Determining the Final Volume of a Gas Under Pressure and Temperature Changes

Understanding the behavior of gases under varying conditions, such as changes in pressure and temperature, is fundamental to both physical chemistry and engineering applications. This article will delve into a specific scenario involving the Ideal Gas Law and the Combined Gas Law, which are essential tools for calculating the volume changes in a fixed mass of gas.

Introduction to the Ideal Gas Law and Combined Gas Law

The Ideal Gas Law, represented by the equation (PV nRT), provides a relationship between the pressure, volume, temperature, and number of moles of a gas. Here, (P) denotes pressure, (V) is volume, (n) is the number of moles, (R) is the ideal gas constant, and (T) is the absolute temperature.

The Combined Gas Law combines Boyle's Law, Charles's Law, and Gay-Lussac's Law into a single equation. The Combined Gas Law is expressed as (frac{P_{1}V_{1}}{T_{1}} frac{P_{2}V_{2}}{T_{2}}), where (P_{1}), (V_{1}), and (T_{1}) represent the initial pressure, volume, and temperature, respectively, and (P_{2}), (V_{2}), and (T_{2}) represent the final pressure, volume, and temperature.

Problem Statement and Solution

Consider the problem of determining the final volume of a fixed mass of a gas when its initial pressure is halved and its absolute temperature is doubled. Let's use the Ideal Gas Law and the Combined Gas Law to solve this problem.

Applying the Combined Gas Law:

Starting from the given Combined Gas Law equation:

[ frac{P_{1}V_{1}}{T_{1}} frac{P_{2}V_{2}}{T_{2}} ]

We can solve for the final volume (V_{2}):

[ V_{2} frac{P_{1}V_{1}T_{2}}{P_{2}T_{1}} ]

Given the problem conditions:

- (T_{2} 2T_{1}) (the final temperature is double the initial temperature)- (P_{2} frac{P_{1}}{2}) (the final pressure is half the initial pressure)

Substituting these values into the equation:

[ V_{2} frac{P_{1}V_{1}(2T_{1})}{left(frac{P_{1}}{2}right)T_{1}} ]

Simplifying the right-hand side:

[ V_{2} frac{P_{1}V_{1}(2T_{1})}{frac{P_{1}T_{1}}{2}} frac{2P_{1}V_{1}T_{1}}{frac{P_{1}T_{1}}{2}} ]

Further simplifying:

[ V_{2} frac{2P_{1}V_{1}T_{1} cdot 2}{P_{1}T_{1}} 4V_{1} ]

Thus, the final volume (V_{2}) is four times the initial volume (V_{1}).

Using the Ideal Gas Law for Verification

To further validate this solution, we can use the Ideal Gas Law. Starting from the initial state:

[ P_{1}V_{1} nRT_{1} ]

And for the final state:

[ P_{2}V_{2} nRT_{2} ]

Substituting (P_{2} frac{P_{1}}{2}) and (T_{2} 2T_{1}):

[ frac{P_{1}}{2}V_{2} nR(2T_{1}) ]

Rearranging for (V_{2}):

[ V_{2} frac{4nRT_{1}}{P_{1}} ]

Using the initial state equation (P_{1}V_{1} nRT_{1}), we get:

[ V_{1} frac{nRT_{1}}{P_{1}} ]

Substituting this into the equation for (V_{2}) gives:

[ V_{2} 4V_{1} ]

This confirms that the final volume is indeed four times the initial volume.

Conclusion

The combined use of the Ideal Gas Law and the Combined Gas Law provides a systematic approach to solving problems related to the behavior of gases under changing conditions. In this specific case, we have shown that halving the initial pressure and doubling the absolute temperature results in a quadrupling of the initial volume.

Key Takeaways

The volume of a gas is directly proportional to its absolute temperature (Charles's Law). The volume of a gas is inversely proportional to its pressure (Boyle's Law). The Combined Gas Law can be used to determine the final state of a gas based on its initial state and the changes in conditions. The Ideal Gas Law provides a more complete picture by incorporating the number of moles and the ideal gas constant.

References

[1] Wikipedia, Ideal Gas Law

[2] Wikipedia, Combined Gas Law