A gas is held in a horizontal
pistonandcylinder device, as shown below. 







A spring is attached to the back of the frictionless
piston. Initially, the spring exerts no force on the piston. 

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 P_{atm} = 100 kPa. 





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 crosssectional area of the piston. 












Diagram: 













Given: 
P_{2} 
650 
kPa 



D_{piston} 
0.0508 
m 


P_{1}
= P_{atm} 
100 
kPa 



k 
3.25 
kN/m 






Find: 
W = 
??? 
kJ 










Assumptions: 
1
 The gas in the cylinder is a closed system. 



2  The process occurs slowly enough that it is a quasiequilibrium process. 



3
 There is no friction
between the piston and
the cylinder wall. 



4
 The spring force
varies linearly with position. 












Equations
/ Data / Solve: 




















For a quasiequilibrium process, boundary or PV work is defined by: 





















Eqn 1 












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. 















Eqn 2 

Where x is the displacement of the piston from its initial position. 













Plug Eqn 2 into Eqn 1 and integrate to get : 






















Eqn 3 













Where x_{2} is the displacement of the spring in the final state. 






So, our next objective is
to determine how far the piston
moved during this process. 















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 4 

P_{1} 
100 
kPa 











The relationship between
force and pressure is: 



Eqn 5 












Where : 

Eqn 6 

A_{piston} 
2.03E03 
m^{2} 







F_{atm} 
0.2027 
kN 










Final
State: 



Eqn 7 












or : 




Eqn 8 












Now, plug numbers into
Eqn 8 : 

F_{2} 
1.1148 
kN 













Because the spring is linear : 





Eqn 9 












or : 







Eqn 10 

















x_{2} 
0.3430 
m 













Finally, substitute
back into Eqn 3 to evaluate
the work done by the gas in the cylinder on its surroundings during this process : 













W = 
0.26070 
kJ 



W 
260.7 
J 


Verify: 
None of the
assumptions can be verified using only the information given in the problem
statement. 



Answers : 
W 
261 
J 











