The law of Dulong and Petit states that the heat capacity of metallic elements is approximately 25 J/mol · °C at 25 °C. In the 19th century, scientists used this relationship to obtain approximate atomic masses of metals, from which they determined the formulas of compounds. Once the formula of a compound of the metal with an element of known atomic mass is known, the mass percentage composition of the compound is used to find the atomic mass of the metal. The following example shows the calculations involved.

(a) Experimentally, the specific heat of a metal is found to be 0.24 J/g · °C. Use the law of Dulong and Petit to calculate the approximate atomic mass of the metal.

(b) An oxide of this element is 6.90% oxygen by mass. Use the molar mass of 16.00 g/mol for oxygen and the approximate atomic mass found in part (a) to determine the subscripts x and y in the formula of the oxide, M_xO_y. (The mole ratio of the elements you find will not be exactly whole numbers, so considerable rounding is needed to obtain whole numbers in the formula.)

(c) From the formula established in part (b), x mol M are combined with y mol O. Calculate the mass of the metal that is combined with y mol O, using the percent composition of the oxide, and find the atomic mass of the metal. What is the element M?