Problem 82 - Chemistry Matter and Change

Question

If the volume of a sample of chlorine gas is 8.2 \(\mathrm{L}\) at 1.2 atm and 306 \(\mathrm{K}\) , what volume will the gas occupy at STP? (Chapter 13\()\)

Step-by-Step Solution

Answer

The gas will occupy approximately 8.77 L at STP.

Step 1: Understand the Problem

We are given the initial conditions of a chlorine gas sample: initial volume \( V_1 = 8.2 \; \mathrm{L} \), initial pressure \( P_1 = 1.2 \; \mathrm{atm} \), and initial temperature \( T_1 = 306 \; \mathrm{K} \). We need to find the final volume \( V_2 \) of the gas at STP (Standard Temperature and Pressure), which is defined as \( P_2 = 1 \; \mathrm{atm} \) and \( T_2 = 273 \; \mathrm{K} \).

Step 2: Apply Combined Gas Law

The combined gas law relates two sets of conditions of pressure, volume, and temperature: \( \frac{P_1 \cdot V_1}{T_1} = \frac{P_2 \cdot V_2}{T_2} \). We'll rearrange this equation to solve for \( V_2 \): \( V_2 = \frac{P_1 \cdot V_1 \cdot T_2}{T_1 \cdot P_2} \).

Step 3: Substitute Known Values

Substitute the given values into the equation: \( V_2 = \frac{1.2 \; \mathrm{atm} \times 8.2 \; \mathrm{L} \times 273 \; \mathrm{K}}{306 \; \mathrm{K} \times 1 \; \mathrm{atm}} \).

Step 4: Calculate the Final Volume

Calculate the expression from the previous step to find \( V_2 \). Simplify the calculation: \( V_2 = \frac{1.2 \times 8.2 \times 273}{306} \approx 8.77 \; \mathrm{L} \).

Key Concepts

Standard Temperature and Pressure (STP)

Standard Temperature and Pressure, commonly abbreviated as STP, is a set of conditions used to describe gases in a consistent way. STP simplifies comparing different gases and their behaviors. The conditions defined as STP are:

• Pressure: 1 atmosphere (atm)
• Temperature: 273 Kelvin (K) or 0 degrees Celsius (6C)

Using STP is crucial because gases expand and contract with changes in temperature and pressure. Therefore, having a standard reference point like STP allows scientists to communicate more effectively using a universal baseline. When dealing with problems in chemistry or physics involving gases, converting conditions to STP may make calculations and comparisons more manageable.

Chlorine Gas

Chlorine is a chemical element symbolized by Cl and it appears as a yellow-green gas under standard conditions. Chlorine gas is a diatomic molecule, meaning each molecule consists of two chlorine atoms bonded together (Cl8). This gas is classified as a halogen, situated in the 17th group of the periodic table, along with other elements such as fluorine and iodine.

Chlorine gas has various industrial and laboratory applications, mainly due to its properties as an oxidizing agent and disinfectant. However, when handling chlorine gas, precautions are necessary because it is highly reactive and can be hazardous to health. Breathing chlorine can irritate the respiratory system, making proper ventilation and protective gear essential.

In chemistry exercises like the one at hand, understanding chlorine as a gas allows us to apply gas laws efficiently to predict how its volume changes under different conditions.

Gas Laws

Gas laws describe how gases behave under various conditions of pressure, volume, and temperature. These laws were developed predominantly in the 17th and 18th centuries and lay the groundwork for understanding gaseous substances.

**Combined Gas Law**:The combined gas law is particularly useful because it integrates three individual gas laws:

• Boyle's Law: Relates pressure and volume at a constant temperature, where pressure inversely affects volume.
• Charles's Law: Describes the direct relationship between volume and temperature at a constant pressure.
• Gay-Lussac's Law: Illustrates the direct relationship between pressure and temperature at a constant volume.

The equation is expressed as \[ \frac{P_1 \cdot V_1}{T_1} = \frac{P_2 \cdot V_2}{T_2} \]This equation helps in calculating how a gas will behave when we change its environment, as in our exercise with chlorine gas. By knowing the initial conditions of the gas and the conditions it will be exposed to, we can find out the new volume the gas will occupy under those circumstances.

Mastering gas laws is essential for tackling problems in both educational settings and practical applications, such as predicting the behavior of natural gases in various industrial processes.