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Step-by-Step Solution
Step 1: Identify the Known Quantities
• Adiabatic index (or ratio of specific heats), \gamma = 1.5
• Initial pressure, p_{1} = 200 \text{ kPa}
• Initial volume, V_{1} = 1200 \text{ cm}^{3}
• Final volume, V_{2} = 300 \text{ cm}^{3}
Step 2: Apply the Adiabatic Condition
In an adiabatic process for an ideal gas, the relation
p V^{\gamma} = \text{constant} holds. Thus,
p_{1} \, V_{1}^{\gamma} = p_{2} \, V_{2}^{\gamma} \,.
We can solve for p_{2} :
p_{2} = p_{1} \times \frac{V_{1}^{\gamma}}{V_{2}^{\gamma}} \,.
Substituting the known values:
p_{2} = 200 \,\text{kPa} \times \left(\frac{1200}{300}\right)^{1.5} \,.
Since \frac{1200}{300} = 4 and 4^{1.5} = 4^{\frac{3}{2}} = 2^3 = 8 \,,
p_{2} = 200 \times 8 = 1600 \,\text{kPa}.
Step 3: Use the Formula for Work Done in an Adiabatic Process
The work done by the gas during an adiabatic change can be found using
W = \frac{p_{2}V_{2} - p_{1}V_{1}}{\gamma - 1} \,.
Note that we are using p in kPa and V in cm ^{3} . The product p \times V in these units converts to Joules by a factor of 10^{-3} (because 1 \,\text{kPa} \cdot 1 \,\text{cm}^{3} = 10^{-3} \,\text{J} ). We will account for this factor at the end.
Step 4: Substitute Numerical Values and Compute
Substitute p_{1} = 200 \,\text{kPa}, V_{1} = 1200 \,\text{cm}^{3}, p_{2} = 1600 \,\text{kPa}, V_{2} = 300 \,\text{cm}^{3}, \gamma = 1.5 :
W = \frac{(1600 \times 300) - (200 \times 1200)}{1.5 - 1} \,.
Calculate the numerator first:
(1600 \times 300) = 480000\,, \quad (200 \times 1200) = 240000 \,,
\text{Difference} = 480000 - 240000 = 240000 \,.
Then,
\gamma - 1 = 1.5 - 1 = 0.5 \,,
so
W = \frac{240000}{0.5} = 480000 \,\text{kPa}\cdot\text{cm}^{3} \,.
Since 1 \,\text{kPa}\cdot\text{cm}^{3} = 10^{-3} \,\text{J} , multiply by 10^{-3} to convert to Joules:
W = 480000 \times 10^{-3} \,\text{J} = 480 \,\text{J}\,.
Step 5: State the Final Answer
The absolute value of the work done by the gas during the adiabatic compression is
\boxed{480 \,\text{J}}.
