Question

An ideal gas originally at \(0.85 \mathrm{~atm}\) and \(66^{\circ} \mathrm{C}\) was allowed to expand until its final volume, pressure, and temperature were \(94 \mathrm{~mL}, 0.60 \mathrm{~atm},\) and \(45^{\circ} \mathrm{C}\), respectively. What was its initial volume?

Short answer

The initial volume is approximately 71.87 mL.

Step-by-step solution

Identify the Given Parameters

We know that we need to find the initial volume \( V_1 \) of the ideal gas. The given initial conditions are pressure \( P_1 = 0.85 \text{ atm} \) and temperature \( T_1 = 66^{\circ} \text{C} \). The final conditions are pressure \( P_2 = 0.60 \text{ atm} \), temperature \( T_2 = 45^{\circ} \text{C} \) and volume \( V_2 = 94 \text{ mL} \).

Convert Temperatures to Kelvin

Convert the initial and final temperatures from degrees Celsius to Kelvin by adding 273.15. Thus, initial temperature \( T_1 = 66 + 273.15 = 339.15 \text{ K} \), and final temperature \( T_2 = 45 + 273.15 = 318.15 \text{ K} \).

Apply the Combined Gas Law

The combined gas law equation is \( \frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2} \). Substitute the known values: \( \frac{0.85 \times V_1}{339.15} = \frac{0.60 \times 94}{318.15} \).

Solve for Initial Volume \( V_1 \)

Rearrange the equation to solve for \( V_1 \): \( V_1 = \frac{0.60 \times 94 \times 339.15}{0.85 \times 318.15} \). Calculate to find \( V_1 \).

Perform the Calculation

Calculate: \( V_1 = \frac{0.60 \times 94 \times 339.15}{0.85 \times 318.15} \approx 71.87 \text{ mL} \).

Key concepts

Combined Gas Law

The combined gas law is a relationship between the pressure, volume, and temperature of a given quantity of gas, assuming that the amount of the gas is constant.
It is especially useful when gas undergoes changes involving three of its properties. The combined gas law formula is given by:

• \( \frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2} \)

In this equation:

• \( P_1 \) and \( P_2 \) are the initial and final pressures, respectively
• \( V_1 \) and \( V_2 \) are the initial and final volumes, respectively
• \( T_1 \) and \( T_2 \) are the initial and final temperatures in Kelvin

The combined gas law tells us how the gas will react to changes in the environment. It combines three separate gas laws: Boyle’s Law (pressure and volume), Charles’s Law (volume and temperature), and Gay-Lussac’s Law (pressure and temperature).
This law helps us solve for one unknown property of the gas when the others are known.

Temperature Conversion

Gases are significantly affected by temperature changes. To accurately use the combined gas law, it is essential to convert temperatures from Celsius to Kelvin.
Kelvin is the preferred unit because it avoids negative temperature values, which can lead to errors when calculating changes in physical properties of gases.

Conversion Steps:

• Add 273.15 to the temperature in Celsius to convert it to Kelvin.
• Example: \( 66^{\circ} \text{C} \) becomes \( 339.15 \text{ K} \).
• Example: \( 45^{\circ} \text{C} \) becomes \( 318.15 \text{ K} \).

Always use Kelvin for any gas law calculations to ensure accuracy.

Pressure and Volume

The relationship between pressure and volume is a fundamental concept in the study of gases. According to Boyle’s Law, for a given mass of gas at constant temperature, the pressure of the gas is inversely proportional to its volume.

• This means as pressure increases, volume decreases if temperature remains constant, and vice versa.

In the combined gas law, both pressure and volume change, which requires adjusting for temperature to accurately predict behavior.

Key Points to Remember:

• A higher pressure indicates a compressed gas, resulting in a smaller volume.
• A lower pressure allows the gas to expand, increasing its volume.

Understanding this relationship helps predict how gas will behave under different environmental conditions.

Gas Expansion

Gas expansion occurs when a gas increases in volume, often as a result of reduced pressure or increased temperature.
According to Charles's Law, if the pressure remains constant, the volume of a gas is directly proportional to its temperature.

• As the temperature rises, the gas molecules gain kinetic energy, causing them to move apart and the gas to expand.
• Lowering the pressure can also lead to expansion by giving molecules more space to spread out.

In the given problem, the gas was allowed to expand while being subjected to decreased pressure and a slightly reduced temperature.
This demonstrates that by changing one or more properties like pressure or temperature, the gas volume can be controlled to fit specific requirements.