The temperatures of equal masses of three different liquids A, B and C are `15^@C`, `-20^@C` and `30^@C`, respectively. When A and B are mixed their equilibrium temperature is `180^@C`. When B and C are mixed, it is `22^@C` .What will be the equilibrium temperature when liquids A and C are mixed

Explanation

Let specific heats of liquids be `S_A`,`S_B` and `S_C`, respectively. Then for mixing of liquids A and B, heat lost by `B=`heat gained by A
`mxxS_Bxx(20-18)=mxxs_Axx(18-15)`
`2S_B=3S_A` ..(i)
For mixing of liquids B and C, we have
Heat lost by `C=` heat gained by B
`mxxS_Cxx(30-22)=mxxS_Bxx(22-20)`
`8S_C=2S_B` ..(ii)
from Eqs. (i) and (ii), we get
`8S_C=3S_A` .(iii)
now if liquids A and C are mixed, let their equilibrium temperature in `T_e`. Then we have
Heat lost by `C=` heat gained by A
`mxxS_Cxx(30-T_e)=mxxS_Axx(T-15)`
or `(3S_A)/(8)xx(20-T_e)=S_Axx(T_e-15)`
or `90-3T_e=8T_e-120`
or `11T_e=210`
or `T_e=(210)/(11)=19.09^@C`