Question

The specific heat of metal is 0.67 J/g. Its equivalent mass is 20. What is its exact atomic mass:
(A) 46
(B) 20
(C) 40
(D) 22

Answer 1

Hint: (1) The specific heat of a substance refers to the amount of heat per unit mass which is required to raise the temperature by one degree Celsius.
(2) Atomic mass of an element refers to the average atomic mass of the atoms of an element measured in atomic mass unit (amu).

Complete step-by-step answer: Given, the specific heat of a metal $\text{ = 0}\text{.67J/g}$
Also given, the equivalent mass of the metal $=20$
To find: the exact atomic mass of the metal.
The relationship between the specific heat and the atomic mass of an element is given by the Dulong-Petit law. According to the Dulong-Petit law, the gram-atomic heat capacity is constant, i.e. the product of the specific heat and the atomic mass of an element is a constant, approximately equal to 6.4. This is the same for all solid elements, about six calories per gram atom.
First, let us convert the specific heat from Joule per gram into calorie per gram.
 We know,
 $$\begin{gathered} \text{1calorie = 4}\text{.184Joule} \\ \Rightarrow \text{1Joule = }{\displaystyle \frac{\text{1}}{\text{4}\text{.184}}}\text{calorie} \end{gathered}$$
So, the specific heat of the metal in calorie per gram $\text{ = 0}\text{.67J/g = }{\displaystyle \frac{\text{0}\text{.67}}{\text{4}\text{.184}}}\text{cal/g = 0}\text{.16cal/g}$
According to Dulong-Petit law, approximate atomic mass × specific heat $=6.4$
Therefore, the approximate atomic mass $\text{ = }{\displaystyle \frac{\text{6}\text{.4}}{\text{specificheat}}}\text{ = }{\displaystyle \frac{\text{6}\text{.4}}{\text{0}\text{.16}}}\text{ = 40g}$
Given, the equivalent mass of the metal $=20$
We know that valency is equal to the approximate atomic mass of the metal divided by the equivalent mass of the metal. So, valency $={\displaystyle \frac{40}{20}}=2$
Therefore, exact atomic mass = valency × equivalent mass $\text{ = 2 times 20 = 40g}$
Hence, the exact atomic mass of the metal is equal to 40g. So, the correct option is (C).

Additional information:
An equivalent statement of Dulong-Petit law is ${\displaystyle \frac{\text{C}}{\text{n}}}\text{ = 3R}$ where C is the heat capacity of the substance, n is the number of moles of the substance and R is the gas constant.

Note: The Dulong-Petit law fails at room temperature for light atoms like Be, B, C etc. because in case of these atoms, the law gives prediction of higher heat capacities than that which are actually found. This difference is because of the high energy vibrational modes that are not populated.