| 4A-2 : | Quasi-Equilibrium Expansion of a Gas | 4 pts |
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| Warm air is contained in a piston-and-cylinder device oriented horizontally, as shown below. The air cools slowly from an initial volume of 0.003 m3 to a final volume of 0.002 m3. During the process, the spring exerts a force that varies linearly with position from an initial value of 900 N to a final value of zero Newtons. The atmospheric pressure is 100 kPa and the area of the piston face is 0.018 m2. Friction between the piston and the cylinder wall can be neglected. For the air, determine the initial and final pressures, in kPa, and work, in kJ. |  |
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| Read : | The key to solving this problem is to determine the slope and intercept for the linear relationship between the force exerted by the spring on the piston and the volume that the gas occupies. This relationship is linear because, for a cylinder of uniform diameter, gas volume varies linearly with respect to the position of the piston. | ||||||||
| Given: | V1 = | 0.003 |
m3 | Patm | 100 |
kPa | |||
| V2 = | 0.002 |
m3 | Apiston | 0.018 |
m2 | ||||
| F1 = | 900 |
N | |||||||
| F2 = | 0 |
N | |||||||
| Find: | P1 = | ??? |
kPa | W = | ??? |
kJ | |||
| P2 = | ??? |
kPa | |||||||
| Assumptions: | - The air in the cylinder is a closed system. | ||||||||
| - The process occurs slowly enough that it is a quasi-equilibrium process. | |||||||||
| - There is no friction between the piston and the cylinder wall. | |||||||||
| - The spring force varies linearly with position. | |||||||||
| Solution : | In the initial and final states, the piston is not accelerating. In fact, it is not moving. Therefore, there is no unbalanced force acting on it. This means that the vector sum of all the forces acting on the piston must be zero. | ||||||||
| Initial State: |  |
Eqn 1 |
P1 = | 150 |
kPa | ||||
| Final State: |  |
Eqn 2 |
P1 = | 100 |
kPa | ||||
| For a quasi-equilibrium process, boundary or PV-work is defined by: |  |
Eqn 3 | |||||||
| Because Fspring varies linearly with the position of the piston AND volume also varies linearly with the position of the piston, we can conclude that Fspring must vary linearly with respect to the volume ! | |||||||||
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Eqn 4 | ||||||||
| m = | 900000 |
N/m3 | b = | -1800 |
N | ||||
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Eqn 5 | ||||||||
| Answer: | W = | -125 | J | ||||||
Example Problem with Complete Solution
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