Definition of a non-polar molecule

[vcsharp2003]

Homework Statement

Which of the following definitions is the correct definition of a non-polar molecule?

(a) A molecule in which the center of mass of positive charges (i.e. protons) coincides with the center of mass of negative charges (i.e. electrons)

(b) A molecule in which the center of the positive charge distribution coincides with the center of the negative charge distribution

Relevant Equations

None

I think definition (a) is not correct since the center of charge distribution rather than mass distribution is important here. The correct definition is the one given in (b).

I am thinking that a distribution of charge will have a center of charge $(x_c,y_c,z_c)$ for -ve charges according to following equations $x_c = \dfrac { \Sigma {q_ix_i}} {\Sigma {q_i}}$, $y_c = \dfrac { \Sigma {q_iy_i}} {\Sigma {q_i}}$ and $z_c = \dfrac { \Sigma {q_iz_i}} {\Sigma {q_i}}$, where $q_i$ is the absolute value of the magnitude of individual -ve charges. The same formulas can be used for center of positive charge distribution provided $q_i$ is a +ve charge. This center of charge distribution according to above formulas need not necessarily coincide with the center of mass of the charges.

I have used the below formulas of center of mass to come up with the formulas for center of charge distribution. The center of mass distribution $(x_c,y_c,z_c)$ would be given by $x_c = \dfrac { \Sigma {m_ix_i}} {\Sigma {m_i}}$, $y_c = \dfrac { \Sigma {m_iy_i}} {\Sigma {m_i}}$ and $z_c = \dfrac { \Sigma {m_iz_i}} {\Sigma {m_i}}$

I am not sure if the formulas for center of charge distribution that I have come up with are correct Also, not sure if definition (a) is an incorrect definition.

 

[andrewkirk]

I agree with you that (a) is incorrect. Your formulas for centre of charge distribution look correct to me, except that they need to restrict the sums to only positive charges for the positive centre, and negative charges for the negative centre. One way to do that is to replace $q_i$ by $\max(0, q_i)$ in both the numerator and denominator in the formula for positive centre, and by $\min(0,q_i)$ for negative centre.

 

[vcsharp2003]

I agree with you that (a) is incorrect. Your formulas for centre of charge distribution look correct to me, except that they need to restrict the sums to only positive charges for the positive centre, and negative charges for the negative centre. One way to do that is to replace $q_i$ by $\max(0, q_i)$ in both the numerator and denominator in the formula for positive centre, and by $\min(0,q_i)$ for negative centre.

After giving it some more thought, it seems to me that center of mass for the +ve charges would coincide with their center of charge distribution. The same would also apply to the -ve charges. The reason why I say this is because the charge to mass ratio $k$ is a constant for a proton and also a constant for an electron, and therefore if one substitutes $q_i= km_i$ in the formulas for the center of charge distribution that I gave in my OP, then the factor $k$ will cancel in the numerator and denominator leaving us with the same formulas as those of the center of mass.

Does above sound correct?

 

[andrewkirk]

After giving it some more thought, it seems to me that center of mass for the +ve charges would coincide with their center of charge distribution. The same would also apply to the -ve charges. The reason why I say this is because the charge to mass ratio $k$ is a constant for a proton and also a constant for an electron, and therefore if one substitutes $q_i= km_i$ in the formulas for the center of charge distribution that I gave in my OP, then the factor $k$ will cancel in the numerator and denominator leaving us with the same formulas as those of the center of mass.

Does above sound correct?

It would certainly be approximately correct. I'm not sure whether it would be exactly correct. In nuclear reactions the number of nucleons does not change but the total weight of the nuclei does. So I am not sure we can assume that protons always weigh the same regardless of what type of nucleus they are in.
I'd stick with the charge-based definition. There's just no need to introduce the concept of mass, and it might cause problems.

 

[Steve4Physics]

After giving it some more thought, it seems to me that center of mass for the +ve charges would coincide with their center of charge distribution. The same would also apply to the -ve charges. . .

Does above sound correct?

Yes, it sounds correct to me. The question seems poorly written as both choices (a) and (b) can be interpreted as true.

 

[DrClaude]

One usually takes the nucleus to be a point charge, so in that case the centre of mass often differs from the centre of positive charge.

 

[vcsharp2003]

One usually takes the nucleus to be a point charge, so in that case the centre of mass often differs from the centre of positive charge.

But, if we use a formula to get the center of charges i.e. protons, won't it coincide with the center of charge distribution of protons? So, we don't start with nucleus being a point charge.

 

[DrClaude]

But, if we use a formula to get the center of charges i.e. protons, won't it coincide with the center of charge distribution of protons? So, we don't start with nucleus being a point charge.

Yes. My point was that this is not how things are usually done (you don't consider individual protons).

I should also have added that I have never seen a definition of a dipole moment that took into account the mass of the charges.