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Questions on calorimetry without phase change

Question

Equal masses of three liquids A, B and C have temperatures 10⁰C, 25⁰C and 40⁰C, respectively. If A and B are mixed, the mixture has a temperature of 15⁰C. If B and C are mixed, the mixture has a temperature of 30⁰C. If A and C are mixed, the mixture will have a temperature of

Moderate

A

16⁰C
B

20⁰C
C

25⁰C
D

29⁰C

Solution

When A and B are mixed,
$m c_{A} (15 - 10) = m c_{B} (25 - 15) or \frac{c_{A}}{c_{B}} = 2$ …(i)
When B and C are mixed,
$m c_{B} (30 - 25) = m c_{C} (40 - 30) or \frac{c_{B}}{c_{C}} = 2$ …(ii)
From Eqs. (i) and (ti), we get
$\frac{c_{A}}{c_{C}} = 4$
Now, let $θ$ be the equilibrium temperature when A and C are mixed.
Then, $m c_{A} (θ - 10) = m c_{C} (40 - θ)$
or $4 (θ - 10) = 40 - θ (Using \frac{c_{A}}{c_{C}} = 4)$
or $θ = 16^{\circ} C$

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