In a cylinder of an automobile engine, just after...

In a cylinder of an automobile engine, just after combustion, the gas is confined to a volume of 50.0 $\mathrm{cm}^{3}$ and has an initial pressure of $3.00 \times 10^{6} \mathrm{Pa}$ . The piston moves outward to a final volume of $300 \mathrm{cm}^{3},$ and the gas expands without energy loss by heat. (a) If $\gamma=1.40$ for the gas, what is the final pressure? (b) How much work is done by the gas in expanding?

Key Concepts

Work of Expansion in an Adiabatic Process

The work done by or on a gas during an adiabatic process can be calculated by integrating the pressure with respect to volume, taking into account the adiabatic relation between pressure and volume. For a reversible adiabatic process of an ideal gas, the work is often given by the formula W = (P_iV_i - P_fV_f) / (? - 1), where P_i and V_i are the initial pressure and volume, and P_f and V_f are the final pressure and volume.

Specific Heat Ratio (?)

The specific heat ratio, ? (gamma), defined as the ratio of the specific heat at constant pressure (c_p) to the specific heat at constant volume (c_v), is a crucial parameter in thermodynamics. It influences how a gas responds to compression or expansion, particularly in determining the steepness of the adiabatic curve (PV^? = constant) and the work involved in such processes.

Adiabatic Process

An adiabatic process is one in which no heat is exchanged with the surroundings. In such processes, any change in the internal energy of the gas is solely due to work done by or on the gas. This is particularly relevant for rapid expansions or compressions, where there is insufficient time for heat transfer, leading to the relation PV^? = constant for an ideal gas.

Pressure-Volume Relationship (PV^? Constant)

For an ideal gas undergoing a reversible adiabatic process, the product of the pressure and the volume raised to the power ? (the ratio of specific heats) remains constant. This relationship, expressed as PV^? = constant, is key to determining the state variables (pressure, volume, temperature) at any point during the process.

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