130 Physical Properties [CH. vi 



to some inverse power of (v v ), and hence it follows that the value of T 

 given by equation (308) must for small values of v v be simply proportional 

 to (v - v ). 



148. We are now in a position to sketch the general isothermals repre- 

 sented by equation (303). 



To draw any single isothermal we notice that for large values of v the 

 coordinates p, v may be supposed to be connected by the relation pv = constant, 

 and the isothermals are, as in fig. 6, rectangular hyperbolas. For the general 

 isothermal T=T lt in which T l is not very great, we first draw the line T = 2\ 

 in fig. 8. The values of v, if any, for which it meets the graph in this 

 figure, give the values of v, if any, at which dp/dv = on the isothermal. At 

 the points obtained in this way, the isothermal is parallel to the axis of v. 

 Also we know that the isothermal approaches v = v asymptotically. Finally, 

 two isothermals can never cross, for if they did, equation (303) regarded as 

 an equation in T would have two roots for T, But the equation is of the 

 well-known form 



e -ax _ l x> 



where x stands for \\T and a, b are positive constants, and it becomes 

 obvious upon drawing the graphs of the two members of this equation that 

 the equation can only have one root. 



149. It will be seen, therefore, that these isothermals are represented, in 

 their main features, in fig. 9. The isothermals corresponding to the highest 

 values of T are those furthest removed from the origin. The isothermal 

 through P corresponds to the maximum value of T for which dp/dv vanishes, 

 this value being of course equal to the maximum ordinate of the graph of 

 fig. 8. The point P, at which dp/dv = 0, is a point of inflexion on this 

 curve, for there are (cf. fig. 8) two coincident values of v for which 



dp/dv = 0. 



For smaller values of T the two values of v at which dp/dv = are not 

 coincident, and there is a range such as QR over which dp/dv is negative. 

 If X is any point within this range, there are two other points Y, Z on the 

 same isothermal as X corresponding to the same pressure as X : three states 

 in all, therefore, for which the pressure and temperature are the same. In 

 the state represented by X a decrease in volume, keeping the temperature 

 constant, is accompanied by a decrease in pressure. This state is therefore 

 unstable: any slight decrease in volume is attended by a tendency to a 

 further decrease in the form of an unbalanced external pressure. Of the two 

 stable states Y, Z, the state Z, corresponding to greater volume, must be the 

 gaseous state, the state Y corresponding to lesser volume is the liquid state. 

 We see therefore that the temperature of the isothermal through P is the 



