A water (H2O) molecule consists of two hydrogen atoms and one oxygen atom. The oxygen atom is located at the origin. One hydrogen atom is between the positive x-axis and positive y-axis and the other hydrogen atom is between the negative x-axis and the positive y-axis. Each hydrogen atom makes an angle of 60º with the positive y-axis and they are both a distance d = 0.0957 Å from the origin. Draw a rough sketch of the water molecule and then calculate the center of mass of the water molecule. The mass of oxygen is 16.0 g and of hydrogen is 1.0 g.
To draw a rough sketch of the water molecule, we can use a Cartesian coordinate system. The oxygen atom is located at the origin (0, 0), while one hydrogen atom is positioned between the positive x-axis and y-axis, and the other hydrogen atom is positioned between the negative x-axis and positive y-axis. Each hydrogen atom makes a 60º angle with the positive y-axis. The distance from the origin to each hydrogen atom is d = 0.0957 Å.
The sketch should look like this:
H
----
\
O
/
----
H
Now let's calculate the center of mass of the water molecule. The center of mass is a weighted average of the positions of the atoms, with the masses of each atom serving as the weights.
First, calculate the x-coordinate of the center of mass (CMx). Since the oxygen atom is located at the origin, its x-coordinate is 0. The x-coordinate of each hydrogen atom can be determined using trigonometry:
HX = d * cos(60º)
HX = 0.0957 Å * cos(60º)
HX = 0.048 Å
Let's denote the mass of the oxygen atom as MO and the mass of the hydrogen atom as MH. Given that MO = 16.0 g and MH = 1.0 g, the total mass of the water molecule (MT) is:
MT = MO + 2 * MH
MT = 16.0 g + 2 * 1.0 g
MT = 18.0 g
Now, calculate the x-coordinate of the center of mass, CMx, using the formula:
CMx = (MO * 0 + MH * HX1 + MH * HX2) / MT
Since we have two hydrogen atoms in the water molecule, we need to calculate their x-coordinates (HX1 and HX2):
HX1 = HX = 0.048 Å
HX2 = -HX = -0.048 Å
Substituting the values into the formula:
CMx = (16.0 g * 0 + 1.0 g * 0.048 Å + 1.0 g * (-0.048 Å)) / 18.0 g
CMx = -0.0016 Å
The x-coordinate of the center of mass is approximately -0.0016 Å.
Next, let's calculate the y-coordinate of the center of mass (CMy). The y-coordinate of each hydrogen atom can also be determined using trigonometry:
HY = d * sin(60º)
HY = 0.0957 Å * sin(60º)
HY = 0.083 Å
Now, calculate CMy using the formula:
CMy = (MO * 0 + MH * HY1 + MH * HY2) / MT
Since we have two hydrogen atoms in the water molecule, we need to calculate their y-coordinates (HY1 and HY2):
HY1 = HY = 0.083 Å
HY2 = HY = 0.083 Å
Substituting the values into the formula:
CMy = (16.0 g * 0 + 1.0 g * 0.083 Å + 1.0 g * 0.083 Å) / 18.0 g
CMy = 0.0046 Å
The y-coordinate of the center of mass is approximately 0.0046 Å.
Therefore, the center of mass of the water molecule is located at approximately (-0.0016 Å, 0.0046 Å).