The guiding principle is the Law of Conservation of Energy:
The sum of all the energy changes must add up to zero.
The formula for the heat #q# gained or lost by a substance is
#color(blue)(bar(ul(|color(white)(a/a)q = mcΔTcolor(white)(a/a)|)))" "#
where
#m# is the mass of the substance.
#c# is its specific heat capacity.
#ΔT = T_"f" - T_"i"# is the change in temperature.
In this problem, there are two heat flows.
#"Heat lost by copper + Heat gained by water" = 0#
#color(white)(mmmmm)q_1 color(white)(mmmll)+color(white)(mmmmm) q_2color(white)(mmmml) = 0#
#color(white)(mmm)m_1c_1ΔT_1 color(white)(mml)+color(white)(mmm) m_2c_2ΔT_2color(white)(mmll) = 0#
The final temperature of the copper will be the same as the final temperature of the water.
In this problem,
#m_1 = "5.61 g";color(white)(mmmm) m_2 = "100 g"#
#c_1 = "0.3844 J·K"^"-1""g"^"-1"; c_2 = "4.184 J·K"^"-1""g"^"-1"#
#ΔT_1 = T_"f"color(white)(l) "- 98.8 °C"; color(white)(ll)ΔT_2 = T_"f" color(white)(l)"- 22.6 °C"#
#m_1c_1ΔT_1 + m_2c_2ΔT_2 = 0#
#5.61 color(red)(cancel(color(black)("g"))) × 0.3844 color(red)(cancel(color(black)("J·K"^"-1""g"^"-1"))) × (T_"f"color(white)(l) "- 98.8 °C")#
#+ 100 color(red)(cancel(color(black)("g"))) ×4.184 color(red)(cancel(color(black)("J·K"^"-1""g"^"-1"))) × (T_"f" color(white)(l)"- 22.6 °C") =0#
#2.156 (T_"f"color(white)(l) "- 98.8 °C") + 418.4(T_"f" color(white)(l)"- 22.6 °C") = 0#
#2.156T_"f" - "213.1 °C" + 418.4T_"f" - "9456 °C" =0#
#420.6 T_"f" = "9669 °C"#
#T_"f" = "9669 °C"/420.6 = "23.0 °C"#
