An unknown metal is found to have a density of 19.300 g/cm^3 and to crystallize in a body-centered cubic lattice. The edge of the unit cell is found to be 0.31627 nm.
Calculate the atomic mass of the metal.
Im not sure how to approach this problem...when i tried doing it i got an answer of 3.1*10^-7..but it was hard...can someone please explain how to do this problem
thank you
Convert a in nm to cm.
Then volume of the unit cell is a^3.
mass of unit cell = volume x density
Calculate mass unit cell since you have volume and density. Substitute for mass of unit cell below.
Then mass of the unit cell =
#atoms in unit cell which is 2 x molar mass/6.022E23
a = (3.1627E-8 cm)^3 = about 3E-23 cc but you need to do it more exactly.
mass = volume x density = about 3E-23 cc x 19.3 g/cc = about 6E-22.
Then 6E-22 = 2*molar mass/6.022E-23 = and solve for molar mass. I get something like 180 but you need to do it more accurately. My guess is W.
Well, approaching problems can be as tricky as trying to juggle with watermelons! But fear not, I'm here to guide you through the metal mystery.
To find the atomic mass of the metal, we need to take a closer look at the information we have. We know that the metal crystallizes in a body-centered cubic lattice, which means that each unit cell contains 2 atoms.
The density of the metal is given as 19.300 g/cm^3, which is the mass per unit volume. Since the unit cell contains 2 atoms, we can say that the mass of each unit cell is equal to twice the atomic mass.
Now, let's start putting the puzzle pieces together. First, we need to convert the given edge length from nm (nanometers) to cm (centimeters). Remember that 1 nm is equal to 1 x 10^-7 cm.
So, the edge length of the unit cell is 0.31627 nm * 1 x 10^-7 cm/nm = 3.1627 x 10^-8 cm.
Next, let's calculate the volume of the unit cell using the equation:
Volume = (edge length)^3
Volume = (3.1627 x 10^-8 cm)^3 = 3.1627 x 10^-8 cm * 3.1627 x 10^-8 cm * 3.1627 x 10^-8 cm = 3.1627 x 10^-8 cm^3
Now, we can finally find the mass of each unit cell using the given density:
Mass = Density * Volume
Mass = 19.300 g/cm^3 * 3.1627 x 10^-8 cm^3 = 6.1099 x 10^-7 g
Since the mass of each unit cell is twice the atomic mass, we can now find the atomic mass of the metal by dividing the mass of the unit cell by 2:
Atomic mass = Mass of unit cell / 2
Atomic mass = (6.1099 x 10^-7 g) / 2 = 3.05495 x 10^-7 g
So, the atomic mass of the metal is approximately 3.055 x 10^-7 g.
Keep in mind that this is just a simple explanation, and the actual calculations can get as complex as juggling chainsaws. But don't worry, you did great in attempting this problem!
To calculate the atomic mass of the metal, we need to use the information given about the density, lattice structure, and unit cell edge length. Here's a step-by-step approach to solve this problem:
Step 1: Determine the volume of the unit cell
For a body-centered cubic (BCC) lattice, the volume of the unit cell can be calculated using the formula:
V = (a^3)/2
Where V is the unit cell volume and a is the edge length of the unit cell. In this case, a = 0.31627 nm = 0.31627 x 10^(-7) cm. Plugging in the values:
V = (0.31627 x 10^(-7))^3 / 2
V ≈ 3.992 x 10^(-23) cm^3
Step 2: Determine the mass of the unit cell
Since the density is given as 19.300 g/cm^3, we can use it to find the mass of the unit cell.
The mass (m) can be calculated using the equation:
m = V x density
Plugging in the values:
m ≈ (3.992 x 10^(-23)) x 19.300 g
m ≈ 7.709 x 10^(-22) g
Step 3: Determine the number of atoms in the unit cell
For a body-centered cubic lattice, there are two atoms per unit cell. So, the mass calculated in the previous step (m) is the combined mass of two atoms.
Step 4: Calculate the atomic mass
To find the atomic mass of a single atom, divide the mass of the unit cell (m) by the number of atoms in the unit cell. In this case, m represents the combined mass of two atoms.
Therefore, the atomic mass (M) can be calculated using the equation:
M = m / 2
Plugging in the values:
M ≈ (7.709 x 10^(-22) g) / 2
M ≈ 3.855 x 10^(-22) g
Thus, the approximate atomic mass of the unknown metal is 3.855 x 10^(-22) g.
To calculate the atomic mass of the metal, you need to follow a series of steps. Let's break it down:
Step 1: Determine the volume of the unit cell.
Given that the edge of the unit cell is 0.31627 nm, we can calculate the volume of the unit cell using the formula:
Volume = (edge length)^3
Volume = (0.31627 nm)^3
Since the atomic mass unit (amu) is usually expressed in grams, we need to convert the volume from nm^3 to cm^3. 1 nm = 10^-7 cm.
So, we have:
Volume = (0.31627 nm)^3 × (10^-7 cm/nm)^3
Step 2: Convert the density from g/cm^3 to g/nm^3.
Given that the density of the metal is 19.300 g/cm^3, we can convert it to g/nm^3 using the conversion factor 1 cm = 10^7 nm.
So, we have:
Density = 19.300 g/cm^3 × (10^7 nm/cm)^3
Step 3: Calculate the mass of the unit cell.
Now, we can use the density and volume to find the mass of the unit cell:
Mass = Density × Volume
Step 4: Calculate the number of atoms in the unit cell.
In a body-centered cubic (BCC) lattice, there are two atoms per unit cell.
So, the number of atoms in the unit cell is 2.
Step 5: Calculate the atomic mass.
Finally, we can calculate the atomic mass of the metal:
Atomic Mass = Mass of Unit Cell / Number of Atoms in Unit Cell
Now, let's perform the calculations:
Step 1:
Volume = (0.31627 nm)^3 × (10^-7 cm/nm)^3
Step 2:
Density = 19.300 g/cm^3 × (10^7 nm/cm)^3
Step 3:
Mass = Density × Volume
Step 4:
Number of atoms in the unit cell = 2
Step 5:
Atomic Mass = Mass of Unit Cell / Number of Atoms in Unit Cell
Using these steps, you can solve the problem and obtain the correct atomic mass of the metal.