A piece of copper of mass `m_1 = 300 g` with initial temperature `t_1 = 97 ^@C` is placed into a calorimeter in which the water of mass `m_2 = 100 g` is at a temperature `t_2 = 7 ^@ C`. Find the entropy increment of the system by the moment the temperature `t_2 = 7 ^@ C`. FInd the entropy increment of the system by the moment the temperatures equalize. The heat capacity of the calorimeter itself is negligibly small.

Explanation

Let `c_1` = specific heat of copper specific heat of water `= c_2`
Then `Delta S = int_(7 + 273)^(T_0) (c_2 m_2 dT)/(T) -int_(T_0)^(97 + 273) (m_1 c_1 dT)/(T) = m_2 c_2 1n (T_0)/(280) - m_1 c_1 1n (370)/(T_0)`
`T_0` is found from
`c_2 m_2 (T_0 - 280) = m_1 c_1 (370 - T_0)` or `T_0 = (280 m_2 c_2 + 370 m_1 c_1)/(c_2 m_2 + m_1 c_1)`
where `c_1 = 0.39 J//g ^@K, c_2 = 4.18 J//g^@K`
`T_0 ~~ 300 ^@K` and `Delta S = 28.4 - 24.5 ~~ 3.9 J//^@K`.