Question

SP7: (20 points) Hydrogen gas is contained in a vertical piston-cylinder device at 140 kPa and 0.3 mº. At this state, a linear spring with a spring constant of k=150,000 N/m touches the top of the piston but exerts no force on it because the spring is at its equilibrium position x=0 (Fspring = kx). The cylinder cross section is circular, and the cross-sectional area of the piston is 0.3 m². Heat transfer occurs to hydrogen gas, pushing the piston upwards against the spring, and the hydrogen gas expands until its volume doubles. (a) What is the total work done (kJ) during the expansion process? (b) What fraction of the work is done to compress the spring? (c) Determine final pressure (kPa) of the hydrogen gas. (d) Show the process on a p-V diagram.

Please show all working so I can learn the material.
Partial answers I have are W1-2 =117KJ, Wspring/W12 = 0.641
Hopefully that helps you verify your answer a bit. Thank you in advance!

Answer 1

Given data :-

• Initial pressure, P1 = 140 kPa
• Initial volume , V1 = 0.3 $m^{3}$
• Spring constant K = 150,000 N/m
• Cross-section area of Piston A = 0.3 $m^{2}$
• Final volume V2 = 2 × V1 = 0.6 $m^{3}$

ܠܠܓ F5=0 Fs K. R Img Pq kPa, P2A V2 = 0.6 m3 140 kPa, T x1 d X2 0.3 m3 & supplied Initial Position Final Position

Now, as shown in the figure at initial condition there is no spring force acting against the piston because it is in equillibrium condition. Now, as we supply heat to the system volume gets double and piston cylinder attains final equillibrium condition as shown in figure.

(1) Total work done during process

Now there are two types of workdone in this process.

1. Displacement workdone ( Pdv workdone )

2. Work done against spring force ( Energy consumed by spring )

Now , Pdv workdone can be found by,

Pdv workdone = P × dv = P1 × ( V2 - V1 )

= 140 × $10^{3}$ × ( 0.6 - 0.3 )

= 42 × $10^{3}$ Joule = 42 kJ

Now, workdone against spring can be given by,

where, x = displacement of piston

W, = -K.. 2

Now for initial condition , position of piston $x_{1}$ ,

$x_{1} = \frac{V1}{A} = \frac{0.3}{0.3} = 1 m$

For final position , position of piston  $x_{2}$,

$x_{2} = \frac{V2}{A} = \frac{0.6}{0.3} = 2 m$

So displacement of piston,  

So, Workdone against the sprong,

$W_{s} = \frac{1}{2}Kx^{2} = \frac{1}{2}\times 150000\times 1^{2} = 75000$  

So,

So, Total workdone in the process ,

= 42 + 75 = 117 kJ

(2) Fraction of work done to compress the spring,

  Now Total work done W = 117 kJ

Workdone to compress the spring = Workdone against the spring

= 75 kJ

So fraction of work done to compress the spring =

= 75 / 117 = 0.641

(3) Final pressure of gas

let final pressure of gas is P2.

Now , consider final state as shown in second figure,

Applying Equillibrium conditions,

Pressure force by gas = Spring force + wieght of piston

P2 × A = Kx + mg where , m = mass of piston ( neglacted )

So, P2 × A = Kx ________________________(1)

So, P2 = Kx/A = 150000× 1/ 0.3

P2 = 500000 Pa = 500 kPa

(4) P-V diagram for the process,

As equation 1 shows, pressure of gas varies linearly with the displacement of piston. And Displacement of piston is linearly propositional to the Volume of the gas,

So, we can say that Pressure of gas is directly propositional to the Volume of the gas, So, Line of P-V diagram will be linear as shown below.

(kPa) 50 140 0.6 0.3 ✓ m3 ) -