A reaction inside a cylindrical container with a movable RAN piston causes the volume to change from 12.0 L to 18.0 L while the pressure outside the container remains constant at 0.975 atm. (The volume of a cylinder is V = pr2h, where h is the height; 1 L # atm = 101.325 J.) (b) The diameter of the piston is 17.0 cm. How far does the piston move?

Verified step by step guidance

1. Calculate the cross-sectional area of the piston using the formula for the area of a circle, \( A = \pi r^2 \), where \( r \) is the radius of the piston. Since the diameter is given as 17.0 cm, first convert this to radius by dividing by 2.
2. Convert the radius from centimeters to meters by dividing by 100, as the standard unit for area in the International System of Units (SI) is square meters (m^2).
3. Substitute the radius in meters into the area formula to find the cross-sectional area of the piston in square meters.
4. Calculate the change in volume, which is the final volume minus the initial volume. Convert this volume from liters to cubic meters by dividing by 1000, since 1 cubic meter equals 1000 liters.
5. Determine the distance the piston moved by dividing the change in volume by the cross-sectional area of the piston. This will give the height change in meters.

Key Concepts

Ideal Gas Law

The Ideal Gas Law relates the pressure, volume, temperature, and number of moles of a gas through the equation PV = nRT. While this question focuses on a specific scenario involving volume change and pressure, understanding the Ideal Gas Law is essential for grasping how gases behave under varying conditions.

Volume of a Cylinder

The volume of a cylinder is calculated using the formula V = πr²h, where r is the radius and h is the height. In this context, knowing how to calculate the volume change of the cylinder as the piston moves is crucial for determining the distance the piston travels.

Work Done by a Gas

When a gas expands or contracts, it does work on its surroundings, which can be calculated using the formula W = PΔV, where W is work, P is pressure, and ΔV is the change in volume. Understanding this concept helps in analyzing the energy changes associated with the movement of the piston in the container.