A 10. g cube of copper at a temperature T1 is placed in an insulated cup containing 10. g of water a — Thermodynamics Chemistry Question

STEPS:

1. Apply the law of conservation of energy: Because the copper and water are placed inside an insulated cup, we can assume a closed thermodynamic system where no heat is lost to or gained from the surroundings. Therefore, the magnitude of heat lost by the hotter copper cube must equal the magnitude of heat gained by the cooler water when they reach thermal equilibrium:
$$|q_{\text{copper}}| = |q_{\text{water}}|$$
2. Relate heat transfer to temperature change: The heat transferred ($q$) is determined by the calorimetry formula:
$$q = mc\Delta T$$
where $m$ is the mass of the substance, $c$ is the specific heat capacity, and $\Delta T$ is the change in temperature.
3. Set up the mathematical equality: Equating the magnitude of heat exchanged for both substances yields:
$$m_{\text{Cu}} \cdot c_{\text{Cu}} \cdot |\Delta T_{\text{Cu}}| = m_{\text{H}_2\text{O}} \cdot c_{\text{H}_2\text{O}} \cdot |\Delta T_{\text{H}_2\text{O}}|$$
4. Simplify the equation using the given values: The mass of both samples is identical ($10.\text{ g}$). Substituting this and the given specific heat capacities into the equation:
$$10.\text{ g} \times 0.385\text{ J/(g}\cdot^\circ\text{C)} \times |\Delta T_{\text{Cu}}| = 10.\text{ g} \times 4.18\text{ J/(g}\cdot^\circ\text{C)} \times |\Delta T_{\text{H}_2\text{O}}|$$
Since the masses are equal, they cancel out, simplifying the relationship to:
$$0.385 \times |\Delta T_{\text{Cu}}| = 4.18 \times |\Delta T_{\text{H}_2\text{O}}|$$
5. Compare the magnitude of temperature changes: Rearranging the simplified equation to find the ratio of the temperature changes:
$$\frac{|\Delta T_{\text{Cu}}|}{|\Delta T_{\text{H}_2\text{O}}|} = \frac{4.18}{0.385} \approx 10.9$$
This mathematical relationship shows that the temperature change of the copper ($|\Delta T_{\text{Cu}}|$) is approximately 11 times greater than the temperature change of the water ($|\Delta T_{\text{H}_2\text{O}}|$).
6. Identify the core concept: Water has an exceptionally high specific heat capacity compared to metals like copper. This means water requires far more energy to change its temperature by $1^\circ\text{C}$ than copper does. Under conditions of equal mass and equal heat exchange, the substance with the lower specific heat capacity (copper) experiences a much larger temperature change, confirming Option A is correct.

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WHY_OTHERS_WRONG:

• B is incorrect: The temperature of the water changes significantly less than that of the copper. Because water has a high specific heat capacity, it resists rapid changes in temperature, acting as a highly efficient thermal buffer.
• C is incorrect: The temperature changes would only be identical if the overall heat capacities ($m \cdot c$) of the two substances were equal. Because the masses are equal but their specific heats differ by a factor of nearly eleven, their temperature changes must also differ by that same factor.
• D is incorrect: The relative changes in temperature can be determined without knowing the exact initial temperatures $T_1$ and $T_2$. Because the heat lost equals the heat gained and the masses are equal, the ratio of their temperature changes is fixed and entirely governed by the ratio of their specific heat capacities.