When pressure is increases at constant temp volume of gas decreases AB rarr gases, BC rarr vpour +liquid, CD rarr liquid critical point: At this point all the physical propeties of liquid phase will be same as the physical properties in vapour such as, density of liquid = density of vapour T_(c) or critical temp: Temperature abive which a gas can not be liquified P_(c) or critical pressure: minimum pressure which must be applied at critical temp to convert the gas into liqid. V_(c) or critical volume: volume occupied by one mole of gas at T_(c) & P_(c) CRITICAL CONSTANT USING VANDERWAAL EQUATIONS: {:((P+(a)/(V_(m)^(2)))(V_(m)-gb)=RT,rArr,(PV_(m)^(2)+a)(V_(m)-b)=RTV_(m)^(2)),(PV_(m)^(3)+aV_(m)-PbV_(m)^(2) -ab -RTV_(m)^(2) =0, rArr, V_(m)^(3)+V_(m)^(2) (b+(RT)/(P))+(a)/(P)(V)/(m)-(ab)/(P) = 0):} since equation is cubic in V_(m) hence there will be three roots of equation of any temperature and pressure. At critical point all three roots will coincide and will given single value of V_(m) = V_(c) at critical point. Vander Waal equation will be V_(m)^(3) - V_(m)^(2) (b+(RT_(C))/(P_(C))) +(a)/(P_(C)) V_(m) - (ab)/(P_(C)) = 0 ...(i) But at critical point all three roots of the equation should be equal, hence equation should be: V_(m) = V_(c) (V_(m) -V_(c))^(3) = 0 V_(m)^(3) - 3V_(m)^(2) V_(c) +3V_(m) V_(c)^(2) -V_(c)^(3) = 0 ..(2) comparing with equation (1) b +(RT_(c))/(P_(c)) = 3V_(c) ...(i) (a)/(P_(c)) = 3V_(c)^(2) ...(ii) (ab)/(P_(c)) = V_(c)^(3) ..(iii) From (ii) and (iii), V_(c) = 3b From (ii) P_(c) = (a)/(3V_(c)^(2)) substituting P_(c) = (a)/(3(3b)^(2)) = (a)/(27b^(2)) From (i) (RT_(c))/(P_(c)) = 3V_(c) -b = 9b -b = 8b rArr T_(c) = (8a)/(27Rb) At critical point, the slope of PV curve (slope of isotherm) will be zero at all other point slope will be negative zero is the maximum value of slope. ((delP)/(delV_(m)))_(TC) =0 ..(i) (del)/(delV_(m)) ((delP)/(delV_(m)))_(TC) = 0 ...(ii) {Mathematically such points an known as point of inflection (where first two derivatives becimes zero)} using the two T_(c)P_(c) and V_(c) can be calculated by A scientist proposed the following equation of state P = (RT)/(V_(m)) - (B)/(V_(m)^(2)) +(C )/(V_(m)^(3)). If this equation leads to teh critical behaviour then critical temperature is: