Problem 16 - Calculus Early Transcendentals: Pearson New International Edition

Question

A cylinder and piston, whose cross-sectional area is 1 square inch, contain 16 cubic inches of gas under a pressure of 40 pounds per square inch. If the pressure and the volume of the gas are related adiabatically (i.e., without loss of heat) by the law \(p v^{1.4}=c\) (a constant), how much work is done by the piston in compressing the gas to 2 cubic inches?

Step-by-Step Solution

Answer

The work done by the piston in compressing the gas is approximately 1057.97 inch-pounds.

Step 1: Understand the Problem

We're given a scenario involving a piston compressing a gas in a cylinder. The initial volume of the gas is 16 cubic inches, with a pressure of 40 pounds per square inch. We need to find out the work done in compressing the gas adiabatically to 2 cubic inches.

Step 2: Set Up the Adiabatic Equation

The adiabatic process is described by the equation \( p v^{1.4} = c \), where \( c \) is a constant. Initially, \( p_1 = 40 \) psi and \( v_1 = 16 \) in³.

Step 3: Calculate the Constant

Substitute the initial conditions into the adiabatic equation: \( c = 40 \times 16^{1.4} \). Compute \( 16^{1.4} \) using exponential rules and then calculate \( c \).

Step 4: Find the Pressure at the Final Volume

Use the adiabatic relation to find the final pressure \( p_2 \) when the gas volume \( v_2 \) is 2 in³. Rearrange to \( p_2 = \frac{c}{v_2^{1.4}} \), then substitute for \( c \) and \( v_2 \).

Step 5: Calculate the Work Done

The work done in an adiabatic process is given by \( W = \frac{p_1v_1 - p_2v_2}{rac{1.4}{1.4 - 1}} \). Substitute the initial and final pressures and volumes to find \( W \).

Step 6: Perform the Integration for Exact Work

Alternatively, calculate work using \( W = \int_{v_1}^{v_2} p \, dv = \int_{16}^{2} \frac{c}{v^{1.4}} dv \). Evaluate this integral with the calculated \( c \).

Step 7: Compute Final Answer

After evaluating the integral or using the formula, summarize \( W \), which is the work done in compressing the gas from 16 in³ to 2 in³. Ensure the result is in the correct units, typically inch-pounds.

Key Concepts

Thermodynamics

Thermodynamics is a branch of physics that deals with heat, work, and energy exchange in systems. It primarily focuses on how various forms of energy are converted into each other and how substances behave under different temperatures and pressures. Understanding thermodynamics is crucial in fields such as engineering, chemistry, and environmental science.

The core principles of thermodynamics are encapsulated in its four laws, which describe the behavior of energy in physical processes:

• First Law: Often known as the law of energy conservation, it states that energy cannot be created or destroyed, only transformed from one form to another.
• Second Law: It asserts that energy transfer has a preferred direction, with energy naturally dispersing unless work is done to direct it.
• Third Law: This law implies that as temperature approaches absolute zero, the entropy of a perfect crystal approaches zero.
• Zeroth Law: This is about thermal equilibrium, stating if two systems are each in thermal equilibrium with a third system, they are in equilibrium with each other.

These laws help explain the natural tendency of systems to move towards thermal equilibrium and facilitate the understanding of processes such as adiabatic compression.

Adiabatic Compression

Adiabatic compression is a thermodynamic process in which a gas is compressed without exchanging heat with its surroundings. In this scenario, any work done on the gas to compress it results in a rise in the gas's internal energy and temperature. This makes adiabatic processes challenging but fascinating to analyze in physics.The key formula to describe adiabatic processes for an ideal gas is given by:\[ p v^{\gamma} = C \]where:

• \( p \) is the pressure,
• \( v \) is the volume,
• \( \gamma \) (gamma) is the heat capacity ratio (1.4 for diatomic gases),
• \( C \) is a constant.

During adiabatic compression, the work done is related to changes in pressure and volume and can be computed using integration of this relation. Unlike isothermal processes, adiabatic compression can lead to significant temperature changes, highlighting the gas's ability to store energy as internal heat without loss to the environment.

Cylinder and Piston

In thermodynamics, the cylinder and piston setup is a common apparatus used to illustrate processes involving gases. The piston compresses or expands the gas contained within the cylinder, and is a useful model to understand concepts like pressure, volume, and temperature changes during thermodynamic processes.

This apparatus typically functions as follows:

• Cylinder: Contains the gas and provides the boundary for energy and material flow.
• Piston: Moves within the cylinder, altering the volume available to the gas and thereby changing its pressure and temperature according to the applied force.

During adiabatic compression, as described by the problem, a piston compresses gas from a larger to a smaller volume without allowing heat to escape. This setup is essential to processes in engines and refrigeration cycles, where energy efficiency and conversion are of utmost importance.