Why does a $\sigma$-MO formed by the overlap of two $2 p_x$ orthitak (Fig. 2.7a) have the label $\sigma_g$ ?

The bond length in HF is $92 \mathrm{pm}$, and the dipole moment is $1.83 \mathrm{D}$. Determine the charge distribution in the molecule.

$$\left[\mathrm{Ans} . \stackrel{+0.41}{\mathrm{H}}-\mathrm{-0.41}^{\mathrm{F}}\right]$$

Given that $D(\mathrm{H}-\mathrm{H}), D(\mathrm{CH}-\mathrm{Cl}), D(\mathrm{Br}-\mathrm{Br})$ and $D(\mathrm{I}-\mathrm{I})$ in $\mathrm{H}_2, \mathrm{Cl}_2, \mathrm{Br}_2$ and $\mathrm{I}_2$ are $436,242,193$ and $151 \mathrm{~kJ} \mathrm{~mol}^{-1}$ respectively, estimate (by the above method) values of $D(\mathrm{H}-\mathrm{X})$ in $\mathrm{HCl}, \mathrm{HBr}$ and $\mathrm{HI}$. $\left[\right.$ Ans. $\left.339 ; 315 ; 294 \mathrm{kJmol}^{-1}\right]$

Using VB theory and the Lewis structure model, determine the bond order in (a) $\mathrm{H}_2$, (b) $\mathrm{Na}_2$, (c) $\mathrm{S}_2$, (d) $\mathrm{N}_2$ and (e) $\mathrm{Cl}_2$. Is there any ambiguity with finding the bond orders by this method?

Within VB theory, the wavefunction that describes the bonding region between two $\mathrm{H}$ atoms in $\mathrm{H}_2$ can be written in the form:

$$\psi_{\text {malonile }}=N\left[\psi_{\text {monalax }}+\left(c \times \psi_{\text {inice }}\right)\right]$$

Explain the meaning of this equation, including the reason why the factor $N$ is included.

Show that the VSEPR model is in agreement with the following molecular shapes:

$$\begin{array}{ll}\mathrm{BF}_3 & \text { trigonal planar } \\ {\left[\mathrm{IF}_5\right]^{2-}} & \text { pentagonal planar } \\ {\left[\mathrm{NH}_4\right]^{+}} & \text {tetrahedral } \\ \mathrm{SF}_6 & \text { octahedral } \\ \mathrm{XeF}_4 & \text { square planar } \\ \mathrm{AsF}_5 & \text { trigonal bipyramidal } \\ {\left[\mathrm{AlCl}_4\right]^{-}} & \text {tetrahedral }\end{array}$$

Show that the VSEPR model is in agreement with the following molecular shapes:

$$\begin{array}{ll}\mathrm{BF}_3 & \text { trigonal planar } \\ {\left[\mathrm{IF}_5\right]^{2-}} & \text { pentagonal planar } \\ {\left[\mathrm{NH}_4\right]^{+}} & \text {tetrahedral } \\ \mathrm{SF}_6 & \text { octahedral } \\ \mathrm{XeF}_4 & \text { square planar } \\ \mathrm{AsF}_5 & \text { trigonal bipyramidal } \\ {\left[\mathrm{AlCl}_4\right]^{-}} & \text {tetrahedral }\end{array}$$

Use electronegativity values in Table $2.2$ to confirm that each of the following molecules is polar. Draw diagrams to show the directions of the molecular dipole moments.

$$\mathrm{Br}-\mathrm{F}$$

S ClC(Cl)Cl

The bond length in HF is $92 \mathrm{pm}$, and the dipole moment is $1.83 \mathrm{D}$. Determine the charge distribution in the molecule.

$$\left[\mathrm{Ans} . \stackrel{+0.41}{\mathrm{H}}-\mathrm{-0.41}^{\mathrm{F}}\right]$$

Using Fig. $2.10$ as a basis, account for the fact that $\left[\mathrm{N}_2\right]^{+}$ is paramagnetic.

Given that $D(\mathrm{H}-\mathrm{H}), D(\mathrm{CH}-\mathrm{Cl}), D(\mathrm{Br}-\mathrm{Br})$ and $D(\mathrm{I}-\mathrm{I})$ in $\mathrm{H}_2, \mathrm{Cl}_2, \mathrm{Br}_2$ and $\mathrm{I}_2$ are $436,242,193$ and $151 \mathrm{~kJ} \mathrm{~mol}^{-1}$ respectively, estimate (by the above method) values of $D(\mathrm{H}-\mathrm{X})$ in $\mathrm{HCl}, \mathrm{HBr}$ and $\mathrm{HI}$. $\left[\right.$ Ans. $\left.339 ; 315 ; 294 \mathrm{kJmol}^{-1}\right]$

Find the Maclaurin polynomials of orders $n=0,1,2$, 3 , and 4 , and then find the $n$th Maclaurin polynomials for the function in sigma notation.

$$x e^x$$

Write the sum without using sigma notation. Do not evaluate.

$$\sum_{k=1}^{10}(k-1)^2$$

Use sigma notation to express the fourth term of the sequence of partial sums of original sequence $\{a_n\} = \{n!\}$. Evaluate your expression.