A 32.8g iron rod, initially at 22.3 C, is submerged into unkown mass of water at 62.7 C, in an insulated container

A 32.8g iron rod, initially at 22.3 C, is submerged into unkown mass of water at 62.7 C, in an insulated container. The final of the mixture upon reaching thermal equilibrium is 58.8 C

The correct answer and explanation is:

Correct Answer: The unknown mass of water is approximately 7.0 grams.

To solve this thermal equilibrium problem, we apply the principle of conservation of energy. This principle states that heat lost by the hot object is equal to heat gained by the cold object, assuming no heat is lost to the surroundings due to the insulated container.

Given:

• Mass of iron rod: 32.8 g
• Initial temperature of iron rod: 22.3 °C
• Final temperature of the system: 58.8 °C
• Initial temperature of water: 62.7 °C
• Specific heat capacity of iron: 0.449 J/g°C
• Specific heat capacity of water: 4.18 J/g°C
• Mass of water: unknown (let’s call it m)

Step-by-step:

1. Heat gained by the iron rod:

qiron=m⋅c⋅ΔT=32.8⋅0.449⋅(58.8−22.3)=32.8⋅0.449⋅36.5≈537.1 Jq_{\text{iron}} = m \cdot c \cdot \Delta T = 32.8 \cdot 0.449 \cdot (58.8 – 22.3) = 32.8 \cdot 0.449 \cdot 36.5 \approx 537.1 \text{ J}

2. Heat lost by the water:

qwater=m⋅4.18⋅(62.7−58.8)=m⋅4.18⋅3.9≈m⋅16.3 Jq_{\text{water}} = m \cdot 4.18 \cdot (62.7 – 58.8) = m \cdot 4.18 \cdot 3.9 \approx m \cdot 16.3 \text{ J}

3. Set heat gained by iron equal to heat lost by water:

537.1=16.3m537.1 = 16.3m m=537.116.3≈32.9 gm = \frac{537.1}{16.3} \approx 32.9 \text{ g}

Correction: This seems reversed. Iron is colder initially and is gaining heat; water is hotter and is losing heat. So:

• Heat gained by iron = heat lost by water

Let’s re-calculate properly with this in mind:

Final temperature is 58.8°C, so:

• ΔT for iron = 58.8 − 22.3 = 36.5°C (gaining heat)
• ΔT for water = 62.7 − 58.8 = 3.9°C (losing heat)

Recalculate with: qiron=32.8⋅0.449⋅36.5=537.1 Jq_{\text{iron}} = 32.8 \cdot 0.449 \cdot 36.5 = 537.1 \text{ J} qwater=m⋅4.18⋅3.9=m⋅16.3q_{\text{water}} = m \cdot 4.18 \cdot 3.9 = m \cdot 16.3 537.1=16.3m⇒m=537.116.3≈32.9 g537.1 = 16.3m \Rightarrow m = \frac{537.1}{16.3} \approx 32.9 \text{ g}

Final Answer: 32.9 grams of water

Explanation (300 words):

This problem is an example of thermal equilibrium, where two substances at different temperatures exchange heat until they reach the same final temperature. The fundamental rule here is conservation of energy — meaning that heat lost by one substance is gained by the other. No energy is lost to the environment because the container is insulated.

In this case, an iron rod at a lower temperature is placed in water that is warmer. The iron absorbs heat and increases in temperature, while the water loses heat and cools down until both reach the same final temperature. To determine the unknown mass of water, one must use the heat transfer formula:
q = m × c × ΔT,
where q is heat, m is mass, c is specific heat capacity, and ΔT is the change in temperature.

We calculate the heat absorbed by the iron rod using its specific heat and temperature change. Next, using the specific heat of water and the water’s temperature drop, we express the heat lost by water in terms of its unknown mass. By equating the heat gained by the iron and the heat lost by the water, we can solve for the unknown mass.

The calculation shows that the water must have a mass of approximately 32.9 grams to result in the observed final temperature. This method assumes no heat is transferred to the surroundings and only the iron and water interact thermally. This kind of calculation is common in calorimetry and helps determine unknown properties in physical systems.