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Chapter 14: Problem 95

A 2.0 -kg block of copper at \(100.0^{\circ} \mathrm{C}\) is placed into $1.0 \mathrm{kg}$ of water in a 2.0 -kg iron pot. The water and the iron pot are at \(25.0^{\circ} \mathrm{C}\) just before the copper block is placed into the pot. What is the final temperature of the water, assuming negligible heat flow to the environment?

Short Answer

Expert verified
Answer: The final temperature of the water is approximately \(7.68^{\circ}C\).

Step by step solution

01

List the known values

- Mass of copper block: \(m_{Cu} = 2.0 \, kg\) - Initial temperature of copper block: \(T_{Cu(initial)} = 100.0^{\circ}C\) - Mass of water: \(m_{water} = 1.0 \, kg\) - Initial temperature of the water and the pot: \(T_{water(initial)} = T_{iron(initial)} = 25.0^{\circ}C\) - Mass of iron pot: \(m_{iron} = 2.0 \, kg\) - Specific heat capacities: \(c_{Cu} = 387 \, J/(kg \cdot K), c_{water} = 4186 \, J/(kg \cdot K), c_{iron} = 449 \, J/(kg \cdot K)\).
02

Apply the conservation of energy principle

As no energy is lost to the environment, the energy gained by water and the iron pot will be equal to the energy lost by the copper block. Therefore, we can write the conservation of energy as: $$m_{Cu}c_{Cu}(T_{Cu(initial)} - T_{final}) = m_{water}c_{water}(T_{final} - T_{water(initial)}) + m_{iron}c_{iron}(T_{final} - T_{iron(initial)})$$
03

Plug in the known values and solve for the final temperature

Substitute the known values into the equation: $$2.0 \, kg \cdot 387 \, J/(kg \cdot K)(100.0^{\circ}C - T_{final}) = 1.0 \, kg \cdot 4186 \, J/(kg \cdot K)(T_{final} - 25.0^{\circ}C) + 2.0 \, kg \cdot 449 \, J/(kg \cdot K)(T_{final} - 25.0^{\circ}C)$$ Now, simplify and solve for \(T_{final}\): $$774(100 - T_{final}) = 4186(T_{final} - 25) + 898(T_{final} - 25)$$ $$77400 - 774T_{final} = 4186T_{final} - 104650 + 898T_{final} - 22450$$ $$-774T_{final} + 4186T_{final} + 898T_{final} = -33100$$ $$4310T_{final} = 33100$$ $$T_{final} \approx 7.68^{\circ}C$$ The final temperature of the water (and the iron pot) will be approximately \(7.68^{\circ}C\).

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Most popular questions from this chapter

A metal rod with a diameter of \(2.30 \mathrm{cm}\) and length of $1.10 \mathrm{m}\( has one end immersed in ice at \)32.0^{\circ} \mathrm{F}$ and the other end in boiling water at \(212^{\circ} \mathrm{F}\). If the ice melts at a rate of 1.32 g every 175 s, what is the thermal conductivity of this metal? Identify the metal. Assume there is no heat lost to the surrounding air.
A piece of gold of mass \(0.250 \mathrm{kg}\) and at a temperature of \(75.0^{\circ} \mathrm{C}\) is placed into a \(1.500-\mathrm{kg}\) copper pot containing \(0.500 \mathrm{L}\) of water. The pot and water are at $22.0^{\circ} \mathrm{C}$ before the gold is added. What is the final temperature of the water?
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