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) \[\frac{{{m}_{1}}{{T}_{1}}+{{m}_{2}}{{T}_{2}}+{{m}_{3}}{{T}_{3}}}{{{c}_{1}}{{T}_{1}}+{{c}_{2}}{{T}_{2}}+{{c}_{3}}{{T}_{3}}}\]
Correct Answer: B
Solution :
| Let the final temperature be \[T{}^\circ C\]. |
| Total heat supplied by the three liquids in coming down to |
| \[0{}^\circ C\] = \[{{m}_{1}}{{c}_{1}}{{T}_{1}}+{{m}_{2}}{{c}_{2}}{{T}_{2}}+{{m}_{3}}{{c}_{3}}{{T}_{3}}\] ..... (i) |
| Total heat used by three liquids in raising temperature from |
| \[0{}^\circ C\] to \[T{}^\circ C\] = \[{{m}_{1}}{{c}_{1}}T+{{m}_{2}}{{c}_{2}}T+{{m}_{3}}{{c}_{3}}T\] .....(ii) |
| By equating (i) and (ii) we get |
| \[({{m}_{1}}{{c}_{1}}+{{m}_{2}}{{c}_{2}}+{{m}_{3}}{{c}_{3}})\,T\] |
| = \[{{m}_{1}}{{c}_{1}}{{T}_{1}}+{{m}_{2}}{{c}_{2}}{{T}_{2}}+{{m}_{3}}{{c}_{3}}{{T}_{3}}\] |
| \[\Rightarrow \] \[T=\frac{{{m}_{1}}{{c}_{1}}{{T}_{1}}+{{m}_{2}}{{c}_{2}}{{T}_{2}}+{{m}_{3}}{{c}_{3}}{{T}_{3}}}{{{m}_{1}}{{c}_{1}}+{{m}_{2}}{{c}_{2}}+{{m}_{3}}{{c}_{3}}}\]. |
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