Discussion
An ideal gas is compressed adiabatically (Adiabatic exponent γ =1.4) from 98 kPa to 480 kPa and the specific volume of the gas at the beginning of the compression stroke is 0.45 m3 kg-1. The specific work done on the gas in kJ kg-1 is _________.
Answer: D
The specific work done on the gas is 63.3 kJ/kg.
For a reversible adiabatic process \(pv^{\gamma}\) is constant, so the final specific volume is
\[ \begin{aligned} v_{2} &= v_{1}(p_{1}/p_{2})^{1/\gamma} \\ &= 0.45 \times (98/480)^{0.714} \\ &= 0.1446\ \text{m}^{3}/\text{kg} \end{aligned} \]
The work done on the gas in a closed (non-flow) adiabatic compression is
\[ \begin{aligned} w &= \frac{p_{2}v_{2} - p_{1}v_{1}}{\gamma - 1} \\ &= \frac{480(0.1446) - 98(0.45)}{0.4} \\ &= \frac{69.4 - 44.1}{0.4} = 63.3\ \text{kJ/kg} \end{aligned} \]
25.4 is close to the numerator \(p_{2}v_{2} - p_{1}v_{1}\) (25.3) left undivided by \(\gamma - 1\), and 12.6 and 18.5 are smaller still; none of them accounts for the full adiabatic work.
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