Gujarati
10-1.Thermometry, Thermal Expansion and Calorimetry
medium

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

$\frac{{{m_1}{T_1} + {m_2}{T_2} + {m_3}{T_3}}}{{{c_1}{T_1} + {c_2}{T_2} + {c_3}{T_3}}}$

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

$({m_1}{c_1} + {m_2}{c_2} + {m_3}{c_3})\,T$

= ${m_1}{c_1}{T_1} + {m_2}{c_2}{T_2} + {m_3}{c_3}{T_3}$

$⇒$ $T = \frac{{{m_1}{c_1}{T_1} + {m_2}{c_2}{T_2} + {m_3}{c_3}{T_3}}}{{{m_1}{c_1} + {m_2}{c_2} + {m_3}{c_3}}}$.

Standard 11
Physics

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