Calculating Maximum Work Done In Isothermal Expansion Of Oxygen

JU07/16/2025 08, 2025 by THE IDEN 64 views

Calculating Maximum Work Done in Isothermal Reversible Expansion of Oxygen

In the realm of thermodynamics, understanding the maximum work done by a system during a process is crucial. This article delves into the calculation of the maximum work done when 32 grams of oxygen gas expands isothermally and reversibly. Isothermal processes, occurring at constant temperature, and reversible processes, proceeding infinitesimally slowly, are fundamental concepts in thermodynamics. This scenario presents a practical application of these principles, allowing us to explore the relationship between pressure, volume, and work done in a gaseous system.

Before we dive into the calculations, it's essential to grasp the key concepts involved:

Isothermal Process: A thermodynamic process where the temperature remains constant. In our case, the expansion of oxygen occurs at a constant temperature of 298 K.
Reversible Process: A process carried out infinitesimally slowly, allowing the system to remain in equilibrium with its surroundings at every step. This ensures that the maximum possible work is done.
Work Done by a Gas: When a gas expands against an external pressure, it performs work. The amount of work done depends on the pressure difference and the change in volume.
Ideal Gas Law: The ideal gas law, PV = nRT, relates the pressure (P), volume (V), number of moles (n), gas constant (R), and temperature (T) of an ideal gas. This law is crucial for calculating volume changes during the expansion.

We are given the following information:

Mass of oxygen (O₂) = 32 g
Number of moles (n) = 1 mol
Temperature (T) = 298 K
Gas constant (R) = 8.314 J K⁻¹ mol⁻¹
Initial pressure (P₁) = 2 bar
Final pressure (P₂) = 1 bar

Our objective is to calculate the maximum work done when this amount of oxygen expands isothermally and reversibly from the initial pressure of 2 bar to the final pressure of 1 bar at a constant temperature of 298 K. This scenario provides a practical application of thermodynamic principles, allowing us to quantify the energy transfer associated with the expansion process. The maximum work done is a key concept in thermodynamics, representing the theoretical limit of energy that can be extracted from a system under specific conditions. Understanding how to calculate this value is crucial for various applications, including designing efficient engines and chemical processes.

For an isothermal reversible process, the maximum work done (_(W_max)) is given by the formula:

${ W_{max} = -nRT \ln\left(\frac{P_2}{P_1}\right) }$

Where:

n = number of moles of gas
R = gas constant (8.314 J K⁻¹ mol⁻¹)
T = temperature in Kelvin
P₁ = initial pressure
P₂ = final pressure

This formula is derived from the fundamental principles of thermodynamics, specifically the relationship between work, pressure, volume, and temperature in a reversible process. The negative sign indicates that the work is done by the system (expansion), which is a convention in thermodynamics. The natural logarithm (_(ln_)) reflects the continuous change in pressure and volume during the reversible process. The maximum work done represents the theoretical limit of energy that can be extracted from the system under these conditions, highlighting the importance of understanding and applying this formula in various thermodynamic calculations.

Let's plug in the given values into the formula:

Identify the values:
- n = 1 mol
- R = 8.314 J K⁻¹ mol⁻¹
- T = 298 K
- P₁ = 2 bar
- P₂ = 1 bar
Substitute the values into the formula: ${ W_{max} = -(1 \text{ mol}) \times (8.314 \text{ J K}^{-1} \text{ mol}^{-1}) \times (298 \text{ K}) \times \ln\left(\frac{1 \text{ bar}}{2 \text{ bar}}\right) }$
Calculate the natural logarithm: ${ \ln\left(\frac{1}{2}\right) = \ln(0.5) \approx -0.693 }$
Multiply the values: ${ W_{max} = -1 \times 8.314 \times 298 \times (-0.693) }$
Calculate the maximum work done: ${ W_{max} \approx 1717.3 ext{ J} }$

Therefore, the maximum work done when 32 g of oxygen expands isothermally and reversibly from 2 bar to 1 bar at 298 K is approximately 1717.3 Joules. This step-by-step calculation demonstrates the practical application of the formula, highlighting the importance of accurate substitution and careful computation. The final result, 1717.3 Joules, represents the theoretical maximum energy that can be extracted from this expansion process, a crucial value for various thermodynamic analyses and applications. Understanding each step in the calculation not only provides the numerical answer but also reinforces the underlying thermodynamic principles and their practical implications. This detailed approach ensures a comprehensive understanding of the problem and its solution.

The maximum work done (_(W_max_)) is approximately 1717.3 J. This result signifies the amount of energy the system can ideally transfer to its surroundings during the isothermal reversible expansion. In practical applications, the actual work done might be less due to factors like friction and irreversibility. However, this calculated value serves as a benchmark for the maximum possible efficiency of the process. The maximum work done is a critical parameter in various engineering and scientific contexts, including the design of engines, turbines, and other energy conversion devices. It also plays a crucial role in understanding the thermodynamics of chemical reactions and phase transitions. The value of 1717.3 J provides a quantitative measure of the energy involved in this specific expansion process, contributing to a deeper understanding of thermodynamic principles and their applications.

Isothermal reversible expansion is a fundamental concept in thermodynamics, serving as an ideal scenario for understanding the behavior of gases. It's significant for several reasons:

Maximum Work: Reversible processes yield the maximum possible work, providing a theoretical benchmark for efficiency.
Thermodynamic Understanding: Studying this process helps elucidate the relationships between pressure, volume, temperature, and work.
Applications: The principles learned are applicable in various fields, including engine design, chemical processes, and energy production.

In the context of real-world applications, understanding isothermal reversible expansion allows engineers and scientists to optimize processes for maximum energy efficiency. For example, in the design of heat engines, striving for conditions that approximate isothermal reversible expansion can lead to improved performance. Similarly, in chemical reactions involving gases, controlling the conditions to approach isothermal reversibility can maximize the yield of desired products. The concept also has implications for understanding atmospheric processes, such as the expansion and cooling of air masses. By studying this idealized process, we gain valuable insights into the fundamental principles governing the behavior of gases and their interactions with the environment. This knowledge is essential for addressing various challenges related to energy, materials, and environmental sustainability.

In conclusion, we have successfully calculated the maximum work done during the isothermal reversible expansion of oxygen using the appropriate thermodynamic formula. The result, approximately 1717.3 J, provides a quantitative measure of the energy involved in this process. Understanding such calculations is crucial for various applications in chemistry, physics, and engineering, where thermodynamic principles play a vital role. The concept of maximum work done in a reversible process serves as a theoretical upper limit, guiding the design and optimization of real-world systems. By grasping these fundamental concepts, we can better analyze and predict the behavior of gases and other thermodynamic systems, leading to advancements in various technological and scientific fields. The ability to calculate and interpret the maximum work done is a cornerstone of thermodynamic analysis, enabling us to make informed decisions in a wide range of applications, from designing efficient engines to developing sustainable energy solutions. This exercise underscores the importance of thermodynamic principles in both theoretical understanding and practical problem-solving.