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Atomic Weight is defined as the mass of a single atom of the element.  Now if you wish to express the mass of a single H atom in gram, the number you will get is :

0.0000000000000000000000017 g

This is a very tedious way to express the mass of a single atom of Hydrogen.  Chemists thus thought of defining a different unit for expressing Atomic Weights.  For this, they chose one of the elements as standard element and thus the mass of other elements can be expressed in terms of that element. 

You can understand this concept by thinking about the old days when weighing balances were used to calculate mass of substances. 

Let us assume, we have 1 kg standard weight.  Now, if for measuring the weight of an object you have to use 5 standard weights of 1 kg each, you mentally calculate :

The object is 5 times as heavy as one standard weight,

i.e. weight of the object = 5 × 1 kg = 5 kg

Similarly for atomic weight, one atom of Hydrogen was chosen as standard and it was assigned mass of 1 a.m.u. ( atomic mass unit). 

Now, if 1 atom of an element is 4 times as heavy as one atom of H, we can say that

Mass of 1 atom of element (Atomic Weight) = 4 × 1 a.m.u.= 4 a.m.u. 

Thus, Atomic Weight = No.  of times an atom of element is heavier than one atom of H.

But the atomic weights calculated by choosing H atom as standard had errors in them.  Thus the standard was changed from Hydrogen to Carbon. 

The newly defined standard is based on C-12 isotope. 

1 atom of C-12 is assigned a mass of 12 a.m.u. 

Thus, 1 a.m.u.  = (1/12)th of mass of C-12 atom

Now, if 1 atom of an element 16 times as heavy as (1/12)th of mass of C-12 atom

Then, Atomic weight of that element = 16 × 1 a.m.u.  = 16 a.m.u. 

According to the new definition,

Atomic Weight = No.  of times an atom of element is heavier than (1/12)th of mass of C-12 atom

The unit of Atomic Weight is atomic mass unit (a.m.u.). 

The conversion of a.m.u.  into grams is : 1 a.m.u.  = 1.66 × 10-24 g