Two liquids A and B have bulk moduli in the ratio 5:2 . They are subjected to pressure changes P_A and P_B respectively, such that P_A : P_B = 1:2 . It is observed that the absolute decrease in volume is identical for both the liquids. If the ratio of the initial volume of liquid A to that of liquid B is x , the value of x is ______.

Step-by-step solution

The bulk modulus B of a liquid is given by the formula: B = P ( V V ) Rearranging this to solve for the initial volume V , we get: V = B V P For liquids A and B , the absolute decrease in volume is the same, so V_A = V_B = V . The ratio of their initial volumes is: V_A V_B = ( B_A V P_A ) ( B_B V P_B ) V_A V_B = ( B_A B_B ) ( P_B P_A ) Given that B_A B_B = 5 2 and P_A P_B = 1 2 (which means P_B P_A = 2 ), we substitute these values: V_A V_B = ( 5 2 ) ( 2 1 ) = 5 Hence, the value of x is 5 . Answer: 5