91Ó°ÊÓ

The Ideal Gas Law is a fundamental equation in thermodynamics that describes the behavior of an ideal gas. It is usually written as \( PV = nRT \) where:

• \( P \) is the pressure of the gas
• \( V \) is the volume it occupies
• \( n \) is the number of moles of the gas
• \( R \) is the universal gas constant
• \( T \) is the temperature in Kelvin

In contexts involving specific volume \( (v) \) and mass \( (m) \), the Ideal Gas Law can be rearranged to: \( P = \frac{R T}{v} \) where:

• \( P \) is the pressure of the gas
• \( R \) is the specific gas constant for the gas
• \( T \) is the temperature in Kelvin
• \( v \) is the specific volume, which is volume per unit mass

For air, you can use this equation to calculate pressure at different states, assuming the gas behaves ideally.

Specific volume \( v \) is an important thermodynamic property that represents the volume occupied by a unit mass of a substance, in this case, air. It is given by the ratio: \( v = \frac{V}{m} \) where:

• \( V \) is the total volume
• \( m \) is the mass

In the provided problem, the specific volumes are given at the initial and final states:

• Initial specific volume, \( v_1 = 0.8 \text{ m}^3/\text{kg} \)
• Final specific volume, \( v_2 = 0.2 \text{ m}^3/\text{kg} \)

Specific volume helps us understand how 'spread out' the air molecules are at different states, and it's crucial for calculations involving the Ideal Gas Law and determining if a process is adiabatic.

Calculating pressure accurately in thermodynamic processes is crucial for further analysis. Using the Ideal Gas Law in its form \( P = \frac{RT}{v} \), we can determine the pressure at different states by substituting specific volume and temperature values given in the problem: For the initial state: \( P_1 = \frac{RT_1}{v_1} = \frac{R \times 300}{0.8} \) For the final state: \( P_2 = \frac{RT_2}{v_2} = \frac{R \times 420}{0.2} \) We have:

• \( T_1 = 300 \text{ K} \)
• \( v_1 = 0.8 \text{ m}^3/\text{kg} \)

**Result**: \( P_1 = \frac{300R}{0.8} \) Similarly, for the final state:

• \( T_2 = 420 \text{ K} \)
• \( v_2 = 0.2 \text{ m}^3/\text{kg} \)

**Result**: \( P_2 = \frac{420R}{0.2} \) These calculations are used to check if the process abides by the adiabatic condition.

The adiabatic index (\( \gamma \) ) is a specific ratio for a gas, defined as the ratio of the specific heats: \( \gamma = \frac{C_p}{C_v} \) where:

• \( C_p \) is the specific heat at constant pressure
• \( C_v \) is the specific heat at constant volume

For air, the adiabatic index is usually taken as \( 1.4 \). In an adiabatic process, where no heat is transferred into or out of the system, the relationship between pressure and volume is given by: \( P_1 v_1^{\gamma} = P_2 v_2^{\gamma} \) To verify if the process is adiabatic, you substitute the initial and final pressures and specific volumes: \( \frac{300R}{0.8} \cdot (0.8)^{1.4} \stackrel{?}{=} \frac{420R}{0.2} \cdot (0.2)^{1.4} \) Solving each side separately determines if they are equal, confirming whether the process can be considered adiabatic. If they are equal, the process is adiabatic, and work done can be calculated using: \( W = C_v (T_1 - T_2) \). If not, the process involves heat transfer.