Adiabatic Compression of a Monatomic Gas: Calculating Pressure Ratios

Understanding the relationship between pressure, volume, and temperature in an adiabatic process is crucial for analyzing the behavior of gases under various conditions. In this scenario, we explore the effects of a sudden adiabatic compression on a monatomic gas, focusing on the ratio of the final pressure to the initial pressure. This process is governed by the equation PV^γ constant, where γ is the adiabatic index, typically equivalent to 5/3 for a monatomic gas.

The Initial Setup and the Adiabatic Equation

Consider a monatomic gas that is suddenly compressed to 1/8th of its initial volume adiabatically. The equation governing this process is:

P1V1γ P2V2γ

Given that γ 5/3 for a monatomic gas, we can rewrite the equation as:

Initial Conditions and Volume Change

Let the initial volume be V1 and the final volume be V2. Since the volume is compressed to 1/8th of the initial volume:

V2 V1 / 8

Deriving the Pressure Ratio

To find the ratio of the final pressure (P2) to the initial pressure (P1), we substitute the given values into the adiabatic equation:

Using PVγ constant, we have:

P2 / P1 (V1 / V2)γ

Substituting V2 V1 / 8:

P2 / P1 (V1 / (V1 / 8))5/3

Further simplifying:

P2 / P1 (8)5/3

Thus, the final pressure ratio is:

Simplifying the Expression

The expression (8)5/3 can be further simplified to find the exact value:

(8)5/3 (23)5/3 25 32

Therefore, the ratio of the final pressure to the initial pressure is 32.

Additional Notes on Adiabatic Processes

In adiabatic processes, the temperature of the gas changes due to the change in volume, but no heat is exchanged with the surroundings. This process is often used in high-pressure and high-performance engineering applications like automobile engines and gas turbines.

Conclusion

The adiabatic compression of a monatomic gas to 1/8th of its initial volume results in a significant increase in pressure according to the adiabatic equation. By understanding the relationship between pressure and volume, we can accurately predict the behavior of gases under such conditions, which is essential for the design and operation of many devices and systems.

References

[1] Ideal Gas Law on Wikipedia

[2] Thermal Physics Classroom - Adiabatic Processes

[3] NASA GRC - Ideal Gas Law Explanation