The temperature of equal masses of three different liquids A,B and C are `12^@C, 19^@C and 28^@C` respectively. The temperature when A and B are mixed is `16^@C` and when B and C are mixed it is `23^@C`. What should be the temperature when A and C are mixed?

Options

A. `18.2^(@)C`
B. `22^(@)C`
C. `20.2^(@)C`
D. `24.2^(@)C`

Answer

C

Explanation

Let `m_(1) = m_(2) = m_(3) m`
Let `s_(1), s_(2), s_(3)` be the respective specific heats of the three liquids A, B and C respectively.
When A and B are mixed, temperature of mixture = `16^(@)C`
As heat gained by A = heat lost by B
`:. ms_(1) (16-12) = ms_(2) (19-16) or 4s_(1) = 3s_(2)` ....(i)
When B and C are mixed, temperature of mixture = `23^(@)C`
As heat gained by B= heat lost by C
`:. ms_(2) (23-19) = ms_(3) (28-23) or 4s_(2) = 5s_(3)` ...(ii)
From eqns (i) and (ii), we get
`s_(1) = (3)/(4) s_(2) = (15)/(16) s_(3)` ....(iii)
When A and C are mixed, let the temperature of mixture be `T^(@)C`. Heat gained by A = heat lost by C
`:. ms_(1) (T- 12) = ms_(3) (28-T)`
or `(15)/(16) s_(3) (T-12) = s_(3) (28-T)` (Using (iii))
or `15T - 180 = 448 - 16T or T= (628)/(31) = 20.2^(@)C`