the temperature remains unchanged. The mathematical expression of Boyle's Law is:

• \[ P_1 \times V_1 = P_2 \times V_2 \]

Where:

• \( P_1 \) and \( V_1 \) are the initial pressure and volume.
• \( P_2 \) and \( V_2 \) are the final pressure and volume.

This law is very useful in calculating the changes in state of a gas as it is compressed or expanded in a closed system. For example, in our problem, we originally used this concept to understand how the change in pressure affects the volume of the gas under constant temperatures, which is part of the larger combined gas law.

Charles's Law

Charles's Law describes how gases tend to expand when heated. This law establishes that the volume of a gas is directly proportional to its temperature, provided that the pressure is constant. The relationship can be expressed mathematically as:

• \[ \frac{V_1}{T_1} = \frac{V_2}{T_2} \]

Where:

• \( V_1 \) and \( V_2 \) are the volumes at temperatures \( T_1 \) and \( T_2 \) respectively.

Remember that temperatures must always be in Kelvin for gas law calculations. By maintaining the pressure constant, Charles's Law helps predict how much a gas will expand as it heats up. In our exercise, understanding Charles's Law helps us calculate how the gas volume changes with temperature. This is particularly applied when we converted temperatures from Celsius to Kelvin to use this proportional relationship.

Temperature Conversion to Kelvin

Converting temperature to Kelvin is crucial for solving gas law problems because gas laws are based on absolute temperature measurements. The Kelvin scale starts at absolute zero, which is -273.15°C, and it eliminates negative numbers that might complicate mathematical calculations.To convert a temperature from Celsius to Kelvin, the formula is straightforward:

• \[ K = C + 273.15 \]

This conversion ensures that the temperatures have an absolute starting point, necessary for maintaining proportional relationships in equations such as Charles's Law. In our exercise, we first converted initial and final temperatures from Celsius to Kelvin before applying the combined gas law.This simple conversion is a fundamental step in working with gas laws, ensuring consistency and accuracy in calculations.