Question

The temperature of equal masses of three different liquids A, B, C are $12{}^\circ C$,$19{}^\circ C$ and $28{}^\circ C$ respectively. The temperature when A and B are mixed is $16{}^\circ C$ and when B and C are mixed is $23{}^\circ C$. What will be the temperature when A and C are mixed?

Answer 1

We know that the number of calories required to raise the temperature of 1 gram of a substance ${{1}^{{}^\circ }}C$, or the number of BTU's per pound per degree F. (originally) the ratio of the thermal capacity of a substance to that of standard material. For example, at a temperature of ${{25}^{{}^\circ }}C$ (the specific heat capacity can vary with the temperature), the heat required to raise the temperature of 1 kg of water by 1 K (equivalent to ${{1}^{{}^\circ }}C$) is 4179. 6 joules, meaning that the specific heat of water is 4179.

Complete step-by step answer:
Let ‘m’ be the mass of each liquid and ${{S}_{A}},\,{{S}_{B}},\,{{S}_{C}}$be specific heats of liquids A, B and C respectively.
When A and B are mixed. The final temperature is $16{}^\circ C$.
$\therefore$ Heat gained by A = heat lost by B
i.e., $m{{S}_{A}}$ $=(16-12)=m{{S}_{B}}(19-16)$
i.e., $m{{S}_{A}}$ $=\dfrac{4}{3}{{S}_{A}}.....(i)$
When B and C are mixed. Heat gained by B = Heat lost by C
i.e., $m{{S}_{B}}=(23-19)=mS_{C}^{{}}(28-23)$
i.e., ${{S}_{B}}=\dfrac{4}{5}{{S}_{B}}.....(ii)$
From equation (i) and (ii)
${{S}_{C}}=\dfrac{4}{5}\times \dfrac{4}{3}{{S}_{A}}=\dfrac{16}{15}{{S}_{A}}$
When A and C are mixed, let the final temperature be $\theta$
$\therefore m{{S}_{A}}(\theta -12)=m{{S}_{C}}(23-\theta )$
i.e., $\theta -12=\dfrac{16}{15}(28-\theta )$
By solving, we get.
$\theta =\dfrac{628}{31}=20.26{}^\circ C.$

Hence, the answer is $20.26{}^\circ C$ .

Note: We know that heat capacity, ratio of heat absorbed by a material to the temperature change. It is usually expressed as calories per degree in terms of the actual amount of material being considered, most commonly a mole (the molecular weight in grams). The heat capacity in calories per gram is called specific heat. Water's high heat capacity is a property caused by hydrogen bonding among water molecules. Specific heat is defined as the amount of heat one gram of a substance must absorb or lose to change its temperature by one degree Celsius. For water, this amount is one calorie, or 4.184 Joules. The heat capacity is a smooth, continuous function of temperature except for a small number of discontinuities. These occur at temperatures where the substance undergoes phase changes.