The Solid State

The Solid State

Question

Explain how you can determine the atomic mass of an unknown metal if you know its mass density and the dimensions of a unit cell of its crystal.

Answer

By knowing the density of an unknown metal and the dimension of its unit cell, the atomic mass of the metal can be determined.

Let ‘a’ be the edge length of a unit cell of a crystal,  ‘d’’  be the density of the metal, ‘m’ be the atomic mass of the metal and ‘z ’ be the number of atoms in the unit cell.
Now comma space density space of space the space unit space cell space equals space fraction numerator Mass space of space the space unit space cell over denominator Volume space of space the space unit space cell end fraction
rightwards double arrow space straight d space equals zm over straight a cubed space space.. left parenthesis straight i right parenthesis
left square bracket Since space mass space of space the space unit space cell space equals space number space of space atoms space in space the space unit space cell space straight x space Atomic right square bracket
left square bracket Volume space of space the space unit space cell equals space left parenthesis edge space length space of space the space unit space cell right parenthesis cubed right square bracket

From space equation space left parenthesis straight i right parenthesis comma space we space have colon space
rightwards double arrow straight d equals space da cubed over straight z space space left parenthesis ii right parenthesis thin space

Now comma space mass space of space the space metal space left parenthesis straight m right parenthesis space equals space fraction numerator Atomic space mass over denominator Avogadro apostrophe straight s space number space left parenthesis straight N subscript straight a right parenthesis end fraction
therefore comma space straight M space equals space fraction numerator da cubed straight N subscript straight a over denominator straight z end fraction space space left parenthesis iiii right parenthesis
If space the space edge space length space are space different space left parenthesis say space straight a comma space straight b space and space straight c right parenthesis comma space space then space equation space left parenthesis ii right parenthesis space becomes colon
rightwards double arrow space straight m equals space fraction numerator straight d left parenthesis abc right parenthesis straight N subscript straight A over denominator straight z end fraction space space space.. space left parenthesis iv right parenthesis space
from space equation space left parenthesis iii right parenthesis space and space left parenthesis iv right parenthesis comma space we space can space determine space the space atomic space mass space of space the space unknown space metal.

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