A water molecule consists of an oxygen atom and two hydrogen atoms. Oxygen has a mass of 16 atomic units (u) and each hydrogen has a mass of 1 u. The hydrogen atoms are each at an average distance of 96 pm ( 96 x 10 -12 m) from the oxygen atom, and are separated from one another by an angle of 104.5o.

Find the center of mass of a water molecule.

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omak

To find the center of mass of a water molecule, we need to determine the position of the center of mass relative to the oxygen atom.

First, let's assign coordinates to the oxygen and hydrogen atoms. Since we are dealing with a 3-dimensional system, we will use x, y, and z coordinates.

Let's place the oxygen atom at the origin of our coordinate system, (0, 0, 0).

The first hydrogen atom is located at a distance of 96 pm from the oxygen atom, and the angle between the oxygen-hydrogen bond and the oxygen-hydrogen-hydrogen bond is 104.5 degrees. To determine the position of the hydrogen atom relative to the oxygen, we can use a spherical coordinate system.

Let's define the polar angle (θ) as the angle between the oxygen-hydrogen bond and the positive z-axis. In this case, θ = 180 - 104.5 degrees.

Next, we define the azimuthal angle (φ) as the angle between the oxygen-hydrogen-hydrogen bond and the positive x-axis. Since we don't have any additional information about the angle φ, we can assume it to be 0 degrees.

Now, we can convert the spherical coordinates to Cartesian coordinates using the following equations:

x = r * sin(θ) * cos(φ)
y = r * sin(θ) * sin(φ)
z = r * cos(θ)

In this case, r = 96 pm = 96 x 10^(-12) m.

Using the given values, we can calculate the Cartesian coordinates of the hydrogen atom relative to the oxygen atom.

x = (96 x 10^(-12) m) * sin((180 - 104.5) degrees) * cos(0 degrees)
y = (96 x 10^(-12) m) * sin((180 - 104.5) degrees) * sin(0 degrees)
z = (96 x 10^(-12) m) * cos((180 - 104.5) degrees)

Simplifying these equations, we find:

x = 4.255532 x 10^(-12) m
y = 0
z = -60.344485 x 10^(-12) m

Now that we have the Cartesian coordinates of the hydrogen atom, we can find the coordinates of the center of mass of the water molecule.

Since the oxygen atom has a mass of 16 atomic units and the hydrogen atoms each have a mass of 1 atomic unit, the total mass of the water molecule is 16 + 1 + 1 = 18 atomic units.

The x-coordinate of the center of mass, Cx, can be calculated using the formula:

Cx = (m1 * x1 + m2 * x2 + m3 * x3) / (m1 + m2 + m3)

In this case, m1 = 16 atomic units, m2 = 1 atomic unit, m3 = 1 atomic unit, x1 = 0 (oxygen atom coordinate), x2 = x (hydrogen atom coordinate), and x3 = x (hydrogen atom coordinate).

Cx = (16 * 0 + 1 * (4.255532 x 10^(-12) m) + 1 * (4.255532 x 10^(-12) m)) / 18
Cx = (8.511064 x 10^(-12) m) / 18
Cx = 0.472837 x 10^(-12) m

The y-coordinate and z-coordinate of the center of mass, Cy and Cz, can be calculated using similar formulas:

Cy = (m1 * y1 + m2 * y2 + m3 * y3) / (m1 + m2 + m3)
Cz = (m1 * z1 + m2 * z2 + m3 * z3) / (m1 + m2 + m3)

In this case, y1 = 0 (oxygen atom coordinate), y2 = y (hydrogen atom coordinate), y3 = 0 (hydrogen atom coordinate), z1 = 0 (oxygen atom coordinate), z2 = z (hydrogen atom coordinate), and z3 = -z (hydrogen atom coordinate).

By calculating these equations, we find:

Cy = 0 / 18 = 0
Cz = (16 * 0 + 1 * (-60.344485 x 10^(-12) m) + 1 * (60.344485 x 10^(-12) m)) / 18
Cz = 0

Therefore, the center of mass of a water molecule is located at (0.472837 x 10^(-12) m, 0, 0).