Q. The temperature of equal masses of three different liquids $x , y$ and $z$ are $10^{\circ} C , 20^{\circ} C$ and $30^{\circ} C$ respectively. The temperature of mixture when $x$ is mixed with $y$ is $16^{\circ} C$ and that when $y$ is mixed with $z$ is $26^{\circ} C$. The temperature of mixture when $x$ and $z$ are mixed will be:
Solution:
X
Y
Z
$m _{1}= m$
$m _{2}= m$
$m _{3}= m$
$T _{1}=10^{\circ} C$
$T _{2}=20^{\circ} C$
$T _{3}=30^{\circ} C$
$s _{1}$
$s _{2}$
$S _{3}$
when $x \,\& \,y$ are mixed, $T_{f_{1}}=16^{\circ} C$
$m _{1} \,s _{1}\,T + m _{2} \,s _{2} \,T _{2}$
$=\left( m _{1} \,s _{1}+ m _{2} \,s _{2}\right) Tf _{1} $
$s _{1} \times 10+ s _{2} \times 20$
$=\left( s _{1}+ s _{2}\right) \times 16 $
$s _{1}=\frac{2}{3} s _{2} \,\,\,\,\ldots .(i)$
when $y \,\& \,z$ are mixex, $T_{f_{2}}=26^{\circ} C$
$m _{2} \,s _{2} \,T + m _{3} \,s _{3} \,T _{3}$
$=\left( m _{3}\, s _{3}+ m _{3} \,s _{3}\right) Tf _{2}$
$s _{2} \times 20+ s _{3} \times 30$
$=\left( s _{2}+ s _{3}\right) \times 26$
$s _{3}=\frac{3}{2} s _{2} \,\,\, \ldots . .(ii)$
when $x \,\&\, z$ are mixex
$m _{1} \,s _{1} \,T _{1}+ m _{3} \,s _{3} \,T _{3}$
$=\left( m _{1} \,s _{1}+ m _{3} \,s _{3}\right) Tf $
$\frac{2}{3} s _{2} \times 10+\frac{2}{3} s _{2} \times 20$
$=\left(\frac{2}{3} s _{2}+\frac{3}{2} s _{2}\right) T _{ f } $
$T _{ f }=23.84{ }^{\circ} C$
| X | Y | Z |
|---|---|---|
| $m _{1}= m$ | $m _{2}= m$ | $m _{3}= m$ |
| $T _{1}=10^{\circ} C$ | $T _{2}=20^{\circ} C$ | $T _{3}=30^{\circ} C$ |
| $s _{1}$ | $s _{2}$ | $S _{3}$ |