At a constant pressure of 0.905 atm, a chemical reaction takes place in a cylindrical container with a movable piston having a diameter of 40.0 cm. During the reaction, the height of the piston drops by 65.0 cm. (The volume of a cylinder is V=pr2h,wherehistheheight;1Latm=101.3J.) (b) What is the value in joules of the work w done during the reaction?

Verified step by step guidance

1. insert step 1> Calculate the initial volume of the cylinder using the formula for the volume of a cylinder: \( V = \pi r^2 h \). Here, the initial height is the original height before the piston drops.
2. insert step 2> Calculate the final volume of the cylinder using the same formula, but with the height reduced by 65.0 cm.
3. insert step 3> Determine the change in volume (\( \Delta V \)) by subtracting the final volume from the initial volume.
4. insert step 4> Use the formula for work done at constant pressure: \( w = -P \Delta V \), where \( P \) is the pressure in atm and \( \Delta V \) is the change in volume in liters.
5. insert step 5> Convert the work from L atm to joules using the conversion factor \( 1 \text{ L atm} = 101.3 \text{ J} \).

Key Concepts

Work in Thermodynamics

In thermodynamics, work is defined as the energy transferred when a force is applied over a distance. For a gas expanding or contracting in a piston, the work done can be calculated using the formula w = -PΔV, where P is the pressure and ΔV is the change in volume. The negative sign indicates that work is done by the system when it expands.

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, the height corresponds to the displacement of the piston, and the radius can be derived from the diameter. Understanding how to calculate the volume change is essential for determining the work done during the reaction.

Pressure-Volume Relationship

The pressure-volume relationship is a fundamental concept in gas laws, indicating how pressure and volume are interrelated in a closed system. At constant pressure, any change in volume directly affects the work done by or on the system. In this scenario, knowing the constant pressure allows for the calculation of work based on the change in volume as the piston moves.