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Derive the work of a reversible isothermal compression of a van der Waals gas. How does it compare to the work needed to compress the ideal gas in the limit of (a) low pressure and (b) high pressure? Calculate how much work is needed to perform the compression in the following scenario -- assuming the isothermal process for (a) 1 L of an ideal gas and (b) 1 L of nitrogen described by van der Waals equation of state:

Consider a cylinder of a gasoline engine at the beginning of the compression cycle, during which a fuel/air mixture (for our purposes mostly composed of nitrogen and oxygen, i.e. an ideal diatomic gas) at 300 K and 1 bar is compressed down to the 1/10th volume (compression ratio of 10:1). Assume that the compression is rapid so no heat exchange occurs with the environment. Calculate the pressure and the temperature of the compressed gas.

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