Calculating the Final Pressure

Oct 13, 2025 by miodeya.com 126 views

Introduction

In thermodynamics, an ideal gas is a hypothetical gas that obeys the ideal gas law, which is a fundamental principle in understanding the behavior of gases. An isothermal expansion is a process where a gas expands while maintaining a constant temperature. In this article, we will explore the thermodynamic process of an ideal gas undergoing an isothermal expansion from an initial volume to a final volume at a constant temperature.

The Ideal Gas Law

The ideal gas law is a fundamental principle in thermodynamics that describes the behavior of an ideal gas. It is given by the equation:

PV = nRT

where:

P is the pressure of the gas
V is the volume of the gas
n is the number of moles of the gas
R is the gas constant
T is the temperature of the gas

Isothermal Expansion

An isothermal expansion is a process where a gas expands while maintaining a constant temperature. In this process, the gas is allowed to expand from an initial volume to a final volume at a constant temperature. The ideal gas law can be used to describe the behavior of the gas during this process.

Given Data

The given data for this problem are:

Initial volume: V1 = 3 m^3
Initial pressure: p1 = 5 atm
Final volume: V2 = 12 m^3
Temperature: T = 320 K

Calculating the Final Pressure

Using the ideal gas law, we can calculate the final pressure of the gas after the isothermal expansion. Since the temperature remains constant, we can use the equation:

P1V1 = P2V2

where:

P1 is the initial pressure
V1 is the initial volume
P2 is the final pressure
V2 is the final volume

Rearranging the equation to solve for P2, we get:

P2 = P1V1 / V2

Substituting the given values, we get:

P2 = (5 atm)(3 m^3) / (12 m^3) P2 = 1.25 atm

Calculating the Number of Moles

Using the ideal gas law, we can calculate the number of moles of the gas. Rearranging the equation to solve for n, we get:

n = PV / RT

Substituting the given values, we get:

n = (5 atm)(3 m^3) / (0.08206 L atm/mol K)(320 K) n = 0.117 mol

Calculating the Final Volume

Using the ideal gas law, we can calculate the final volume of the gas after the isothermal expansion. Rearranging the equation to solve for V, we get:

V = nRT / P

Substituting the given values, we get:

V = (0.117 mol)(0.08206 L atm/mol K)(320 K) / (1.25 atm) V = 3.04 m^3

Conclusion

In this article, we have explored the thermodynamic process of an ideal gas undergoing an isothermal expansion from an initial volume to a final volume at a constant temperature. We have used the ideal gas law to calculate the final pressure, number of moles, and final volume of the gas. The results show that the final pressure is 1.25 atm, the number of moles is 0.117 mol, and the final volume is 3.04 m^3.

References

Cengel, Y. A., & Boles, M. A. (2015). Thermodynamics: An Engineering Approach. McGraw-Hill Education.
Halliday, D., Resnick, R., & Walker, J. (2014). Fundamentals of Physics. John Wiley & Sons.

FAQs

Q: What is an isothermal expansion? A: An isothermal expansion is a process where a gas expands while maintaining a constant temperature.
Q: What is the ideal gas law? A: The ideal gas law is a fundamental principle in thermodynamics that describes the behavior of an ideal gas.
Q: How can we calculate the final pressure of a gas after an isothermal expansion? A: We can use the ideal gas law to calculate the final pressure of the gas after an isothermal expansion.
Frequently Asked Questions (FAQs) About Isothermal Expansion and the Ideal Gas Law =====================================================================================

Q: What is an isothermal expansion?

A: An isothermal expansion is a process where a gas expands while maintaining a constant temperature. This means that the temperature of the gas remains the same throughout the expansion process.

Q: What is the ideal gas law?

A: The ideal gas law is a fundamental principle in thermodynamics that describes the behavior of an ideal gas. It is given by the equation:

PV = nRT

where:

P is the pressure of the gas
V is the volume of the gas
n is the number of moles of the gas
R is the gas constant
T is the temperature of the gas

Q: How can we calculate the final pressure of a gas after an isothermal expansion?

A: We can use the ideal gas law to calculate the final pressure of the gas after an isothermal expansion. Since the temperature remains constant, we can use the equation:

P1V1 = P2V2

where:

P1 is the initial pressure
V1 is the initial volume
P2 is the final pressure
V2 is the final volume

Rearranging the equation to solve for P2, we get:

P2 = P1V1 / V2

Q: How can we calculate the number of moles of a gas?

A: We can use the ideal gas law to calculate the number of moles of a gas. Rearranging the equation to solve for n, we get:

n = PV / RT

Q: What is the relationship between the initial and final volumes of a gas during an isothermal expansion?

A: During an isothermal expansion, the initial and final volumes of a gas are related by the equation:

V1 / V2 = P2 / P1

where:

V1 is the initial volume
V2 is the final volume
P1 is the initial pressure
P2 is the final pressure

Q: Can we calculate the final volume of a gas after an isothermal expansion?

A: Yes, we can calculate the final volume of a gas after an isothermal expansion using the ideal gas law. Rearranging the equation to solve for V, we get:

V = nRT / P

Q: What is the significance of the ideal gas law in thermodynamics?

A: The ideal gas law is a fundamental principle in thermodynamics that describes the behavior of an ideal gas. It is used to calculate the pressure, volume, and number of moles of a gas, and is essential in understanding the behavior of gases in various thermodynamic processes.

Q: Can we apply the ideal gas law to real gases?

A: While the ideal gas law is a fundamental principle in thermodynamics, it is not applicable to real gases. Real gases do not behave ideally and deviate from the ideal gas law due to intermolecular forces and other factors.

Q: What are some common applications of the ideal gas law?

A: The ideal gas law has numerous applications in various fields, including:

Chemical engineering: The ideal gas law is used to design and optimize chemical processes, such as distillation and reaction systems.
Biological engineering: The ideal gas law is used to model the behavior of gases in biological systems, such as respiration and gas exchange.
Materials science: The ideal gas law is used to study the behavior of gases in materials, such as porous materials and nanomaterials.

Q: Can we use the ideal gas law to calculate the work done by a gas during an isothermal expansion?

A: Yes, we can use the ideal gas law to calculate the work done by a gas during an isothermal expansion. The work done by a gas is given by the equation:

W = nRT ln(V2 / V1)

where:

W is the work done by the gas
n is the number of moles of the gas
R is the gas constant
T is the temperature of the gas
V1 is the initial volume
V2 is the final volume

Conclusion

In this article, we have answered some frequently asked questions about isothermal expansion and the ideal gas law. We have discussed the significance of the ideal gas law in thermodynamics, its applications, and its limitations. We have also provided equations and examples to illustrate the use of the ideal gas law in calculating the final pressure, number of moles, and work done by a gas during an isothermal expansion.