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Chapter 1: Problem 21

For a process taking place in a closed system containing gas, the volume and pressure relationship is \(p V^{1.4}=\) constant. The process starts with initial conditions, \(p_{1}=1.5\) bar, \(V_{1}=0.03 \mathrm{~m}^{3}\) and ends with final volume, \(V_{2}=0.05 \mathrm{~m}^{3}\). Determine the final pressure \(p_{2}\) in bar.

Short Answer

Expert verified
The final pressure \( p_{2} \) is 0.528 bar.

Step by step solution

01

Understand the Equation and Given Data

The relationship between pressure and volume for the process is given by the equation: \[ p V^{1.4} = \text{constant} \] The initial conditions are \( p_{1} = 1.5 \text{ bar} \) and \( V_{1} = 0.03 \text{ m}^{3} \). The final volume is given as \( V_{2} = 0.05 \text{ m}^{3} \). The goal is to find the final pressure \( p_{2} \).
02

Write the Equation with Initial Conditions

Using the initial conditions, write the equation as: \[ p_{1} V_{1}^{1.4} = \text{constant} \] Substitute the given values: \[ 1.5 \text{ bar} \times (0.03 \text{ m}^3)^{1.4} = \text{constant} \]
03

Calculate the Constant

Calculate the value of the constant: \[ \text{constant} = 1.5 \times (0.03)^{1.4} \] Using a calculator: \[ \text{constant} \ = 1.5 \times (0.03)^{1.4} \ = 1.5 \times 0.006621 \ = 0.0099315 \text{ bar} \text{ m}^{1.4} \]
04

Use the Constant with Final Volume

Now use the constant value with the final volume to find the final pressure \( p_{2} \): \[ p_{2} V_{2}^{1.4} = \text{constant} \] Substitute the known values: \[ p_{2} \times (0.05)^{1.4} = 0.0099315 \text{ bar} \text{ m}^{1.4} \]
05

Solve for Final Pressure

Rearrange the equation to solve for \( p_{2} \): \[ p_{2} = \frac{0.0099315 \text{ bar} \text{ m}^{1.4}}{(0.05)^{1.4}} \] Calculate the value: \[ p_{2} = \frac{0.0099315}{0.01879} \] \[ p_{2} = 0.528 \text{ bar} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Closed System
A closed system is one of the main types of thermodynamic systems. In a closed system, no mass enters or leaves the system, but energy can be exchanged with the surroundings. This is different from an open system, where both mass and energy can transfer across the system boundaries.
For example, consider a sealed container of gas. In our exercise, the process occurs within such a closed system, where the gas inside cannot escape.
When dealing with real-life problems, recognizing whether you're working with a closed system is crucial because it determines the applicable laws and equations.
Pressure-Volume Relationship
In many thermodynamic processes involving gases, there is a particular relationship between pressure (p) and volume (V). In our exercise, we follow an equation that governs this relationship: \[ p V^{1.4} = \text{constant} \] This equation suggests that the product of pressure and volume raised to a power is constant throughout the process.
This type of relationship often occurs in adiabatic processes, where no heat is exchanged with the environment.
Understanding this equation helps us predict how one variable changes when the other is altered, as seen in the step-by-step solution.
Keep in mind that this relationship is specific to certain types of processes and might change if applied to different scenarios.
Isentropic Process
An isentropic process is a special type of adiabatic process that is also reversible. In such a process, the entropy of the system remains constant. Isentropic means 'same entropy'.
This property makes these processes ideal in theory because there is no loss due to friction, unrestrained expansion, or other irreversible effects.
In the context of our problem, the relationship between pressure and volume adheres to the characteristic of an isentropic process, expressed by: \[ p V^{1.4} = \text{constant} \] Recognizing an isentropic process allows us to use specific mathematical relationships, which simplify calculations and helps us determine critical system values, such as final pressure, given volume changes.
Calculation Steps
The step-by-step solution breaks down the process into understandable parts. Here's a detailed explanation:
  • Step 1: Understand the given equation and the initial data. This involves recognizing the pressure-volume relationship and identifying given values.
  • Step 2: Write the equation with initial conditions. Substituting known initial values helps us define the relationship.
  • Step 3: Calculate the constant. This is done by plugging in the initial values and solving to find the constant for the entire process.
  • Step 4: Use the constant with the final volume. Substitute the final volume into the equation to prepare to solve for the final pressure.
  • Step 5: Solve for the final pressure. Rearrange the equation to isolate the final pressure variable and compute its value.
This structured approach ensures that every step logically follows from the previous one, reinforcing a solid understanding of the process.
Breaking down problems in this manner can significantly help in tackling complex thermodynamic problems.

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