Answer
Verified
409.5k+ views
Hint: FCC is also called as face centered cubic structure. In FCC, atoms are arranged at the corners and center of each cube face. To solve this question, we will use the density formula and equate the two formulae of number of moles.
Formula used: The formulae used in this equation are as follows,
$ {\text{Density = }}\dfrac{{{\text{Molar mass }} \times {\text{ Z}}}}{{{{\text{N}}_a} \times {\text{ }}{{\text{a}}^3}{\text{ }}}} $
$ \dfrac{{{\text{Given mass}}}}{{{\text{Molar mass}}}}{\text{ = }}\dfrac{{{\text{Number of atoms}}}}{{{{\text{N}}_a}}} $ .
Complete Step by step solution:
The density of the FCC crystal lattice is given as 10g $ {\text{c}}{{\text{m}}^{{\text{ - 3}}}} $ . Let a be the edge length of the unit cell of FCC lattice. The edge length of the unit cell is 100pm. The Avogadro number is represented by $ {N_a} $ . Z is the coordination number and for FCC lattice, the value of Z is 4. Moles can be given by the number of atoms divided by the Avogadro number. It can also be represented as a given mass divided by molar mass.
The given mass of the crystal lattice is 1 g. The number of atoms present in 1 g of crystal lattice is given as follows,
$ {\text{Density = }}\dfrac{{{\text{Molar mass }} \times {\text{ Z}}}}{{{{\text{N}}_a} \times {\text{ }}{{\text{a}}^3}{\text{ }}}} $ …(i)
$ \dfrac{{{\text{Given mass}}}}{{{\text{Molar mass}}}}{\text{ = }}\dfrac{{{\text{Number of atoms}}}}{{{{\text{N}}_a}}} $
Hence, $ {\text{Molar mass = }}\dfrac{{{\text{Given mass }} \times {\text{ }}{{\text{N}}_a}}}{{{\text{Number of atoms}}}} $ …(ii)
Substituting (ii) in (i),
$ {\text{Density = }}\dfrac{{{\text{Given mass }} \times {\text{ }}{{\text{N}}_a}{\text{ }} \times {\text{ Z}}}}{{{\text{Number of atoms }} \times {\text{ }}{{\text{N}}_a} \times {\text{ }}{{\text{a}}^3}{\text{ }}}} $
$ {\text{Number of atoms = }}\dfrac{{{\text{Given mass }} \times {\text{ Z}}}}{{{{\text{a}}^3}{\text{ }} \times {\text{ density}}}} $
$ {\text{Number of atoms = }}\dfrac{{1 \times 4}}{{{{\left( {100 \times {{10}^{ - 10}}} \right)}^3} \times 10}}{\text{ = }}\dfrac{{4 \times 1}}{{10 \times {{10}^{ - 24}}}}{\text{ = 4}} \times {\text{1}}{{\text{0}}^{23}}{\text{ atoms}} $
Therefore, the number of atoms in 1 g of FCC crystal lattice is $ 4 \times {10^{23}} $ .
Note:
The smallest repeating unit of a crystal structure is known as the unit cell.
Coordination number is the number of the particles that each particle in a crystalline solid contacts. In other words, it is the number of the nearest neighbouring atoms or the ions surrounding an atom or an ion.
Formula used: The formulae used in this equation are as follows,
$ {\text{Density = }}\dfrac{{{\text{Molar mass }} \times {\text{ Z}}}}{{{{\text{N}}_a} \times {\text{ }}{{\text{a}}^3}{\text{ }}}} $
$ \dfrac{{{\text{Given mass}}}}{{{\text{Molar mass}}}}{\text{ = }}\dfrac{{{\text{Number of atoms}}}}{{{{\text{N}}_a}}} $ .
Complete Step by step solution:
The density of the FCC crystal lattice is given as 10g $ {\text{c}}{{\text{m}}^{{\text{ - 3}}}} $ . Let a be the edge length of the unit cell of FCC lattice. The edge length of the unit cell is 100pm. The Avogadro number is represented by $ {N_a} $ . Z is the coordination number and for FCC lattice, the value of Z is 4. Moles can be given by the number of atoms divided by the Avogadro number. It can also be represented as a given mass divided by molar mass.
The given mass of the crystal lattice is 1 g. The number of atoms present in 1 g of crystal lattice is given as follows,
$ {\text{Density = }}\dfrac{{{\text{Molar mass }} \times {\text{ Z}}}}{{{{\text{N}}_a} \times {\text{ }}{{\text{a}}^3}{\text{ }}}} $ …(i)
$ \dfrac{{{\text{Given mass}}}}{{{\text{Molar mass}}}}{\text{ = }}\dfrac{{{\text{Number of atoms}}}}{{{{\text{N}}_a}}} $
Hence, $ {\text{Molar mass = }}\dfrac{{{\text{Given mass }} \times {\text{ }}{{\text{N}}_a}}}{{{\text{Number of atoms}}}} $ …(ii)
Substituting (ii) in (i),
$ {\text{Density = }}\dfrac{{{\text{Given mass }} \times {\text{ }}{{\text{N}}_a}{\text{ }} \times {\text{ Z}}}}{{{\text{Number of atoms }} \times {\text{ }}{{\text{N}}_a} \times {\text{ }}{{\text{a}}^3}{\text{ }}}} $
$ {\text{Number of atoms = }}\dfrac{{{\text{Given mass }} \times {\text{ Z}}}}{{{{\text{a}}^3}{\text{ }} \times {\text{ density}}}} $
$ {\text{Number of atoms = }}\dfrac{{1 \times 4}}{{{{\left( {100 \times {{10}^{ - 10}}} \right)}^3} \times 10}}{\text{ = }}\dfrac{{4 \times 1}}{{10 \times {{10}^{ - 24}}}}{\text{ = 4}} \times {\text{1}}{{\text{0}}^{23}}{\text{ atoms}} $
Therefore, the number of atoms in 1 g of FCC crystal lattice is $ 4 \times {10^{23}} $ .
Note:
The smallest repeating unit of a crystal structure is known as the unit cell.
Coordination number is the number of the particles that each particle in a crystalline solid contacts. In other words, it is the number of the nearest neighbouring atoms or the ions surrounding an atom or an ion.
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
Mark and label the given geoinformation on the outline class 11 social science CBSE
When people say No pun intended what does that mea class 8 english CBSE
Name the states which share their boundary with Indias class 9 social science CBSE
Give an account of the Northern Plains of India class 9 social science CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
Trending doubts
Difference Between Plant Cell and Animal Cell
Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
Give 10 examples for herbs , shrubs , climbers , creepers
10 examples of evaporation in daily life with explanations
Write a letter to the principal requesting him to grant class 10 english CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
Name 10 Living and Non living things class 9 biology CBSE