| A gas is held in a horizontal
piston-and-cylinder device, as shown below. |
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| A spring is attached to the back of the frictionless
piston. Initially, the spring exerts no force on the piston. |
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| The gas is heated
until the pressure
inside the cylinder is 650 kPa. Determine the boundary
work done by the gas on the piston. Assume Patm = 100 kPa. |
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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 pressure within the gas. This
relationship is linear
because the pressure
within the cylinder is
atmospheric pressure plus the spring force divided by the cross-sectional area of the piston. |
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| Diagram: |
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| Given: |
P2 |
650 |
kPa |
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Dpiston |
0.0508 |
m |
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P1
= Patm |
100 |
kPa |
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k |
3.25 |
kN/m |
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| Find: |
W = |
??? |
kJ |
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| Assumptions: |
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- The gas in the cylinder is a closed system. |
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2 - The process occurs slowly enough that it is a quasi-equilibrium process. |
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3
- There is no friction
between the piston and
the cylinder wall. |
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4
- The spring force
varies linearly with position. |
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| Equations
/ Data / Solve: |
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For a quasi-equilibrium process, boundary or PV work is defined by: |
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Eqn 1 |
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It is critical to note that the gas must overcome the force due to atmospheric pressure AND the force
of the spring during
this expansion process. Because the force exerted by the linear spring on the piston increases linearly as the gas expands, we can
write the following equation relating the force exerted by the gas
on the piston to the displacement of the piston from its original, unstretched position. |
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Eqn 2 |
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Where x is the displacement of the piston from its initial position. |
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Plug Eqn 2 into Eqn 1 and integrate to get : |
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Eqn 3 |
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Where x2 is the displacement of the spring in the final state. |
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So, our next objective is
to determine how far the piston
moved during this process. |
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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. |
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Initial State: |
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Eqn 4 |
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P1 |
100 |
kPa |
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The relationship between
force and pressure is: |
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Eqn 5 |
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Where : |
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Eqn 6 |
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Apiston |
2.03E-03 |
m2 |
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Fatm |
0.2027 |
kN |
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Final
State: |
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Eqn 7 |
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or : |
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Eqn 8 |
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Now, plug numbers into
Eqn 8 : |
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F2 |
1.1148 |
kN |
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Because the spring is linear : |
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Eqn 9 |
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or : |
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Eqn 10 |
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x2 |
0.3430 |
m |
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Finally, substitute
back into Eqn 3 to evaluate
the work done by the gas in the cylinder on its surroundings during this process : |
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W = |
0.26070 |
kJ |
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W |
260.7 |
J |
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| Verify: |
None of the
assumptions can be verified using only the information given in the problem
statement. |
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| Answers : |
W |
261 |
J |
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