# Copper and water heat capacity concern

What mass of copper, originally at $50\ ^{\circ}\text{C}$ must be added to $1\ \mathrm{kg}$ of $10\ ^{\circ}\text{C}$ water to raise the water temperature by $10\ ^{\circ}\text{C}$?

We can work out that the necessary energy gain by the water is $41.8\ \mathrm{kJ}$, but I'm not sure how to work out the mass of copper, since both the mass and the temperature change is unknown in the equation $\Delta E=mc\Delta T$? Or do we assume that the copper cools to $20\ ^{\circ}\text{C}$ also?

You are doing an exercise of thermodynamics and this is not mention that the equilibrium is not reached. So if your water heats up only by $10\ \mathrm{^\circ C}$ at the end even in one million years if your system is adiabatic, your solution will still be at $20\ \mathrm{^\circ C}$. And so at the equilibrium the copper must have the same temperature than the water.
So if you use $\Delta H=mc_p\Delta T$, for water $\Delta T=10\ \mathrm{^\circ C}$ and for copper $\Delta T=30\ \mathrm{^\circ C}$.