Problem 29 - Calculus: Early Transcendentals

Question

When gas expands in a cylinder with radius \( r \), the pressure at any given time is a function of the volume: \( P = P(V) \). The force exerted by the gas on the piston (see the figure) is the product of the pressure and the area: \( F = \pi r^2 P \). Show that the work done by the gas when the volume expands from volume \( V_1 \) to volume \( V_2 \) is $$ W = \int_{V_1}^{V_2} P dV $$

Step-by-Step Solution

Answer

The work done is given by the integral \( \int_{V_1}^{V_2} P \, dV \).

Step 1: Understand the Relationship between Force and Pressure

The problem states that the force exerted by the gas on the piston is given by the formula \( F = \pi r^2 P \). Here, \( P \) represents the pressure and \( \pi r^2 \) is the area of the piston. This relationship shows that force is dependent on both the pressure exerted by gas and the area it acts upon.

Step 2: Define the Formula for Work Done

The work done by a force is calculated as the product of the force and the displacement in the direction of the force. In the context of gas expanding in a cylinder, this displacement is along the axis of the cylinder and relates to a change in volume, denoted as \( dV \). The formula for infinitesimal work done is \( dW = F \, dx \).

Step 3: Relate Displacement to Volume Change

The relationship between displacement \( dx \) and a change in volume \( dV \) is given by the cylinder volume formula. Thus, \( dV = \pi r^2 \, dx \). Substituting this into the formula for work gives us \( dW = F \, dx = \pi r^2 P \, dx = P \, dV \).

Step 4: Integrate to Find Total Work Done

To find the total work done as the gas expands from volume \( V_1 \) to \( V_2 \), integrate the expression for infinitesimal work:\[W = \int_{V_1}^{V_2} dW = \int_{V_1}^{V_2} P \, dV.\]This integral gives the cumulative work done by the gas during the expansion process, accounting for the changing pressure and volume.

Key Concepts

Pressure-Volume Relationship

When we discuss the expansion of gas in a cylinder, one key concept that arises is the pressure-volume relationship. This relationship is depicted mathematically by the function \( P = P(V) \), which shows that pressure \( P \) changes as the volume \( V \) of the gas changes. Each specific volume corresponds to a specific pressure, and this relationship is crucial for understanding how gas behaves when it expands.

In thermodynamics, the pressure-volume relationship helps predict the behavior of gases under various conditions. For instance, if a gas is compressed, the molecules are forced closer together, resulting in an increased pressure. Conversely, when a gas expands and takes up more volume, the pressure exerted by the gas typically decreases, assuming temperature remains constant.

Understanding this relationship allows us to calculate the work done by the gas as it expands or contracts within a system. It's vital because it gives us insight into the internal energy changes and allows us to predict how much energy will be required or released in real-world applications, like engines or refrigeration cycles.

Force Exerted by Gas

The force exerted by the gas in the piston-cylinder system is based entirely on the pressure and the surface area of the piston upon which the gas is acting. The formula for this force is given by \( F = \pi r^2 P \), where \( r \) is the radius of the circular piston and \( P \) is the pressure exerted by the gas.

This equation highlights an important principle: the force exerted by a gas is proportional to its pressure and the area over which it acts. If either the pressure increases or the piston’s area increases, the force exerted by the gas will increase. This principle is why a larger cylinder or a high-pressure environment leads to greater force outputs in mechanical systems.

Such understanding is particularly powerful in designing systems where force needs to be precisely controlled, such as hydraulic systems, automotive engines, and pneumatic devices. It allows engineers to predict how changes in conditions like pressure or piston size will affect the overall functioning of the device.

Piston-Cylinder System

A piston-cylinder system is a fundamental mechanical setup used to convert energy provided by gas expansion into useful mechanical work. In this system, a piston moves within a cylindrical chamber. As the gas expands, it pushes against the piston, causing it to move.

The key components of this system include:

• The **cylinder**, which contains the gas and provides a boundary for the force exerted by the gas.
• The **piston**, which is a movable component within the cylinder that transfers the force exerted by the gas to the outside mechanical system.
• The **gas**, which serves as the medium that performs the work through expansion.

The movement of the piston due to gas expansion is utilized in countless machines. For instance, in an internal combustion engine, the expansion of gases from a fuel-air mixture forces the pistons to move, which then turns the crankshaft, ultimately powering the vehicle.

This system harnesses thermodynamic principles to translate microscopic energy changes at the molecular level into macroscopic mechanical work. Understanding the dynamics of a piston-cylinder system allows us to develop efficient machines and engines that form the backbone of modern technology and industry.