Finding the equilibrium temperature between two substances in a constant pressure calorimeter
Assuming the heat absorption of the calorimeter is negligible (so if it's made of an insulating material, the only changes in heat will be of the contents themselves).

Substance 1 (the one at the higher initial temperature) has a heat = $q_1$

Substance 2 has a heat of $q_2$

Since substance 1 will lose heat until it comes to equilibrium with substance 2, the heat lost by 1 will be absorbed by 2, so $q_1 = -q_2$

If $q_1 = m_1c_{p1}(T_{{1}_{initial}} - T_{{1}_{final}})$ and $q_2 = m_2c_{p2}(T_{{2}_{initial}} - T_{{2}_{final}})$ and $T_{1_{final}} = T_{2_{final}} = T_f$ you can solve for $T_f$.

If the heat absorption of the calorimeter was not negligible, then $q_1 + q_2 + q_{cal} = 0$