A mass $m_{1}$ of a substance of specific heat capacity $c_{1}$ at temperature $t_{1}$ is mixed with a mass $m_{2}$ of other substance of specific heat capacity $c_{2}$ at a lower temperature $t_{2}$ . Deduce the expression for the temperature of the mixture. State the assumption made, if any.

\frac{m_{1} t_{1} + m_{2} t_{2}}{m_{1} + m_{2}}

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\frac{m_{1} t_{1} + m_{2} t_{2}}{m_{1} c_{1} + m_{2} c_{2}}

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\frac{m_{1} c_{1} + m_{2} c_{2}}{m_{1} + m_{2}}

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\frac{m_{1} c_{1} t_{1} + m_{2} c_{2} t_{2}}{m_{1} c_{1} + m_{2} c_{2}}

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Solution

The correct option is D $\frac{m_{1} c_{1} t_{1} + m_{2} c_{2} t_{2}}{m_{1} c_{1} + m_{2} c_{2}}$
Let the temperature of the mixture be $t$ .
let $t_{2} < t < t_{1}$ . So the heat supplied by $m_{1}$ is $m_{1} c_{1} (t_{1} - t)$ .

Similarly, the heat absorbed by $m_{2}$ is $m_{2} c_{2} (t - t_{2})$ .
As heat supplied is equal to the heat absorbed. So,
$m_{1} c_{1} (t - t_{1}) = m_{2} c_{2} (t_{2} - t) \Rightarrow t = \frac{m_{1} c_{1} t_{1} + m_{2} c_{2} t_{2}}{m_{1} c_{1} + m_{2} c_{2}}$

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