Understanding Ideal Gas Laws in Thermodynamic Processes

In the study of thermodynamics, the behavior of gases is often analyzed using the ideal gas law. One key application is determining the final pressure of a gas when its temperature and volume changes. This article will delve into how to calculate the final pressure of a gas when cooled under constant volume conditions, using the combined gas law and Gay-Lussac's law.

Principles of Ideal Gas Law and Combined Gas Law

The ideal gas law states that PV nRT, where P is the pressure, V is the volume, n is the number of moles, R is the gas constant, and T is the temperature in Kelvin. When dealing with thermodynamic processes where the volume is kept constant (V constant), the law simplifies to P/T constant. This principle is the basis for solving problems involving changes in temperature and pressure under constant volume conditions.

Solving for Final Pressure

Let's consider a specific scenario: if a gas is cooled from 300K to 250K and the volume is kept constant, what would the final pressure result if the original pressure was 720.0 mm Hg?

Given:

Initial pressure (P?): 720.0 mm Hg Initial temperature (T?): 300K Final temperature (T?): 250K Volume is constant

Using the relation P?/T? P?/T? (the combined gas law for constant volume), we can solve for the final pressure (P?):

P? P? × T? / T?

Substituting the given values:

P? 720.0 mm Hg × (250K / 300K)

P? 720.0 mm Hg × 0.8333

P? 600 mm Hg

Explaining the Calculation Step-by-Step

Start with the given initial and final temperatures and the initial pressure: Use the equation P?/T? P?/T?. Rearrange the equation to solve for P?: P? P? × T? / T?. Substitute the given values into the equation. Perform the calculation, ensuring the correct units are used (mm Hg for pressure and Kelvin for temperature). Finish by simplifying the result, rounding to the appropriate number of significant figures (4 in this case).

Alternative Method - Using Gay-Lussac's Law

Gay-Lussac's Law states that for a constant volume, the pressure of a gas is directly proportional to its temperature in Kelvin. This is expressed as P?/T? P?/T?. Using this law directly for our scenario:

P? P? × T? / T?

Substituting the given values:

P? 720.0 mm Hg × (250K / 300K)

P? 720.0 mm Hg × 0.8333

P? 600 mm Hg

Conclusion

Understanding and applying the ideal gas law, the combined gas law, and Gay-Lussac's Law is crucial for solving thermodynamic problems involving gases. In this case, the final pressure of the gas, when cooled from 300K to 250K under constant volume, is 600 mm Hg.

Key Takeaways

The ideal gas law is PV nRT. The combined gas law for constant volume is P?/T? P?/T?. Gay-Lussac's Law is P?/T? P?/T? under constant volume conditions.

By mastering these principles and applying them correctly, one can effectively solve a wide range of thermodynamic problems involving gases.