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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?

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There are all elements in your exercise to solve it expect the heat capacities but this is easy to find them.

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 $\Delta T=30\ \mathrm{^\circ C}$.

Are you able to finish now ? :-)

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