In the HCl molecule, the separation between the nuclei of the two atoms is about 1.27 A (1 A = 10-10m). Find the approximate location of the CM of the molecule, given that a chlorine atom is about 35.5 times as massive as a hydrogen atom and nearly all the mass of an atom is concentrated in its nucleus.

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To tackle this problem, let's first understand the concept of the center of mass (CM) of a molecule. The center of mass of a system is the point where you can consider the entire mass of the system to be concentrated for the purpose of analyzing its motion. In a diatomic molecule like HCl, the center of mass lies along the line joining the two nuclei, and it divides this line in a ratio inversely proportional to the masses of the atoms.

Given that the separation between the nuclei of the two atoms is about 1.27 Å (1 Å = 10^-10m), and chlorine is approximately 35.5 times as massive as hydrogen, we can use the formula for the center of mass:

xCM=m1⋅x1+m2⋅x2m1+m2xCM=m1+m2m1⋅x1+m2⋅x2

Where:

• xCMxCM is the position of the center of mass,
• m1m1 and m2m2 are the masses of the two atoms (in this case, hydrogen and chlorine),
• x1x1 and x2x2 are the positions of the two atoms (nuclei).

Let's denote:

• m1m1 as the mass of hydrogen (let's denote it as mHmH),
• m2m2 as the mass of chlorine (let's denote it as mClmCl),
• x1x1 as the position of hydrogen,
• x2x2 as the position of chlorine.

Since nearly all the mass of an atom is concentrated in its nucleus, we can directly equate the masses of the atoms to their respective nuclei.

Now, substituting the given values:

mH=mmH=m mCl=35.5mmCl=35.5m x1=0x1=0 x2=1.27×10−10x2=1.27×10−10

xCM=m⋅0+(35.5m)⋅(1.27×10−10)m+35.5mxCM=m+35.5mm⋅0+(35.5m)⋅(1.27×10−10)

xCM=35.5×1.27×10−1036.5xCM=36.535.5×1.27×10−10

xCM≈4.4985×10−1036.5xCM≈36.54.4985×10−10

xCM≈1.23×10−11xCM≈1.23×10−11

So, the approximate location of the center of mass of the HCl molecule is 1.23×10−111.23×10−11 meters away from the hydrogen nucleus, along the line joining the two nuclei.