A particular metallic element, M, has a heat capacity of 0.36 J*g*K, and it forms an oxide that contains 2.90 grams of M per gram of oxygen.
(a) Using the law of Dulong and Petit (Cp x M = 25 J*mol*K), estimate the molar mass of the metal M [Answer: _____ g/mol]
(b) From the composition of the oxide (2.90 g M/g O) and the molar mass of oxygen (16 g/mol), determine the mass of M that combines with each mole of oxygen [Answer: _____ g M/mol O]
(c) Use your estimate of the atomic mass from question (a) and the information from question (b) to determine the empirical formula of the oxide (the moles of M that combine with each mole of O) [Answer: _____]
(d) Now that you know the composition of the oxide and its formula, what is the accurate value of the atomic mass of the metal M? What is the identity of M? [Answer: _____ g/mol]
To find the answers, let's go step by step:
(a) According to the law of Dulong and Petit, the product of the heat capacity (Cp) and the molar mass (M) of an element is approximately a constant value of 25 J*mol*K. We are given that the heat capacity of metal M is 0.36 J*g*K, so we need to convert it to J*mol*K:
0.36 J*g*K * (1 mol / Molar mass of M) = 0.36 J / (M/Molar mass of M)
Since the product of Cp and M is a constant, we can solve for the molar mass of M:
0.36 J / (M/Molar mass of M) = 25 J*mol*K
Solving for Molar mass of M, we get:
M/Molar mass of M = 0.36 J / 25 J*mol*K
Molar mass of M = 0.36 J / 25 J*mol*K * M
Therefore, the molar mass of metal M is given by the expression 0.36 J / 25 J*mol*K * M.
(b) We're given that the oxide contains 2.90 grams of M per gram of oxygen. We need to determine the mass of M that combines with each mole of oxygen.
Since the molar mass of oxygen is 16 g/mol, we can calculate the mass ratio of M to O:
Mass ratio of M to O = 2.90 g M / 1 g O
To determine the mass of M that combines with each mole of oxygen, we multiply the mass ratio by the molar mass of oxygen:
Mass of M/mol O = (2.90 g M / 1 g O) * (16 g O / 1 mol O)
Therefore, the mass of M that combines with each mole of oxygen is given by the expression (2.90 g M / 1 g O) * (16 g O / 1 mol O).
(c) To determine the empirical formula of the oxide, we need to know the moles of M that combine with each mole of O.
Using the molar mass of M determined in part (a) and the mass of M/mol O determined in part (b), we can calculate the ratio of moles of M to O:
Moles of M/mol O = (Mass of M/mol O) / Molar mass of M
Therefore, the moles of M that combine with each mole of O is given by the expression (Mass of M/mol O) / Molar mass of M.
(d) Now that we know the composition of the oxide and its formula, we can determine the accurate value of the atomic mass of the metal M.
Knowing the empirical formula allows us to find the empirical formula weight of the compound. In this case, we can use the moles of M and O determined in parts (b) and (c) respectively, and the molar mass of oxygen to calculate the empirical formula weight:
Empirical formula weight = (Molar mass of M * Moles of M) + (Molar mass of O * Moles of O)
Once we have the empirical formula weight, we compare it to the actual formula weight of the compound to find the multiple of the empirical formula. This multiple corresponds to the number of M atoms in one molecule of the compound and leads us to the accurate value of the atomic mass of M.
Therefore, the identity of the metal M can be determined by finding its atomic mass and comparing it to known elements. Once the atomic mass is determined, it can be used to identify the metal M.