Question

A reaction inside a cylindrical container with a movable piston causes the volume to change from \(12.0 \mathrm{~L}\) to \(18.0 \mathrm{~L}\) while the pressure outside the container remains constant at \(0.975\) atm. (The volume of a cylinder is \(V=\pi r^{2} h\), where \(h\) is the height; \(1 \mathrm{~L} \cdot \mathrm{atm}=101.325 \mathrm{~J} .)\) (a) What is the value in joules of the work \(w\) done during the reaction? (b) The diameter of the piston is \(17.0 \mathrm{~cm}\). How far does the piston move?

Short answer

(a) -593.35 J, (b) 0.265 meters.

Step-by-step solution

Identify the Task

The problem is asking us to calculate two things: (a) the work done in joules when the volume of a gas inside a cylindrical container changes, and (b) the distance the piston moves when the volume changes, given the diameter of the piston.

Convert Volumes from Liters to Cubic Meters

We know the change in volume is from 12.0 L to 18.0 L. To use the work formula appropriately with joules, we first need to convert these volumes to cubic meters.\[\text{Volume in cubic meters} = \frac{\text{Volume in liters}}{1000}\]So, - 12.0 L = 0.012 m³ - 18.0 L = 0.018 m³ The change in volume, \( \Delta V = 0.018 \text{ m}^3 - 0.012 \text{ m}^3 = 0.006 \text{ m}^3 \).

Calculate Work Done

To find the work done by the reaction, we use the formula:\[ w = - P_{\text{ext}} \cdot \Delta V \]Where \( P_{\text{ext}} \) is the external pressure and \( \Delta V \) is the change in volume. Given \( P_{\text{ext}} = 0.975 \) atm, first convert this pressure to Pascals (Pa):\[ 1 \text{ atm} = 101325 \text{ Pa} \implies 0.975 \text{ atm} = 0.975 \times 101325 \text{ Pa} = 98891.875 \text{ Pa} \]Substituting the values into the formula gives:\[ w = - (98891.875 \text{ Pa}) \times (0.006 \text{ m}^3) = -593.35125 \text{ J} \]Thus, the work done is approximately -593.35 J. The negative sign indicates work is done by the system (expansion).

Calculate the Movement of the Piston

To find out how far the piston moves, we use the relation of volume and height in a cylinder: \[ V = \pi r^2 h \]Where \( r \) is the radius. The diameter of the piston is 17.0 cm, thus:\[ r = \frac{17.0 \text{ cm}}{2} = 8.5 \text{ cm} = 0.085 \text{ m} \]The change in volume is still 0.006 m³. Rearrange the volume formula to solve for \( h \):\[ h = \frac{\Delta V}{\pi r^2} \]\[ h = \frac{0.006}{\pi (0.085)^2} \approx 0.265 \text{ m} \]Thus, the piston moves approximately 0.265 meters.

Key concepts

Cylinder Volume Calculation

When dealing with a cylindrical container, such as the one described in our problem, calculating the volume of the cylinder is crucial to understanding how much space the gas inside will occupy. The volume of a cylinder is given by the formula:\[ V = \pi r^2 h \]Where:

• \(V\) is the volume
• \(\pi\) is a constant (approximately 3.1416)
• \(r\) is the radius of the cylinder's base
• \(h\) is the height of the cylinder

In the exercise, the volume change is caused by the movement of the piston, and we need to calculate how this affects the gas inside. For this particular problem, we're given the diameter of the piston, which helps us find the radius \(r\) by dividing by two and converting from centimeters to meters. Remember to always keep your units consistent, which is why converting to meters throughout the calculation is essential.

Pressure Conversion in Physics

Pressure is a central factor in many physics problems, particularly those dealing with gases. In this exercise, we're concerned with converting pressure units to those compatible with the work and energy equations. The problem specifies the external pressure in atmospheres (atm) and instructs us to calculate it in pascals (Pa), suitable for energy calculations. One atmosphere is equal to 101,325 Pa. Thus, converting 0.975 atm to pascals involves a simple multiplication:- \(0.975 \text{ atm} \times 101,325 \text{ Pa/atm} = 98,891.875 \text{ Pa}\)This conversion is essential as it allows us to apply the pressure value directly into work equations without encountering unit incompatibility. Accuracy in unit conversion ensures all aspects of the mathematical models used in physics are correctly aligned, leading to accurate results.

Piston Movement Calculation

The movement of the piston corresponds directly to the change in internal volume within the cylinder. Knowing how far the cylinder moves is critical in understanding how volume expansions or contractions affect other variables. When the piston moves, it alters the cylinder’s height, thus affecting volume. To find this movement, we rearrange the cylinder volume formula to solve for the height \(h\):\[ h = \frac{\Delta V}{\pi r^2} \]Given:

• The diameter of the piston is 17.0 cm, making the radius 0.085 m
• The change in volume \(\Delta V = 0.006 \text{ m}^3 \)

Substitute these values in to find \(h\):- \( h \approx \frac{0.006}{\pi (0.085)^2} \approx 0.265 \text{ m} \)Understanding the piston movement and its effects on volume aids in visualizing how gases behave during thermodynamic processes. This step is particularly meaningful when analyzing systems' responses to internal and external influences.

Work-Energy Principle in Thermodynamics

The work-energy principle is a fundamental concept in thermodynamics, playing a pivotal role in understanding energy exchange. In the exercise, the work done by the system when the gas expands is calculated using this principle. Work, in this context, is expressed as:\[ w = - P_{\text{ext}} \times \Delta V \]Where:

• \(w\) is the work done
• \(P_{\text{ext}}\) is the external pressure
• \(\Delta V\) is the change in volume

The negative sign indicates that work is performed by the system (gas expansion) instead of on it. Substituting the external pressure in pascals (\(98,891.875 \text{ Pa}\)) and the change in volume (\(0.006 \text{ m}^3\)), we calculate the work:- \( w = - (98,891.875) \times (0.006) \approx -593.35 \text{ J} \)This means the system expends approximately 593.35 joules of energy in doing work on the surroundings, a typical behavior when gases expand under constant external pressure.