Problem 8 - Chemistry and Chemical Reactivity

Question

A 5.0 -mL sample of \(\mathrm{CO}_{2}\) gas is enclosed in a gastight syringe (Figure 10.2 ) at \(22^{\circ} \mathrm{C}\). If the syringe is immersed in an ice bath \(\left(0^{\circ} \mathrm{C}\right),\) what is the new gas volume, assuming that the pressure is held constant?

Step-by-Step Solution

Answer

The new volume of the gas is 4.63 mL.

Step 1: Identify the Gas Law

In this problem, we will use Charles's Law, which states that the volume of a gas is directly proportional to its temperature in Kelvin, as long as the pressure is held constant. The formula is: \( \frac{V_1}{T_1} = \frac{V_2}{T_2} \).

Step 2: Convert Temperatures to Kelvin

Convert the initial and final temperatures from degrees Celsius to Kelvin by adding 273.15. \( T_1 = 22^{\circ}C + 273.15 = 295.15 \text{ K} \) and \( T_2 = 0^{\circ}C + 273.15 = 273.15 \text{ K} \).

Step 3: Plug Values into Charles's Law

Using the equation \( \frac{V_1}{T_1} = \frac{V_2}{T_2} \), substitute the known values: \( \frac{5.0 \text{ mL}}{295.15 \text{ K}} = \frac{V_2}{273.15 \text{ K}} \).

Step 4: Solve for the New Volume \(V_2\)

Rearrange the equation to solve for \( V_2 \): \( V_2 = \frac{5.0 \text{ mL} \times 273.15 \text{ K}}{295.15 \text{ K}} \). Calculate \( V_2 \) to find the new volume.

Step 5: Calculate the New Volume

Perform the calculation: \( V_2 = \frac{5.0 \times 273.15}{295.15} = 4.63 \text{ mL} \).

Key Concepts

Gas Laws

Gas laws are foundational principles in chemistry that describe the behavior of gases under various conditions. One important gas law is Charles's Law, which illustrates the relationship between the volume of a gas and its temperature at constant pressure. According to Charles's Law, when the temperature of a gas increases, its volume increases. Conversely, when the temperature decreases, the volume decreases as well. The formula that represents this relationship is expressed as:

• \( \frac{V_1}{T_1} = \frac{V_2}{T_2} \)

Where:

• \( V_1 \) and \( V_2 \) are the initial and final volumes.
• \( T_1 \) and \( T_2 \) are the initial and final temperatures in Kelvin.

This law helps us understand how a gas will respond to changes in temperature, which is crucial in predicting gas behavior in various scientific and real-world applications.
By applying Charles's Law, we can determine how a gas will expand or contract when subjected to temperature changes, provided the pressure remains constant.

Temperature Conversion

Temperature conversion is a necessary step in applying gas laws accurately. In the context of gas laws, temperatures must be in Kelvin to utilize formulas correctly. This is because Kelvin is an absolute temperature scale, starting from absolute zero, where molecular motion theoretically ceases.
To convert a temperature from Celsius to Kelvin, you simply add 273.15:

• \( K = ^\circ C + 273.15 \)

For example, converting the initial temperature of \(22^{\circ}C\) to Kelvin gives \(295.15 \text{ K}\), and \(0^{\circ}C\) becomes \(273.15 \text{ K}\).
Using Kelvin ensures that calculations involving ratios and proportions will yield accurate, meaningful results. This is pivotal in applying Charles's Law and other gas laws where temperature plays a critical role.

Volumetric Expansion

Volumetric expansion refers to the change in volume of a gas with changes in temperature. According to Charles's Law, the relationship between volume and temperature is direct, meaning as temperature changes, the gas volume adjusts proportionally.
When performing experiments, it is essential to measure the initial and final volumes and temperatures to evaluate how much expansion or contraction has occurred. In the example problem, as the gas is cooled from \(22^{\circ}C\) to \(0^{\circ}C\), we notice a reduction in volume from 5.0 mL to approximately 4.63 mL.
Understanding how volumetric expansion works helps in many fields, including meteorology, respiratory applications, and engineering processes, where temperature changes may impact gas volumes.

Kelvin Scale

The Kelvin scale is a thermodynamic temperature scale that is pivotal for scientific calculations involving gases. Unlike the Celsius scale, Kelvin begins at absolute zero (0 K), the point where it is theorized that atoms cease motion. This makes it ideal for situations where proportional comparisons of temperature are necessary, such as in gas law calculations.
The Kelvin scale eliminates negative temperature values, which is crucial when dealing with mathematical equations involving ratios, such as Charles's Law. For instance, in our exercise, both initial and final temperatures were converted to Kelvin for practical application:

• Initial Temperature: \(295.15 \text{ K}\) from \(22^{\circ}C\).
• Final Temperature: \(273.15 \text{ K}\) from \(0^{\circ}C\).

Using Kelvin ensures consistent and reliable results when analyzing relationships like that between temperature and volume. It is the most widely used temperature scale in scientific research due to its precision and practicality.