A certain balloon maintains an internal gas pressure of $$P0=100$$ kPa until the volume reaches $$V0=20$$ $$m3$$ . Beyond a volume of $$20$$ $$m3$$ , the internal pressure varies as $$P=P0+2k(V$$−$$V0)2$$ where P is in kPa, V is in $$m3$$ and k is a constant (k=1kPa/m3) . Initially the balloon contains helium gas at $$200C$$ , $$100$$ kPa with a $$15m3$$ volume. The balloon is then heated until the volume becomes $$25$$ $$m3$$ and the pressure is $$150$$ kPa. Assume ideal gas behaviour for helium. The work done by the balloon for the entire process in kJ is

Question

A  certain balloon maintains an internal gas pressure of $$P0=100$$      kPa      until the volume  reaches $$V0=20$$  $$m3$$     . Beyond a volume of $$20$$  $$m3$$     , the internal pressure varies as $$P=P0+2k(V$$−$$V0)2$$      where P      is in kPa,  V      is in $$m3$$      and k      is a constant (k=1kPa/m3)     . Initially the balloon contains helium gas at $$200C$$     , $$100$$  kPa      with a $$15m3$$      volume. The balloon  is then heated until the volume becomes $$25$$ $$m3$$      and the pressure is  $$150$$ kPa. Assume ideal gas behaviour for helium. The work done by the balloon  for the entire process in kJ      is

Infinity Learn · Accepted Answer

Work  done by the balloon $$W=$$∫$$15$$ $$m320$$ $$m3P0dV+$$∫$$20$$ $$m325$$ $$m3pdV$$        $$=$$∫$$15$$ $$m320$$ $$m3P0dV+$$∫$$20$$ $$m325$$ $$m3$$[$$P0+2(V$$−$$V0)2$$]dV                             $$=$$[$$100$$×$$5+100$$×$$5+2$$×$$(25$$−$$20)33$$]$$kJ=1083$$  kJ

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