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 is 1831.65 in-lb (on the gas).

Step 1: Understand the Variables and Constants

We start by identifying the variables and constants given in the problem. The initial volume \( V_1 = 16 \) cubic inches, the initial pressure \( p_1 = 40 \) pounds per square inch, and the final volume \( V_2 = 2 \) cubic inches. It's also important to note that \( pV^{1.4} = c \) is a given constant in adiabatic processes.

Step 2: Finding the Constant 'c'

Use the initial conditions to find the constant \( c \). Substitute \( p = 40 \) psi and \( V = 16 \) cubic inches into the equation: \[40 \times 16^{1.4} = c\]Calculate \( 16^{1.4} \approx 45.2548 \). Therefore, \[c = 40 \times 45.2548 = 1810.192\].

Step 3: Find Pressure at Final Volume

Using the constant \( c \) found in Step 2, and the final volume, find the final pressure \( p_2 \). Set up the equation: \[p_2 \times 2^{1.4} = 1810.192\]Calculate \( 2^{1.4} \approx 2.639 \). Solving for \( p_2 \), we get \[p_2 = \frac{1810.192}{2.639} \approx 686.33 \text{ psi}\].

Step 4: Calculate Work Done by the Piston

The work done in an adiabatic process is given by the formula:\[W = \frac{p_1 V_1 - p_2 V_2}{\gamma - 1}\]where \( \gamma = 1.4 \). Substitute the known values:\[W = \frac{40 \times 16 - 686.33 \times 2}{1.4 - 1}\]Calculate:\[40 \times 16 = 640 \text{ and } 686.33 \times 2 = 1372.66\]Thus, \[W = \frac{640 - 1372.66}{0.4} = -1831.65 ext{ in-lb}.\] The negative sign indicates work done on the gas.

Key Concepts

Cylinder and Piston

A cylinder and piston system is a core element in thermodynamics, especially in the study of gases and their behavior under different conditions. Imagine the cylinder as a tube or chamber closed at one end, within which a movable piston can slide up and down. This piston acts as a boundary separating different volumes of the gas inside the cylinder.
The piston can move to compress or expand the gas, which directly involves changes in pressure, volume, and temperature. In this kind of setup, the volume of the gas can change as the piston moves in or out.
Key features involve:

• The cross-sectional area of the piston, which defines the amount of force applied, usually given in square inches or square meters.
• The initial volume of gas, which in our example starts at 16 cubic inches.
• The initial pressure, which is at 40 pounds per square inch in this scenario.

Understanding these features is essential as they impact the energy and work calculations performed when the piston compresses or expands the gas.

Work Done by Piston

The concept of 'work done' by a piston in a thermodynamic system is a measure of energy transfer. When a piston compresses or expands gas within a cylinder, work is performed. This is quantifiable and often expressed in terms of energy units, such as inch-pounds or Joules.
During adiabatic processes, where no heat is exchanged with the environment, the work done can be determined by specific equations. In our example, the piston compresses the gas from an initial 16 cubic inches to a final volume of 2 cubic inches. The work done in this scenario is calculated using:

• The initial and final pressures, obtained from the adiabatic process formula. In this example, starting from 40 psi to a calculated pressure of 686.33 psi at the end.
• The adiabatic index or \( \gamma \), which is a specific heat ratio. Here, \( \gamma = 1.4 \).
• The formula \( W = \frac{p_1 V_1 - p_2 V_2}{\gamma - 1} \), showcasing the transition of states in terms of energy exerted.

While calculations show work as negative (-1831.65 in-lb), it denotes that work was needed to compress the gas, aligning with energy getting stored or altered during compression.

Adiabatic Law

The adiabatic law describes a process occurring within a system where no heat is transferred to or from the surroundings. It's a type of thermodynamic process that ensures the internal energy change is only due to the work done by or on the gas.
In an adiabatic process, pressure and volume are related by the expression \( p v^{1.4} = c \), where:

• \( p \) is the pressure of the gas.
• \( v \) is the specific volume of the gas.
• \( c \) is a constant specific to the particular setup and amount of substance involved.

Using this equation, if the volume of a gas changes, its pressure changes inversely, i.e., when the volume decreases, as seen in our example, the pressure increases. The adiabatic index or exponent, here 1.4, dictates the variation and depends on the specific heat capacities of the gas involved.
In practical application, this law helps to predict and calculate how energy transformations occur within engines and other systems involving gases, by defining the relationship between pressure and volume without external heat exchange.