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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
$\frac{{{c_1}{T_1} + {c_2}{T_2} + {c_3}{T_3}}}{{{m_1}{c_1} + {m_2}{c_2} + {m_3}{c_3}}}$
$\frac{{{m_1}{c_1}{T_1} + {m_2}{c_2}{T_2} + {m_3}{c_3}{T_3}}}{{{m_1}{c_1} + {m_2}{c_2} + {m_3}{c_3}}}$
$\frac{{{m_1}{c_1}{T_1} + {m_2}{c_2}{T_2} + {m_3}{c_3}{T_3}}}{{{m_1}{T_1} + {m_2}{T_2} + {m_3}{T_3}}}$
$\frac{{{m_1}{T_1} + {m_2}{T_2} + {m_3}{T_3}}}{{{c_1}{T_1} + {c_2}{T_2} + {c_3}{T_3}}}$
Solution
$(b)$ Let the final temperature be $T\, °C.$
Total heat supplied by the three liquids in coming down to
$0°C$ = ${m_1}{c_1}{T_1} + {m_2}{c_2}{T_2} + {m_3}{c_3}{T_3}$ ….. $(i)$
Total heat used by three liquids in raising temperature from
$0°C$ to $T°C$ = ${m_1}{c_1}T + {m_2}{c_2}T + {m_3}{c_3}T$ …..$(ii)$
By equating $(i)$ and $(ii)$ we get
$({m_1}{c_1} + {m_2}{c_2} + {m_3}{c_3})\,T$
= ${m_1}{c_1}{T_1} + {m_2}{c_2}{T_2} + {m_3}{c_3}{T_3}$
$⇒$ $T = \frac{{{m_1}{c_1}{T_1} + {m_2}{c_2}{T_2} + {m_3}{c_3}{T_3}}}{{{m_1}{c_1} + {m_2}{c_2} + {m_3}{c_3}}}$.