Explain Calorimetry and derive an expression for final temperature when two thermodynamic systems are mixed.

Explanation

Calorimetry: Calorimetry means the measurement of the amount of heat released or absorbed by thermodynamic system during the heating process. When a body at higher temperature is brought in contact with another body at lower temperature, the heat lost by the hot body is equal to the heat gained by the cold body. No heat is allowed to escape to the surroundings. It can be Anomalous expansion of water in lakes mathematically expressed as
`Q_("gain") = -Q_("lost")`
`Q_("gain") + Q_("lost") = 0`

Heat gained or lost is measured with a calorimeter. Usually the calorimeter is an insulated container of water A sample is heated at high temperature `(T_(1))` and immersed into water at room temperature `(T_(2))` in the calorimeter. After some time both sample and water reach a final equilibrium temperature `T_(f)`. Since the calorimeter is insulated, heat given by the hot sample is equal to heat gained by the water.
`Q_("gain") = -Q_("lost")`
Note the sign convention. The heat lost is denoted by negative sign and heat gained is denoted as positive. From the definition of specific heat capacity

`Q_("gain")=m_(2)s_(2)(T_(f)-T_(2))`
`Q_("lost")=m_(1)s_(1)(T_(f)-T_(1))`
Here `s_(1) and s_(2)` specific heat capacity of hot sample and water respectively.
So we can write
`m_(2)s_(2)(T_(f)-T_(2))=-m_(1)s_(1)(T_(f)-T_(1))`
`m_(2)s_(2)T_(f)-m_(2)s_(2)T_(2)=-m_(1)s_(1)T_(f)+m_(1)s_(1)T_(1)`
`m_(2)s_(2)T_(f)+m_(1)s_(1)T_(f)=m_(2)s_(2)T_(2)+m_(1)s_(1)T_(1)`
`"The final temperature "T_(f)=(m_(1)s_(1)T_(1)+m_(2)s_(2)T_(2))/(m_(1)s_(1)+m_(2)s_(2))`