Understanding Adiabatic Compression and its Impact on Temperature

In this article, we will delve into the process of adiabatic compression of an ideal gas, specifically how to calculate the final temperature when a gas is compressed adiabatically.

Introduction

Imagine a scenario where you have a gas at an initial temperature of 10deg;C and a pressure of 1.013times;105 pascals. If this gas is compressed adiabatically to half of its original volume, how can we determine the final temperature? This article will guide you through the process.

Theoretical Background

For an ideal gas undergoing adiabatic compression, the relationship between the initial and final temperatures can be described by the equation:

(frac{T_2}{T_1} left( frac{V_1}{V_2} right)^{gamma - 1})

T1 is the initial temperature in Kelvin. T2 is the final temperature in Kelvin. V1 is the initial volume. V2 is the final volume. (gamma) is the ratio of specific heats, given as 1.4.

Step-by-Step Calculation

Step 1: Convert the Initial Temperature to Kelvin

The initial temperature T1 is given as 10deg;C. To convert this to Kelvin:

T1 10 273.15 283.15 K

Step 2: Determine the Volume Ratio

Since the gas is compressed to half of its volume:

(frac{V_1}{V_2} 2)

Step 3: Apply the Adiabatic Relation

Now we can substitute the values into the adiabatic relation:

(frac{T_2}{T_1} left( frac{V_1}{V_2} right)^{gamma - 1} 2^{1.4 - 1} 2^{0.4})

Step 4: Calculate (2^{0.4})

Using a calculator or approximating:

(2^{0.4} approx 1.3195)

Step 5: Calculate (T_2)

Now we can find (T_2):

T2 T1 times; 20.4 283.15 times; 1.3195 approx 374.23 K

Step 6: Convert Back to Celsius

Finally, convert (T_2) back to Celsius:

T2deg;C T2K - 273.15 approx 374.23 - 273.15 approx 101.08deg;C

Conclusion

The final temperature after the adiabatic compression is approximately 101.08deg;C. This process and the outcome demonstrate the importance of the adiabatic process in thermodynamics and its practical applications.

Application in Real-life Scenarios

Understanding adiabatic compression is crucial in various fields such as engineering, meteorology, and even automotive design. For example, in engines, the compression stroke is an adiabatic process, and knowing the final temperature can help optimize engine performance.

Related Concepts and Keywords

Adiabatic compression Gas temperature calculation Ratio of specific heats

Keywords: adiabatic compression, gas temperature calculation, ratio of specific heats