How to determine change in internal energy?

Question

A gas is contained in a piston-cylinder has the ff initial properties: 3000 kPa, 0.03 m^3. The gas is compressed during a process where pV^1.4=C to a pressure of 9000 kPa. The heat transfer from the gas is 3kJ. Determine the change in internal energy, neglect changes in kinetic and potential energies.

Answer 1

Change of internal energy is heat transferred to plus work done on the gas:

ΔU = Q + W

Heat is transferred from:

Q = -3kJ  

Work done is given by the integrals

W = - ∫ |V₁→V₂|  p dV

p∙V^1.4 = constant = p₁∙V₁^1.4

=>

p = p₁ ∙ V₁^1.4 / V^-1.4 = p₁ ∙ V₁^1.4 ∙ V^-1.4

Hence:

W = - ∫ |V₁→V₂|  p₁∙V₁^1.4 ∙ V^-1.4 dV

W = - p₁ ∙ V₁^1.4 ∙∫ |V₁→V₂|   V^-1.4 dV

= - p₁ ∙ V₁^1.4 ∙ (1/-0.4) ∙ ( V₂^-0.4 -  V₁^-0.4 )

= 2.5 ∙ p₁ ∙ V₁^1.4 ∙ ( V₂^-0.4 -  V₁^-0.4 )

= 2.5 ∙ p₁ ∙ V₁ ∙ ( (V₁/V₂)^0.4 -  1 )

V₂ is not known but you can express the volume ratio in terms of the pressure ratio:

p₁ ∙ V₁^1.4 = p₂ ∙ V₂^1.4

<=>

(V₁/V₂) = (p₂/p₁)^(1/1.4)

W = 2.5 ∙ p₁ ∙ V₁ ∙ ( (p₂/p₁)^(0.4/1.4) -  1 )

= 2.5 ∙ p₁ ∙ V₁ ∙ ( (p₂/p₁)^(2/7) -  1 )

=  2.5 ∙ 3000kPa ∙ 0.03m³ ∙ ( (9000kPa/3000kPa)^(2/7) -  1 )

= 83kJ

Hence the change of internal energy of the gas is:

ΔU = -3kJ + 83kJ = 80kJ