Thermodynamics: Adiabatic Expansion Work Calculation

This problem involves calculating the work done during an adiabatic expansion of a gas. We are given the initial state ($P_1, V_1, T_1$), the final pressure ($P_2$), and relationships for specific heats ($c_p, c_v$). The key is to use the properties of adiabatic processes.

Identify Key Parameters

• Initial Pressure: $\( P_1 = 8 \text{ bar} = 8 \times 10^5 \text{ Pa} \)
• Initial Volume: $\( V_1 = 0.15 \text{ m}^3 \)
• Final Pressure: $\( P_2 = 1 \text{ bar} = 1 \times 10^5 \text{ Pa} \)
• Specific Heat Relationship: \( c_p = 3R \), \( c_v = 2R \)
• Process Type: Adiabatic Expansion

Calculate Adiabatic Index ($\gamma$)

The adiabatic index, $\gamma$, is the ratio of specific heats:

$ \gamma = \frac{c_p}{c_v} = \frac{3R}{2R} = 1.5 $

Determine Final Volume ($V_2$)

For an adiabatic process, the relationship $P V^\gamma = \text{constant}$ holds. Therefore:

$ P_1 V_1^\gamma = P_2 V_2^\gamma $

Rearranging to solve for $V_2$:

$ V_2 = V_1 \left( \frac{P_1}{P_2} \right)^{1/\gamma} $

Substituting the values:

$ V_2 = 0.15 \text{ m}^3 \times \left( \frac{8 \text{ bar}}{1 \text{ bar}} \right)^{1/1.5} = 0.15 \times (8)^{2/3} = 0.15 \times 4 = 0.60 \text{ m}^3 $

Calculate Work Done ($W$)

The work done by a gas during an adiabatic process is given by:

$ W = \frac{P_1 V_1 - P_2 V_2}{\gamma - 1} $

Plugging in the known values:

$ W = \frac{(8 \times 10^5 \text{ Pa})(0.15 \text{ m}^3) - (1 \times 10^5 \text{ Pa})(0.60 \text{ m}^3)}{1.5 - 1} $

$ W = \frac{1.2 \times 10^6 \text{ J} - 0.6 \times 10^6 \text{ J}}{0.5} $

$ W = \frac{0.6 \times 10^6 \text{ J}}{0.5} = 1.2 \times 10^6 \text{ J} $

Converting Joules to kilojoules (kJ):

$ W = 1200 \text{ kJ} $

The work done during this adiabatic expansion is 1200 kJ.