Problem 24 - Chemistry The Science in Context

Question

An expanding gas does \(150.0 \mathrm{J}\) of work on its surroundings at a constant pressure of 1.01 atm. If the gas initially occupied \(68 \mathrm{mL},\) what is the final volume of the gas?

Step-by-Step Solution

Answer

Answer: The final volume of the gas is approximately 66.53 mL.

Step 1: Convert pressure and volume to standard units

First, convert the given pressure of 1.01 atm to Pascals (Pa) using the conversion factor: 1 atm = 101325 Pa 1.01 atm × (101325 Pa / 1 atm) = 102339.25 Pa Next, convert the initial volume of 68 mL to cubic meters (m³) using the conversion factor: 1 mL = 1 × 10⁻⁶ m³ 68 mL × (1 × 10⁻⁶ m³ / 1 mL) = 6.8 × 10⁻⁵ m³

Step 2: Rearrange the formula for work to solve for final volume

The formula for work done on surroundings by an expanding gas at constant pressure is: Work = -P × (V_final - V_initial) Rearrange the formula to solve for the final volume (V_final): V_final = (Work / -P) + V_initial

Step 3: Plug in the values and calculate the final volume

Now, plug the given values into the formula and calculate the final volume of the gas: V_final = (150 J / -102339.25 Pa) + 6.8 × 10⁻⁵ m³ V_final ≈ -1.467 × 10⁻⁶ m³ + 6.8 × 10⁻⁵ m³ V_final ≈ 6.653 × 10⁻⁵ m³

Step 4: Convert the final volume back to mL

Finally, convert the final volume from cubic meters (m³) back to milliliters (mL) using the conversion factor: 1 m³ = 10⁶ mL 6.653 × 10⁻⁵ m³ × (10⁶ mL / 1 m³) ≈ 66.53 mL The final volume of the gas is approximately 66.53 mL.

Key Concepts

Work Done by Gas

When a gas expands, it does work on its surroundings. This is quite similar to when you push something, using energy. The main way this work by a gas is measured in physics is by knowing the pressure at which the gas expands and how much it changes in volume. The formula for calculating the work done by an expanding gas is:

• \[ \text{Work} = -P \times (V_{\text{final}} - V_{\text{initial}}) \]

Here,

• \( P \) is the pressure,
• \( V_{\text{final}} \) is the final volume,
• and \( V_{\text{initial}} \) is the initial volume of the gas.

The negative sign indicates that work is done by the gas. This formula applies when the pressure remains constant during the process. As the gas does work to expand, its energy decreases, which also means it can affect its temperature and other thermodynamic properties. Understanding how gases do work help us predict how they will behave when placed in different containers or situations.

Volume Conversion

Volume conversion is an essential part of understanding how gases expand or contract in different conditions. In scientific calculations, it's common to convert volume measurements to standard units so that calculations become consistent and accurate. Here's how that's usually done:

• Convert volume from milliliters (mL) to cubic meters (m³), as 1 mL is equal to \( 1 \times 10^{-6} \) m³.
• This conversion helps to standardize our computations, especially in thermodynamics calculations, where using m³ is typical.

For instance, in our problem, the initial volume of the gas was 68 mL, which converts to \( 6.8 \times 10^{-5} \) m³ in standard units. It's important to always go back to the original units once the calculations are complete to ensure you interpret the results in a context that's easy to understand, like turning cubic meters back to milliliters, as we did at the end of the solution.

Ideal Gas Law

The Ideal Gas Law is a fundamental principle used to analyze the behavior of gasses. It's represented by the equation:

• \[ PV = nRT \]

Where:

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

The Ideal Gas Law simplifies the relationships between these variables. However, it assumes ideal conditions where gases don't have any intermolecular forces and occupy no space. In many real-world situations, the Ideal Gas Law provides a good approximation of gas behavior. By understanding these relationships, scientists and engineers can predict how a change in one condition, like temperature or pressure, will affect the behavior of a gas.