50. A 50.0 g sample of Fe at 100°C is added to 500.0 mL of water at 35°C in a perfectly insulated co — Thermodynamics Chemistry Question

STEPS:

1. Identify the system and its constraints: The problem states that the iron (Fe) and water are mixed inside a perfectly insulated container. In thermodynamics, a perfectly insulated container represents a closed, adiabatic system where no heat can be exchanged with the outside surroundings.
2. Apply the First Law of Thermodynamics (Conservation of Energy): According to the law of conservation of energy, energy cannot be created or destroyed, only transferred. Therefore, inside this insulated calorimeter, any thermal energy (heat, $q$) released by the cooling system must be completely and exclusively absorbed by the warming system:
$$q_{\text{lost}} = -q_{\text{gained}}$$
3. Determine the direction of heat flow: Since the iron sample is initially at a higher temperature ($100^\circ\text{C}$) than the water ($35^\circ\text{C}$), thermal energy will spontaneously transfer from the hot iron to the colder water until they reach thermal equilibrium.
4. Relate the heat quantities: Because no thermal energy escapes the insulated container, the total magnitude of heat energy lost by the cooling iron must be exactly equal to the magnitude of heat energy gained by the warming water. This directly identifies Option D as the true statement.

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WHY_OTHERS_WRONG:

• Option A is incorrect: Because thermal energy spontaneously transfers from the hotter substance ($100^\circ\text{C}$ iron) to the colder substance ($35^\circ\text{C}$ water), the water will absorb heat. This absorption of heat causes the temperature of the water to increase, not decrease.
• Option B is incorrect: The magnitude of temperature change ($\Delta T$) depends on mass ($m$), specific heat capacity ($c$), and heat exchanged ($q$) through the relationship $q = mc\Delta T$. Although the heat exchanged ($q$) is equal for both substances, water has a much larger specific heat capacity ($\approx 4.18\text{ J/g}^\circ\text{C}$) than iron ($\approx 0.45\text{ J/g}^\circ\text{C}$) and is present in a much larger mass ($500.0\text{ g}$ vs. $50.0\text{ g}$). Consequently, the water's temperature will increase only slightly, while the iron's temperature will drop dramatically.
• Option C is incorrect: If the heat lost by the Fe were less than the heat gained by the water, it would mean that additional energy was spontaneously created inside the container, violating the fundamental law of Conservation of Energy.