Two different metals, A and B, have identical densities and identical atomic radii. Metal A crystallises in a face centred cubic (fcc) lattice, while metal B crystallises in a body centred cubic (bcc) lattice. The ratio of the molar mass of metal A to the molar mass of metal B ( M_ A M_ B ) is:

Step-by-step solution

The density of a unit cell is given by = Z M N_ A a³ . Rearranging for molar mass M : M = N_ A a³ Z The ratio of the molar masses is: M_ A M_ B = _ A N_ A a_ A ³ Z_ A Z_ B _ B N_ A a_ B ³ Since the densities are identical ( _ A = _ B ), this simplifies to: M_ A M_ B = Z_ B Z_ A ( a_ A a_ B )³ For metal A (fcc lattice): Z_ A = 4 a_ A = 2 2 r For metal B (bcc lattice): Z_ B = 2 a_ B = 4 3 r The ratio of the number of atoms per unit cell is: Z_ B Z_ A = 2 4 = 1 2 The ratio of the edge lengths is: a_ A a_ B = 2 2 r 4 3