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Derive the values of critical constants in terms of van der Waals constants. |
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Answer» Solution :Derivation of critical constants from van DER Waals constant : The van der Waals equation for n moles is `(P+(an^(2))/(V^(2)))(V-nb)=nRT"".......(1)` For 1 mole `(P+(a^(2))/(V^(2)))(V-B)=RT"".......(2)` From the equation we can derive the values of critical canstants `P_(e),V_(e)` and `T_(e)` in TERMS of a and b, the van der Waals constants, On expanding the above equation `PV+(a)/(V)-Pb-(ab)/(V^(2))-RT=0""......(3)` Multiply equation (3) by `V^(2)//P` `(V^(2))/(P)(PV+(a)/(V)-Pb-(ab)/(V^(2))-RT)=0` `V^(3)+(aV)/(P)+""-bV^(2)-( ab)/(V^(2))-(RTV^(2))/(P)""......(4)` When the above equation is rearranged in powers of V. `V^(3)-[(RT)/(P)+b]V^(2)+[(a)/(P)]V-[(ab)/(P)]=0""......(5)` The equation (5) is a cubic equation in V. On solving this equation, we will get three solutions. At the critical point all these three solutions of V are equal to the critical volume `V_(C)`. The pressure and temperature becomes `P_(e)andT_(e)` respectively i.e., `V=V_(C)` `V-V_(C)=0` `(V-V_(C))^(3)=0` `V^(3)-3V_(C)V^(2)+3V_(C)""^(2)V-V_(C)^(3)=0"".......(6)` As equation (5) is identical with equation (6), we can equate the coefficients of `V_(2)`, V and constant terms in (5) and (6). `-3V_(C)V^(2)=-[(RT_(C))/P_(C)+b]V^(2)` `3V_(C)=(RT_(C))/(P_(C))+b""......(7)` `3V_(C)^(2)=(a)/(P_(C))"".......(8)` `3V_(C)^(2)=(ab)/(P_(C))"".......(9)` Divide equation (9) by equation (8) `(V_(C)^(3))/(3V_(C)^(2))=(ab//P_(C))/(a//P_(C))` `(V_(C))/(3)=b` i.e. `V_(C)=3b"".......(10)` When equation (10) is substituted in (8) `3V_(C)^(2)=(a)/(P_(C))` `P_(C)=(a)/(3V_(C)^(2))=(a)/(3(3b^(2)))=(a)/(3xx9b^(2))=(a)/(3xx9b^(2))=(a)/(27b^(2))` `P_(C)=(a)/(27b^(2))""......(11)` substituting the values of `V_(C)andP_(C)` in equation (7), `3VC=b+(RT_(C))/(P)` `3(3b)=b+(RT_(C))/((a/(27b^(2))))` `9b-b=((RT_(C))/a)=27b^(2)` `9b-b=((RT_(C))/a)=27b^(2)` `8b=(T_(C)R27b^(2))/(a)` `:.T_(C)=(8ab)/(27Rb^(2))=(8a)/(27Rb)` `T_(C)=(8a)/(27Rb)"".......(12)` The critical constants can be calculated using the values of van der waals constant of a gas and vice versa. `a=3V_(C)^(2)P_(C)ANDB=(V_(C))/(3)`. |
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