A gas initially at 1 MPa and 500^∘ C is contained in a piston and cylinder arrangement with an i

step 1 Here by knowing the process and the states of 1 and 2 we can find the relation between the press

A gas initially at 1 MPa and 500^∘ C is contained in a...

A gas initially at $1 \mathrm{MPa}$ and $500^{\circ} \mathrm{C}$ is contained in a piston and cylinder arrangement with an initial volume of $0.1 \mathrm{m}^{3}$. The gas is then slowly expanded according to the relation $P V=$ constant until a final pressure of $100 \mathrm{kPa}$ is reached. Determine the work for this process.

Key Concepts

Integration of the Pressure-Volume Relationship

When pressure and volume are linked by an inverse relationship—as in processes described by P·V = constant—the work done by the system during expansion or compression can be computed by substituting the pressure expression into the work integral, yielding an integral of the form ?(constant/V) dV. This integration results in a logarithmic function, which is a critical mathematical tool in obtaining the work done in processes where the product of pressure and volume remains constant.

Polytropic Processes

A polytropic process is one in which the pressure and volume of a gas follow the relation P·V^n = constant, where n is the polytropic index. When n = 1, this relationship simplifies to P·V = constant, which is typically associated with an isothermal process for an ideal gas. Understanding the polytropic process enables one to apply the appropriate integration techniques and thermodynamic relationships to determine work, heat transfer, and other state properties during the process.

Work in Quasi?Static Processes

In thermodynamics, work is defined as the energy transfer associated with a force acting over a distance. For quasi?static (infinitely slow) processes, the system remains nearly in equilibrium at all times so that the work done can be precisely calculated by integrating the pressure with respect to volume, i.e., W = ?P dV. This concept is fundamental when analyzing piston-cylinder systems or any process where volume changes occur under controlled pressure conditions.

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