A vertical cylinder of cross-sectional area A is fitted with...

A vertical cylinder of cross-sectional area $A$ is fitted with a tight-fitting, frictionless piston of mass $m$ (Fig. PI9. 52$)$ . The piston is not restricted in its motion in any way and is supported by the gas at pressure $P$ below it. Atmospheric pressure is $P_{0}$ . We wish to find the height $h$ in Figure $\mathrm{P} 19.52$ . (a) What analysis model is appropriate to describe the piston? (b) Write an appropriate force equation for the piston from this analysis model in terms of $P, P_{0}, m, A,$ and $g$ . (c) Suppose $n$ moles of an ideal gas are in the cylinder at a temperature of T. Substitute for $P$ in your answer to part $(\mathrm{b})$ to find the height $h$ of the piston above the bottom of the cylinder.

A vertical cylinder of cross-sectional area $A$ is fitted with a tight-fitting, frictionless piston of
mass $m$ (Fig. PI9. 52$)$ . The piston is not restricted in its motion in any way and is supported by the gas at pressure $P$ below it. Atmospheric pressure is $P_{0}$ . We wish to find the height $h$ in Figure $\mathrm{P} 19.52$ . (a) What analysis model is appropriate to describe the piston? (b) Write an appropriate force equation for the piston from this analysis model in terms of $P, P_{0}, m, A,$ and $g$ . (c) Suppose $n$ moles of an ideal gas are in the cylinder at a temperature of T. Substitute for $P$ in your answer to part $(\mathrm{b})$ to find the height $h$ of the piston above the bottom of the cylinder.

Key Concepts

Static Equilibrium

Static equilibrium refers to the condition where an object is at rest with all forces acting on it canceling out, resulting in a net force of zero. In the context of the piston problem, the upward force due to the gas pressure must balance the sum of the downward forces from the atmospheric pressure and the gravitational force (weight of the piston) to maintain equilibrium.

Free Body Diagram

A free body diagram is a schematic representation that shows all the forces acting on an object. For the piston, the diagram would include the upward force from the gas pressure (calculated as pressure times area), the downward force due to atmospheric pressure (also pressure times area), and the weight of the piston (mass times gravitational acceleration), serving as a foundational tool for setting up the force balance equation.

Ideal Gas Law

The ideal gas law describes the relationship among pressure, volume, temperature, and the number of moles of a gas, expressed as P = nRT/V. In this problem, it is used to substitute for the gas pressure below the piston, linking the thermodynamic state of the contained gas (in terms of moles, temperature, and volume) to the mechanical equilibrium of the piston.

Pressure Force (Force per Unit Area)

The concept that a force can be generated by a pressure acting over an area is critical in this analysis. The force exerted by a gas on the piston is calculated as the product of the gas pressure and the cross-sectional area of the piston, which directly informs the force balance with external forces such as atmospheric pressure and gravity.

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