Suppose that 93.3 g of an unknown metal at 98.6°C is immersed in 200.0g of water at 20.5°C. After coming to thermal equilibrium, the final temperature ofthe metal and the water is 22.7°C. Given the approximate heat capacity for water of 4.18 J/g°C, calculate the heat capacity of the metal. Also, calculate the approximate molar mass of the metal using the Law of Dulong & Petitand identify the element. Show your work. Explain why your numerical answers are limited to 2 significant figures.

1. Heat capacity: 24J/ºC. This is the full heat capacity. See below for explanation and the specific heat capacity.

2. The approximate molar mass is 25g/mol and the element is magnesium.

3. Tha calculations are limited to two significant figures for two reasons:

i) the temperature difference 22.7ºC - 20.5ºC = 2.2ºC contains two significant figures, and

ii) Dulong & Petit's Law is an approximation.

Explanation:

1. Heat capacity

Assuming that the two metals are isolated from the surroundings, by the law of conservation of energy, the heat released by the hotter substance is equal to the heat absorbed by the colder substance:

Heat released by the metal = Heat absorbed by water

Heat = mass * specific heat * change in temperature

Call Sh the specific heat:

93.3g * Sh * (98.6°C - 22.7°C) = 200.0g * 4.18 J/g°C * (22.7°C - 20.5°C)

Compute and clear Sh:

Sh = 0.2597 g/J'C ≈ 0.26J/gºC

Notice that the temperature difference 22.7 - 20.5 = 2.2 only contains two significant figures which makes that your final calculation is limited to two significant figures.

The full heat capacity is the product of the specific heat capacity by the mass:

Full heat capacity = 0.2597J/gºC * 93.3g ≈ 24 J/ºC ← answer

Note that the question is the heat capacity, not the specific heat capacity.

2. Approximate molar mass of the metal

Dulong & Petit's Law states an approximate form to estimate the molar mass of a solid by relating the specifig heat capacity and the molar mass:

Specific heat capacity * molar mass ≈ 6.4

Then:

Molar mass ≈ 6.4 / specific heat Molar mass ≈ 6.4 / 0.26 ≈ 24.6 Molar mass ≈ 25 g/mol

Dulong&Petit's Law is just an approximation.

From a periodic table, the metal element whose atomic mass (the same as molar mass for elements) is closer to 25g/mol is magnesium, with an atomic mass of 24.3g/mol. Thus, your element most likely is magnesium.