Animated Solution for Chemistry - States of Matter: A copper complex crystallising in a ccp lattice with a cell edge of $0.4518\text{ nm}$ has been revealed by employing X-ray diffraction studies. The density of a copper complex is found to be $7.62{\text{ g cm}}^{-3}$. The molar mass of copper complex is ...... ${\text{g mol}}^{-1}$. (Nearest integer) [Given : $N_A=6.022\times 10^{23}{\text{ mol}}^{-1}$]

The problem of finding the molar mass of a substance from its crystal structure is a classic application of solid-state chemistry. It beautifully bridges the gap between the macroscopic world we can measure—like density—and the microscopic world of atoms and unit cells.

Imagine you are shrinking down to the atomic level and looking at a crystal of this copper complex. You would see a highly organized, repeating pattern of atoms. The smallest repeating unit of this pattern is called the unit cell.

In this problem, we are told the copper complex crystallizes in a cubic close-packed (ccp) lattice. Geometrically, a ccp lattice is identical to a face-centered cubic (fcc) lattice. This is a crucial piece of information because it tells us how many atoms effectively belong to one unit cell. In an fcc lattice, there are atoms at all eight corners (each shared by 8 adjacent cells) and atoms at the centers of all six faces (each shared by 2 adjacent cells).

Calculating the effective number of atoms, $Z$:

So, for our ccp lattice, $Z=4$.

How do we connect this microscopic picture to the macroscopic density given as $7.62{\text{ g cm}}^{-3}$? We use the master equation for crystal density:

Here, $d$ is the density, $Z$ is the number of atoms per unit cell, $M$ is the molar mass, $N_A$ is Avogadro's number, and $a^3$ is the volume of the cubic unit cell (where $a$ is the edge length).

Our goal is to find the molar mass, $M$. Let's rearrange the formula to isolate $M$:

Before we plug in the numbers, we must be extremely careful with our units. This is where many students make a silly mistake!

The density is given in ${\text{g cm}}^{-3}$, but the edge length $a$ is given in nanometers ($0.4518\text{ nm}$). We must convert the edge length to centimeters to ensure consistency.

Recall that $1\text{ nm}=10^{-9}\text{ m}$ and $1\text{ cm}=10^{-2}\text{ m}$. Therefore, $1\text{ nm}=10^{-7}\text{ cm}$.

Now, we have all our pieces ready. Let's substitute them into our rearranged formula:

First, let's calculate the volume of the unit cell, $a^3$:

Now, substitute this back into the equation:

Notice how the powers of 10 simplify nicely: $10^{23}\times 10^{-21}=10^2=100$.

The question asks for the nearest integer. Rounding off $105.79$, we get our final answer: $106{\text{ g mol}}^{-1}$.

Through careful unit conversion and application of the density formula, we've successfully unveiled the molar mass of the copper complex!