Question

Hydrogen is contained in a piston–cylinder device at 100 kPa and 0.4 m3. At this state, a linear spring (F α x) with a spring constant of 20,300 N/m is touching the piston but exerts no force on it. The cross-sectional area of the piston is 0.3 m2. Heat is transferred to the hydrogen, causing it to expand until its volume doubles. Determine (a) the final pressure, (b) the total work done by the hydrogen, and (c) the fraction of this work done against the spring. Also, show the process on a P-V diagram.

Answer 1

Step-by-step explanation:

P1 = 100 kPa, V1 = 0.4 m³.

Spring constant k = 20,300 N/m.

Piston Area A = 0.3 m².

We assume that the expansion of Hydrogen gas is done isothermally. Temperature is maintained a constant.

V2 = 2 × V1 = 0.88 m³.

Applying the ideal gas law: P1 × V1 = P2 × V2

=> P2 = P1 × V1 / V2 = 100 × 0.4/0.8 = 50 kPa.

P×V = P1 × V1 = P2 × V2 = 40 kJ = constant

=> P = 40,00/V Pascal.

Compression x in the spring = displacement of the piston

x = (V2 - V1)/A = (0.8 - 0.4) / 0.3 = 4/3 m.

Potential Energy U stored in the spring due to the compression

= 1/2 k x² = 1/2 × 20,300 × (4/3)² J

U = 162,400/9 = 18, 044.44 J

Work W done in expansion of Hydrogen from V1 to V2 isothermally

= integral from V = V1 to V2 of P dV

= integral from V = 0.4m³ to 0.8m³ of 40,000/V dV

W = 40,000 [ Ln V2 - Ln V1 ]

= 40,000 [ Ln 0.8/0.4 ] Joules

= 27, 725.88 Joules

Total work done by the system= W + U

= 45,770.32 Joules

Fraction of the total work done against the spring

= 18, 044.44 / 45,770.32 = 0.39