Question

A titanium cube contains 2.86×1023 atoms. What is the edge length l of the cube? The density of titanium is 4.50 g/cm3 . (The volume of a cube is V=l3.) Express the length in centimeters to three significant figures.

Answer 1

I = 1.716 cm

Step-by-step explanation:

We want the length of the cube. In this case, we know that the volume of a cube is:

V = I³

From here we can solve for I:

I = ∛V (1)

Now, the volume can be obtained using data of density. The expression for density is:

d = m/V

Solvinf for V:

V = m/d (2)

The mass of the cube, can be obtained with the moles and molar mass of titanium:

m = n * MM (3)

And finally, the moles (n) can be obtained using avogadro number:

1 mol ----> 6.02x10²³ atoms (4)

Therefore, to get the length we need to calculate with the atoms of titanium, the moles, then mass, volume and finally the length. Let's calculate the moles of titanium using the ratio of (4):

nTi = 2.86x10²³ / 6.02x10²³ = 0.475 moles

Now, let's calculate the mass using (3) and the molecular mass of titanium which is 47.867 g/mol:

m = 0.475 * 47.867 = 22.737 g

Let's replace this value in (2) to get the volume:

V = 22.737 / 4.5 = 5.053 cm³

Finally, the length using (1):

I = ∛5.053

I = 1.716 cm

This is the length of the cube