Question

Three liquids with masses $m_{1},m_{2},m_{3}$ are thoroughly mixed. If their specific heats are $c_{1},c_{2},c_{3}$ and their temperatures $T_{1},T_{2},T_{3}$ respectively, then the temperature of the mixture is

• A. $\frac{c_{1}T_{1} + c_{2}T_{2} + c_{3}T_{3}}{m_{1}c_{1} + m_{2}c_{2} + m_{3}c_{3}}$
• B. $\frac{m_{1}c_{1}T_{1} + m_{2}c_{2}T_{2} + m_{3}c_{3}T_{3}}{m_{1}c_{1} + m_{2}c_{2} + m_{3}c_{3}}$
• C. $\frac{m_{1}c_{1}T_{1} + m_{2}c_{2}T_{2} + m_{3}c_{3}T_{3}}{m_{1}T_{1} + m_{2}T_{2} + m_{3}T_{3}}$
• D. $\frac{m_{1}T_{1} + m_{2}T_{2} + m_{3}T_{3}}{c_{1}T_{1} + c_{2}T_{2} + c_{3}T_{3}}$

Answer 1

Answer: (B)

$\frac{m_{1}c_{1}T_{1} + m_{2}c_{2}T_{2} + m_{3}c_{3}T_{3}}{m_{1}c_{1} + m_{2}c_{2} + m_{3}c_{3}}$

Answer 2

Let the final temperature be T ⁰C.

Total heat supplied by the three liquids in coming down to 0⁰C = $m_{1}c_{1}T_{1} + m_{2}c_{2}T_{2} + m_{3}c_{3}T_{3}$ ..... (i)

Total heat used by three liquids in raising temperature from 0⁰C to T⁰C = $m_{1}c_{1}T + m_{2}c_{2}T + m_{3}c_{3}T$ .....(ii)

By equating (i) and (ii) we get $(m_{1}c_{1} + m_{2}c_{2} + m_{3}c_{3})T$ = $m_{1}c_{1}T_{1} + m_{2}c_{2}T_{2} + m_{3}c_{3}T_{3}$

⇒ $T = \frac{m_{1}c_{1}T_{1} + m_{2}c_{2}T_{2} + m_{3}c_{3}T_{3}}{m_{1}c_{1} + m_{2}c_{2} + m_{3}c_{3}}$.