Appendix F — Problem Set 4 | DIRECTION: Vectors and Bonds

Companion to §2.1–2.2 — direction and magnitude

A. Recognition

1. Scalar or vector?

Classify each quantity and justify one borderline case.

  1. temperature
  2. molecular dipole moment
  3. bond length
  4. force on a nucleus
  5. activation energy
  6. concentration gradient
  7. reaction rate in \mathrm{mol\,L^{-1}\,s^{-1}}

2. What operation is needed?

Choose vector addition, subtraction, scalar multiplication, magnitude, or normalization.

  1. Find a bond vector from two atomic positions.
  2. Combine three bond dipoles.
  3. Retain direction but discard magnitude.
  4. Compare the length of a computed displacement with experiment.
  5. Reverse a force while doubling its strength.

B. Components and geometry

3. Bond vector and unit vector

In Cartesian coordinates, \mathbf r_C=(1.20,-0.40,0.10)\ \text{Å} and \mathbf r_O=(2.35,0.25,-0.55)\ \text{Å}.

  1. Find the vector from C to O.
  2. Find the C–O bond length.
  3. Find the unit vector pointing from C to O.
  4. Confirm that the unit vector has magnitude 1 within rounding.

4. Coordinate translation test

Translate both positions in Problem 3 by \mathbf a=(-4.0,2.0,1.5)\ \text{Å}.

  1. Write the translated coordinates.
  2. Recalculate the C-to-O vector.
  3. State and verify the invariant.

5. Resultant molecular dipole

Three bond-dipole contributions, in debye, are \boldsymbol\mu_1=(1.20,0,0),\quad \boldsymbol\mu_2=(-0.35,0.95,0),\quad \boldsymbol\mu_3=(-0.35,-0.20,0.60).

  1. Calculate the total molecular dipole vector.
  2. Calculate its magnitude.
  3. A model reports only the sum of the three magnitudes. Explain why that is generally wrong.

6. Force from a potential

Along one coordinate, U(x)=\tfrac12k(x-x_0)^2, with k=500\ \mathrm{N\,m^{-1}} and x-x_0=2.0\times10^{-12}\ \mathrm m.

  1. Use F_x=-dU/dx to derive the force.
  2. Calculate its value and direction.
  3. Repeat for x-x_0=-2.0\times10^{-12}\ \mathrm m.
  4. What physical role does the minus sign play?

C. Chemical models

7. Water from two bond vectors

Model two equal O–H bond dipoles of magnitude \mu_b=1.50\ \mathrm D, separated by 104.5^\circ. Place their bisector along +y.

  1. Write both dipoles in components using half the bond angle.
  2. Show which components cancel.
  3. Calculate the resultant magnitude.
  4. Predict the effect of changing the angle toward 180^\circ while holding \mu_b fixed.

8. Carbon dioxide cancellation

Represent the two equal C=O bond dipoles of linear CO_2 as vectors.

  1. Show algebraically why their sum is zero.
  2. Explain why “CO_2 has polar bonds” does not imply “CO_2 has a molecular dipole.”
  3. If one bond dipole were 1.0% larger, express the residual dipole in terms of \mu_b.

9. Center of mass as a weighted vector sum

For three atoms with masses m_i and positions \mathbf r_i, derive \mathbf R_{CM}=\frac{\sum_i m_i\mathbf r_i}{\sum_i m_i}. Then calculate \mathbf R_{CM} for atoms of mass 12, 1, and 16 u at (0,0,0), (1,0,0), and (0,1,0) Å.

10. Error diagnosis

A student normalizes \mathbf v=(3,4,0) by dividing every component by 3+4+0.

  1. Test whether the result has unit magnitude.
  2. Correct the normalization.
  3. Explain the difference between component sum and Euclidean magnitude.