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Three kg of gas in a piston–cylinder assembly undergo a process during which the relationship between pressure and specific volume is Pv0.5=constant. The process begins with P1=250 kPa and V1=1.5 m3 and ends with P2=100 kPa. Determine the final specific volume, in m3/kg.

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The final specific volume in m³/kg is 0.0313 m³/kg.

Since [tex]Pv^{0.5} = constant[/tex] where P = pressure and v = specific volume, we have

Since [tex]P_{1} v_{1}^{0.5} = P_{2} v_{2}^{0.5}[/tex] where

  • P₁ = initial pressure = 250 kPa,
  • v₁ = initial specific volume = V₁/m where
  • V₁ = initial volume of gas = 1.5 m³ and
  • m = mass of gas = 3 kg. so, v₁ = V₁/m = 1.5 m³/3 kg = 0.5 m³/kg,
  • P₂ = final pressure = 100 kPa,
  • v₂ = final specific volume of gas

Final Specific volume

Making v₂ subject of the formula, we have

[tex]v_{2} = v_{1} ( \frac{P_{1} }{P_{2} }) ^{2}[/tex]

Substituting the values of the variables into the equation, we have

[tex]v_{2} = v_{1} ( \frac{P_{1} }{P_{2} }) ^{2}[/tex]

[tex]v_{2} = 0.5 \frac{m^{3} }{kg} ( \frac{250 kPa }{100 kPa }) ^{2}\\v_{2} = 0.5 \frac{m^{3} }{kg} ( \frac{25}{10}) ^{2}\\v_{2} = 0.5 \frac{m^{3} }{kg} ( 0.25) ^{2}\\v_{2} = 0.5 \frac{m^{3} }{kg} (0.0625)\\v_{2} = 0.0313 \frac{m^{3} }{kg}[/tex]

The final specific volume in m³/kg is 0.0313 m³/kg.

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