Calculating Work Done Isothermal Reversible Expansion Of Gas
Introduction
In thermodynamics, understanding the behavior of gases under various conditions is crucial. This article focuses on calculating the work done during an isothermal reversible expansion of a gas. Specifically, we will explore a scenario where one cubic meter (1 m³) of gas, initially at 5 atm and 273.2 K with a given heat capacity at constant volume (Cv = 3R), expands isothermally and reversibly to a final pressure of 1 atm. This detailed analysis provides a comprehensive understanding of the underlying principles and calculations involved in such thermodynamic processes.
Understanding Isothermal Reversible Expansion
To accurately calculate the work done, it is essential to first grasp the concepts of isothermal and reversible processes. An isothermal process occurs at a constant temperature. In our case, the gas expands while maintaining a consistent temperature of 273.2 K. This means that any energy input into the system, such as heat, is immediately used to do work, preventing a change in temperature. The term "reversible" implies that the process occurs in a series of infinitesimal steps, allowing the system to remain in equilibrium at all times. In simpler terms, a reversible process is an idealized scenario where the system can be returned to its initial state by reversing the conditions without any net change in the system or its surroundings.
Isothermal reversible expansion is a fundamental concept in thermodynamics, often used as a theoretical benchmark for real-world processes. The isothermal condition simplifies calculations because the internal energy change of an ideal gas depends only on temperature. Since the temperature remains constant, the change in internal energy is zero. However, work is still being done by the gas as it expands against the external pressure. This work is directly related to the heat absorbed by the gas from the surroundings to maintain the constant temperature. The reversibility aspect ensures that the expansion occurs in a controlled manner, maximizing the work output. This is because at each infinitesimal step, the external pressure is only infinitesimally smaller than the internal pressure of the gas, ensuring maximum efficiency.
Understanding these principles allows us to apply the appropriate thermodynamic equations and methodologies to accurately compute the work done. The combination of isothermal and reversible conditions provides a clear framework for analyzing the energy transfers and state changes involved in the expansion process. By exploring these concepts in detail, we can better appreciate the theoretical underpinnings of thermodynamics and their practical applications in various engineering and scientific contexts.
Given Conditions and Parameters
Before diving into the calculation, let's reiterate the given conditions and parameters that define our system. These parameters are crucial for setting up the problem and selecting the correct approach. We have the following information:
- Initial Volume (V1): 1 m³
- Initial Pressure (P1): 5 atm
- Final Pressure (P2): 1 atm
- Constant Temperature (T): 273.2 K
- Heat Capacity at Constant Volume (Cv): 3R, where R is the ideal gas constant
These values provide a clear picture of the system's initial and final states, allowing us to trace the expansion process accurately. The initial volume and pressure set the stage for the expansion, while the final pressure indicates the endpoint of the process. The constant temperature is a key factor in isothermal processes, simplifying calculations by maintaining a consistent thermal state throughout the expansion. The heat capacity at constant volume (Cv) is essential for determining the thermodynamic properties of the gas, as it relates the heat added to the gas to the change in its temperature under constant volume conditions. In this case, Cv = 3R implies that the gas is likely monatomic or has a simple molecular structure, as this value is typical for such gases.
With these parameters defined, we can now proceed with the necessary steps to calculate the work done. The isothermal and reversible nature of the expansion, combined with the given conditions, dictates the use of specific thermodynamic formulas and principles. The ideal gas law, along with the work equation for reversible isothermal processes, will be central to our calculations. Ensuring a clear understanding of these parameters is the foundation for accurately quantifying the work done during the expansion process.
Calculating the Work Done
To calculate the work done during an isothermal reversible expansion, we need to use the appropriate thermodynamic formula. For an ideal gas undergoing a reversible isothermal process, the work done (W) is given by:
W = -nRT ln(V2/V1)
Where:
- n is the number of moles of the gas
- R is the ideal gas constant (8.314 J/mol·K)
- T is the constant temperature (in Kelvin)
- V1 is the initial volume
- V2 is the final volume
First, we need to determine the number of moles (n) of the gas. We can use the ideal gas law, which states:
PV = nRT
Using the initial conditions (P1 = 5 atm, V1 = 1 m³, T = 273.2 K), we can rearrange the formula to solve for n:
n = (P1V1) / (RT)
However, we need to ensure all units are consistent. Let's convert the pressure from atm to Pascals (Pa) using the conversion factor 1 atm = 101325 Pa:
P1 = 5 atm * 101325 Pa/atm = 506625 Pa
Now, we can substitute the values into the equation for n:
n = (506625 Pa * 1 m³) / (8.314 J/mol·K * 273.2 K)
n ≈ 223.2 moles
Next, we need to find the final volume (V2). Since the process is isothermal, we can use Boyle's Law, which states:
P1V1 = P2V2
Rearranging to solve for V2:
V2 = (P1V1) / P2
Substituting the given values (P1 = 5 atm, V1 = 1 m³, P2 = 1 atm):
V2 = (5 atm * 1 m³) / 1 atm
V2 = 5 m³
Now we have all the values needed to calculate the work done:
W = -nRT ln(V2/V1)
W = -223.2 moles * 8.314 J/mol·K * 273.2 K * ln(5 m³ / 1 m³)
W ≈ -223.2 * 8.314 * 273.2 * ln(5)
W ≈ -223.2 * 8.314 * 273.2 * 1.609
W ≈ -828508.7 J
W ≈ -828.5 kJ
The negative sign indicates that work is done by the system (expansion), which is expected in an expansion process.
Detailed Step-by-Step Calculation
To ensure clarity and understanding, here’s a detailed step-by-step calculation of the work done during the isothermal reversible expansion. This breakdown allows for a clear understanding of each phase of the calculation, enhancing comprehension and accuracy.
Step 1: Calculate the Number of Moles (n)
We begin by determining the number of moles of gas using the ideal gas law, PV = nRT. Rearranging the equation to solve for n, we get n = (P1V1) / (RT). We are given P1 = 5 atm, V1 = 1 m³, and T = 273.2 K. To use the ideal gas constant R in SI units (8.314 J/mol·K), we need to convert the pressure from atmospheres to Pascals. 1 atm is equivalent to 101325 Pa, so:
P1 = 5 atm * 101325 Pa/atm = 506625 Pa
Now, substituting the values into the equation:
n = (506625 Pa * 1 m³) / (8.314 J/mol·K * 273.2 K)
n = 506625 / (8.314 * 273.2)
n ≈ 223.2 moles
Step 2: Calculate the Final Volume (V2)
Since the expansion is isothermal, we can apply Boyle's Law, P1V1 = P2V2. We need to find V2, given P1 = 5 atm, V1 = 1 m³, and P2 = 1 atm. Rearranging the equation to solve for V2:
V2 = (P1V1) / P2
V2 = (5 atm * 1 m³) / 1 atm
V2 = 5 m³
Step 3: Calculate the Work Done (W)
The formula for work done during an isothermal reversible expansion is W = -nRT ln(V2/V1). We have n ≈ 223.2 moles, R = 8.314 J/mol·K, T = 273.2 K, V1 = 1 m³, and V2 = 5 m³. Substituting these values:
W = -223.2 moles * 8.314 J/mol·K * 273.2 K * ln(5 m³ / 1 m³)
W = -223.2 * 8.314 * 273.2 * ln(5)
The natural logarithm of 5 (ln(5)) is approximately 1.609. So:
W ≈ -223.2 * 8.314 * 273.2 * 1.609
W ≈ -828508.7 J
To convert the work done to kilojoules, divide by 1000:
W ≈ -828.5 kJ
Step 4: Final Result and Interpretation
The final result for the work done during the isothermal reversible expansion is approximately -828.5 kJ. The negative sign indicates that the work is done by the system, which is expected in an expansion process. This means the gas expands and pushes against the external pressure, doing work on the surroundings.
This detailed step-by-step calculation provides a clear and thorough understanding of the process, ensuring that each step is logically followed and accurately computed. Understanding each step enhances comprehension and reduces the likelihood of errors in future calculations.
Significance of the Result
The calculated work done during the isothermal reversible expansion, approximately -828.5 kJ, carries significant implications in the context of thermodynamics. This result quantifies the amount of energy the gas expends as it expands from a higher pressure (5 atm) to a lower pressure (1 atm) at a constant temperature (273.2 K). The negative sign of the work done is crucial, as it indicates that the work is done by the system (the gas) rather than on the system. In simpler terms, the gas is performing work on its surroundings as it expands.
Understanding the magnitude of the work done is essential in various practical applications. For instance, in engineering, this calculation is vital in designing engines and turbines, where the expansion of gases is used to generate mechanical work. The efficiency of such devices depends heavily on the amount of work that can be extracted from the expanding gas. A higher magnitude of negative work indicates a more efficient process, meaning more energy is converted into useful work.
Furthermore, the fact that the expansion is isothermal implies that the gas absorbs heat from its surroundings to maintain a constant temperature. This heat input compensates for the energy expended as work, ensuring the temperature remains constant. The concept of reversibility, while an idealized condition, provides a theoretical maximum for the work that can be obtained from the expansion. Real-world processes are often irreversible, meaning they involve inefficiencies and result in less work output. However, the reversible process serves as a benchmark for evaluating the performance of actual systems.
The value of the work done also reflects the thermodynamic properties of the gas, including its heat capacity at constant volume (Cv = 3R). This parameter influences how the gas responds to changes in volume and pressure. The work done is intricately linked to the gas's ability to transfer energy and its state changes during the expansion process. By calculating the work done, we gain a deeper understanding of the gas's behavior and its thermodynamic characteristics.
In summary, the result of -828.5 kJ not only provides a numerical value but also offers insights into the energy dynamics and efficiency of the isothermal reversible expansion. It highlights the interplay between work, heat, and the thermodynamic properties of the gas, making it a cornerstone in the analysis and design of various thermodynamic systems.
Conclusion
In conclusion, we have successfully calculated the work done during an isothermal reversible expansion of one cubic meter of gas from 5 atm to 1 atm at a constant temperature of 273.2 K. The step-by-step calculation, which yielded a result of approximately -828.5 kJ, underscores the importance of understanding and applying thermodynamic principles. This negative value indicates that the work is done by the system, a characteristic feature of expansion processes.
Throughout the calculation, we emphasized the critical role of various parameters and formulas. The ideal gas law helped us determine the number of moles of gas, while Boyle's Law allowed us to find the final volume after expansion. The work equation for isothermal reversible processes provided the framework for quantifying the energy exchange during the expansion. Each step was carefully outlined to ensure clarity and accuracy, demonstrating the methodical approach required in thermodynamic calculations.
Moreover, we highlighted the significance of the result in practical applications, particularly in engineering and the design of thermodynamic systems. The efficiency of engines and turbines, for instance, depends heavily on the work that can be extracted from expanding gases. Understanding these processes is crucial for optimizing energy conversion and minimizing losses.
This analysis also reinforces the theoretical underpinnings of thermodynamics, such as the concepts of isothermal and reversible processes. While real-world processes may deviate from these idealized conditions, they provide a valuable benchmark for evaluating system performance and identifying potential improvements. The combination of theoretical knowledge and practical application is essential for advancing our understanding of thermodynamics and its role in various scientific and engineering domains.
The comprehensive approach taken in this article, from defining the problem to interpreting the results, serves as a valuable resource for anyone seeking to deepen their understanding of thermodynamic principles and calculations. By meticulously exploring each aspect of the problem, we have demonstrated the importance of precision, clarity, and a thorough understanding of the underlying concepts in thermodynamics.
