Problem 45 - Chemistry for Allied Health

Question

What is the final volume of a gas that was originally at \(0.75 \mathrm{~L}\) at \(25^{\circ} \mathrm{C}\) and a final temperature of \(50^{\circ} \mathrm{C}\) ? Assume constant pressure and moles.

Step-by-Step Solution

Answer

The final volume of the gas is approximately 0.813 L.

Step 1: Identify the Known Variables

We have an initial volume \( V_1 = 0.75 \mathrm{~L} \), an initial temperature \( T_1 = 25^{\circ} \mathrm{C} \), and a final temperature \( T_2 = 50^{\circ} \mathrm{C} \).

Step 2: Convert Temperatures to Kelvin

Since the temperature must be in Kelvin for gas law calculations, convert the Celsius temperatures: \( T_1 = 25 + 273.15 = 298.15 \mathrm{~K} \) and \( T_2 = 50 + 273.15 = 323.15 \mathrm{~K} \).

Step 3: Understand the Formula

Use Charles's Law, which states \( \frac{V_1}{T_1} = \frac{V_2}{T_2} \), where \( V_2 \) is the final volume.

Step 4: Rearrange Formula to Solve for \( V_2 \)

Rearrange the equation to find the final volume: \( V_2 = V_1 \times \frac{T_2}{T_1} \).

Step 5: Plug in the Values and Solve

Substitute the known values into the equation: \( V_2 = 0.75 \mathrm{~L} \times \frac{323.15 \mathrm{~K}}{298.15 \mathrm{~K}} \approx 0.813 \mathrm{~L} \).

Key Concepts

Understanding Gas Laws

Gas laws help us understand how gases behave under different conditions, particularly in terms of pressure, volume, and temperature. One of the fundamental gas laws is Charles's Law. This law describes the relationship between the volume and temperature of a gas at constant pressure:

• Charles's Law states that for a given amount of gas at constant pressure, the volume is directly proportional to its absolute temperature (in Kelvin).
• The formula is expressed as \( \frac{V_1}{T_1} = \frac{V_2}{T_2} \), where \( V \) represents volume and \( T \) represents temperature.
• This means if the temperature of the gas increases, its volume increases as well, provided the pressure remains constant.

Understanding this relationship is crucial in solving problems related to gas volume changes. It's a powerful tool for predicting how the gas will behave when temperatures change, as long as no other conditions (like pressure or the amount of gas) change.

Temperature Conversion

To use any gas law, temperatures must be in the same unit, particularly Kelvin, the SI unit of temperature. Converting Celsius to Kelvin is straightforward:

• Use the formula: \( K = ^\circ C + 273.15 \).
• This conversion is necessary because the Kelvin scale reflects absolute temperature, starting from absolute zero, the point where particle motion virtually stops.
• In our example, the initial temperature of \(25^{\circ} \mathrm{C}\) becomes \(298.15 \mathrm{~K}\), and the final temperature of \(50^{\circ} \mathrm{C}\) becomes \(323.15 \mathrm{~K}\).

This ensures consistent results across all calculations and applies to all equations involving thermodynamic processes with gases. Remember, Kelvin is always used for gas laws, as negative temperatures in Celsius could yield non-physical results in calculations.

Volume Calculation Using Charles's Law

Once temperatures are converted to Kelvin and you understand the basic gas law involved, calculating the new volume is straightforward. For volume calculations using Charles's Law:

• Rearrange the Charles's Law formula to solve for the final volume: \( V_2 = V_1 \times \frac{T_2}{T_1} \).
• Plug in the known values: Initial volume \( V_1 = 0.75 \mathrm{~L} \), initial temperature \( T_1 = 298.15 \mathrm{~K} \), and final temperature \( T_2 = 323.15 \mathrm{~K} \).
• Thus, the formula becomes: \( V_2 = 0.75 \mathrm{~L} \times \frac{323.15}{298.15} \).
• Calculate the result: \( V_2 \approx 0.813 \mathrm{~L} \).

This step-by-step approach helps you determine how the volume of a gas changes with temperature, ensuring accurate and reliable results every time. This process highlights the importance of understanding and correctly applying the underlying principles of gas laws.