1. **State the problem:** We have a 2.0 kg metal block with specific heat capacity $c = 450$ J/kg°C. It is heated from 30°C to 80°C, doing 400 J of work on the surroundings during expansion. We need to find how much heat is released when it cools back from 80°C to 30°C without doing work. 2. **Calculate the heat added during heating:** The heat added to increase the temperature is given by $$ Q = mc\Delta T $$ where - $m = 2.0$ kg - $c = 450$ J/kg°C - $\Delta T = 80 - 30 = 50$ °C Substitute values: $$ Q = 2.0 \times 450 \times 50 = 45000 \text{ J} $$ 3. **Calculate internal energy change during heating:** From the first law of thermodynamics: $$ \Delta U = Q - W $$ where - $Q = 45000$ J (heat added) - $W = 400$ J (work done by the block) So, $$ \Delta U = 45000 - 400 = 44600 \text{ J} $$ 4. **Cooling process (return to 30°C) without doing work:** Since no work is done during cooling, from the first law: $$ \Delta U = Q \quad \Rightarrow \quad Q = \Delta U $$ The internal energy change for cooling back is: $$ \Delta U = -44600 \text{ J} $$ (negative because the block is losing energy) Therefore, $$ Q = -44600 \text{ J} $$ 5. **Interpretation:** Heat released during cooling is positive in magnitude: $$ \text{Heat released} = 44600 \text{ J} $$