A copper calorimeter weighting 106g is filled with 100g of water at 700 C and the time taken for the temperature to fall to 650 C is 5min. When filled with another liquid to the same level the weight of the liquid is 80g and time for the same fall of temperature is 2.7 mins. Find the specific heat of the liquid. [ Specific heat capacity of copper = 400J Kg-K-]

Solution:

Mass of Calorimeter (m₁) = 106g=0.106Kg

Mass of Water (m₂) = 100gm=0.1kg

Specific Heat Capacity of Copper (S₁) = 400 JKg^-1K^-1

Specific Heat Capacity of Water (S₂) = 4200 JKg^-1K^-1

Let S_l be the specific heat capacity of the liquid.

For water time taken to fall from 70⁰ C to 65⁰ C = 5 min = 300seconds

For water time taken to fall from 70⁰ C to 65⁰ C = 2.7 min = 162seconds

The average heat loss of calorimeter and Water 

$\begin{array}{l} = \frac{{{m_1}{s_1}\Delta \theta  + {m_2}{s_2}\Delta \theta }}{t}\\ = ({m_1}{s_1} + {m_2}{s_2})\frac{{\Delta \theta }}{t}\\ = (0.016 \times 400 + 0.1 \times 4200) \times \frac{5}{{300}}\\ = 7.1066\end{array}$

And the average heat lost by the calorimeter and liquid 

$\begin{array}{l} = \frac{{{m_2}{s_2}\Delta \theta  + {m_3}{s_l}\Delta \theta }}{t}\\ = ({m_1}{s_1} + {m_2}{s_l})\frac{{\Delta \theta }}{t}\\ = (0.016 \times 400 + 0.08 \times {s_l})\frac{5}{{162}}\end{array}$

Since the rate of heat loss in identical condition is same, so two rates equal and we have:

7.1066 $ = (0.016 \times 400 + 0.08 \times {s_l})\frac{5}{{162}}$

Or, S_l = 2591.2

Thus, the specific heat of the liquid is 2591.2 JKg^-1K^-1