91Ó°ÊÓ

The ideal gas law is a fundamental equation that describes the behavior of an ideal gas. It relates the pressure, volume, and temperature of the gas. The ideal gas law is expressed as \( PV = nRT \), where:

• \( P \) is the pressure of the gas in Pascals (Pa).
• \( V \) is the volume in cubic meters (m³).
• \( n \) is the number of moles of gas.
• \( R \) is the universal gas constant \( 8.314 \) J/mol·K.
• \( T \) is the temperature in Kelvin (K).

In this exercise, we're dealing with an adiabatic compression, which means the ideal gas law is not applied directly to relate final and initial states, but it's important to understand that gas behavior in any state would conform to this law.

An adiabatic process is one where no heat is exchanged between the system and its surroundings. This means the entire energy transfer within the system is work. In the context of gases, an adiabatic process signifies a change in state without gain or loss of thermal energy.
During adiabatic compression, the product of pressure times volume raised to the power of \( \gamma \) is constant, i.e., \( P_1 V_1^\gamma = P_2 V_2^\gamma \). Here, \( \gamma \) (the heat capacity ratio), is the crucial factor influencing how pressure and volume interact.
This principle is the backbone of the solution in calculating final pressure and understanding the nature of work done and temperature change.

Work done in an adiabatic process is an important concept in thermodynamics. Work done \( W \) by the gas, or on it, can be calculated using the equation:
\[ W = \frac{P_1 V_1 - P_2 V_2}{\gamma - 1} \]Here, it's crucial to note that because there is no heat exchange, internal energy must also change.
If the work done is negative, it tells us that work is done on the gas rather than by the gas. This simply implies that during the compression, the energy goes into changing the volume rather than escaping as heat.

In an adiabatic process, the temperature of the gas changes due to the work done. This change is given by the relation:
\[ \frac{T_2}{T_1} = \left( \frac{V_1}{V_2} \right)^{\gamma - 1} \]This shows that the temperature is linked to the change in volume raised to the power of \( \gamma - 1 \).

• If the gas is compressed (volume decreases), temperature increases.
• If the gas expands, temperature decreases.

In this exercise, since the gas is compressed, the final temperature is higher than the initial temperature, which is expected.

Pressure change in an adiabatic process can be calculated using:
\[ P_2 = P_1 \left( \frac{V_1}{V_2} \right)^\gamma \]This formula highlights that pressure change is directly related to how the volume changes, influenced by the value of \( \gamma \).
The increase in pressure with reduced volume during adiabatic compression results from the gas molecules being forced together, which naturally elevates pressure.
Understanding this concept is vital for solving various problems in gas physics, as it ties together the ideas of pressure, temperature, and volume in a closed system.