Problem 100 - Chemistry Matter and Change

Question

If the volume of a sample of chlorine gas is 4.5 \(\mathrm{L}\) at 0.65 atm and \(321 \mathrm{K},\) what volume will the gas occupy at STP? (Chapter 13\()\)

Step-by-Step Solution

Answer

The gas will occupy approximately 2.49 L at STP.

Step 1: Understand the Problem

We're given the initial conditions of a gas: volume \( V_1 = 4.5 \, \text{L} \), pressure \( P_1 = 0.65 \, \text{atm} \), and temperature \( T_1 = 321 \, \text{K} \). We need to find the volume \( V_2 \) at standard temperature and pressure (STP), which is \( P_2 = 1 \, \text{atm} \) and \( T_2 = 273 \, \text{K} \).

Step 2: Use the Ideal Gas Law

The ideal gas law relates pressure, volume, and temperature: \( \frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2} \). This is called the Combined Gas Law. Our goal is to solve for \( V_2 \).

Step 3: Rearrange the Formula

Rearrange the combined gas law formula to solve for \( V_2 \): \( V_2 = \frac{P_1V_1T_2}{P_2T_1} \).

Step 4: Substitute Known Values

Substitute the known values into the rearranged formula: \( V_2 = \frac{0.65 \, \text{atm} \times 4.5 \, \text{L} \times 273 \, \text{K}}{1 \, \text{atm} \times 321 \, \text{K}} \).

Step 5: Calculate the Result

First, calculate the numerator: \( 0.65 \times 4.5 \times 273 = 798.525 \). Then calculate \( V_2 \) by dividing by the denominator: \( \frac{798.525}{321} \approx 2.488 \, \text{L} \).

Step 6: Conclude the Solution

The volume of the chlorine gas at STP is approximately \( 2.49 \, \text{L} \).

Key Concepts

Combined Gas Law

The Combined Gas Law is a fundamental principle that combines three basic gas laws: Boyle's Law, Charles's Law, and Gay-Lussac's Law. These laws correlate pressure, volume, and temperature of a gas sample. By merging these relations, we arrive at a more flexible tool for solving problems where conditions change.

The formula for the Combined Gas Law is:

• \[ \frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2} \]

This equation helps us determine the final state of a gas when its pressure, volume, or temperature changes. In our exercise, we use this equation to find the new volume of a chlorine gas sample when moved to conditions of Standard Temperature and Pressure (STP).

It is important to always use the same units throughout your calculations to avoid any errors. Temperature must be in Kelvin, while pressure and volume can be in any consistent units like atm and liters.

The Combined Gas Law is ideal for calculating changes in a gas's condition. It assumes that the gas behaves ideally, which is often a good approximation unless at very high pressures or low temperatures.

Ideal Gas Law

The Ideal Gas Law is a cornerstone of gas chemistry, representing an idealized model of how gases behave. It is succinctly expressed as

• \[ PV = nRT \]

where

• \( P \) is the pressure,
• \( V \) is the volume,
• \( n \) is the number of moles of gas,
• \( R \) is the universal gas constant, and
• \( T \) is the temperature in Kelvin.

This law is fundamental because it links all four variables into a single equation. It allows us to predict any one of the parameters if the others are known.

In our context, the Ideal Gas Law supports understanding how volume adjusts under different conditions. Although our exercise focused primarily on the Combined Gas Law, the Ideal Gas Law provides a broader understanding. It allows for calculations involving the amount of gas if needed, making it an immensely powerful tool.

In real life situations, gases closely follow the Ideal Gas Law when they are at high temperatures and low pressures, situations where the gas molecules are far apart and interactions are minimal.

Standard Temperature and Pressure (STP)

Standard Temperature and Pressure (STP) is a standardized set of conditions often used in chemistry to create a common reference point for scientists. At STP, the temperature is set to \( 273 \, \text{K} \) (which is equivalent to 0°C), and the pressure is \( 1 \, \text{atm} \).

These conditions allow for consistent comparison of gas behaviors across different experiments and calculations. In our exercise, knowing these specific conditions is crucial for calculating the new volume of chlorine gas when it equilibrates to STP.

At STP, the molar volume of an ideal gas is about \( 22.4 \, \text{L} \), meaning that one mole of an ideal gas occupies this volume. This assumption allows us to determine moles, volume, or mass of a gas under these conditions.

STP serves as a convenient standard for conducting gas law calculations, ensuring uniformity and simplicity in chemical equations and experiments. It is particularly useful when determining gas behaviors or reactions in ideal conditions, offering a baseline to judge against real-world observations.