Rank the following molecules in order of increasing London dispersion forces: \(\mathrm{CO}_{2}, \mathrm{SO}_{2}\), and \(\mathrm{CS}_{2}\).

Understanding molecular size is crucial in determining London dispersion forces.
London dispersion forces arise from temporary shifts in electron density, and larger molecules typically have larger electron clouds.
This leads to stronger dispersion forces. When comparing molecular sizes of \(\text{\footnotesize{}\text{CO}_2}\), \(\text{\footnotesize{}\text{SO}_2}\), and \(\text{\footnotesize{}\text{CS}_2}\), it’s helpful to know the general shape and size of each molecule.
\(\text{\footnotesize{}\text{CO}_2}\) is a linear molecule with three atoms, while \(\text{\footnotesize{}\text{SO}_2}\) and \(\text{\footnotesize{}\text{CS}_2}\) are bent and linear respectively.
Without calculating specific dimensions, larger molecules with more atoms tend to have more substantial electron clouds.
This suggests that molecular size, even intuitively, points to greater dispersion forces in larger molecules.

Molar mass plays a vital role in determining the extent of London dispersion forces.
Let's calculate the molar masses: \[\text{\footnotesize{}\text{CO}_{2}: 12 \text{(C)} + 16 \times 2 \text{(O)} = 44 \text{g/mol}}\] \[\text{\footnotesize{}\text{SO}_{2}: 32 \text{(S)} + 16 \times 2 \text{(O)} = 64 \text{g/mol}}\] \[\text{\footnotesize{}\text{CS}_{2}: 12 \text{(C)} + 32 \times 2 \text{(S)} = 76 \text{g/mol}}\] \(\text{\footnotesize{}\text{CS}_2}\) has the highest molar mass, followed by \(\text{\footnotesize{}\text{SO}_2}\), and \(\text{\footnotesize {}\text{CO}_2}\) has the lowest.
The ranking of molar masses informs us that the London dispersion forces will likely follow the order of increasing molar mass:
\[\text{CO}_{2} < \text{SO}_{2} < \text{CS}_{2}\]

Polarizability refers to a molecule's ability to have its electron cloud distorted.
Large molecules with bigger electron clouds are more polarizable.
This leads to stronger London dispersion forces because temporary dipoles are easier to induce.
Thus, molecules like \(\text{\footnotesize {}\text{CS}_2}\) with higher molar mass are more polarizable than smaller ones like \(\text{\footnotesize {}\text{CO}_2}\).
To summarize:

• Molecules with higher molar mass and bigger size are generally more polarizable.
• More polarizable molecules exhibit stronger London dispersion forces.

Therefore, \(\text{\footnotesize {}\text{CS}_2}\) ends up having greater dispersion forces compared to \(\text{\footnotesize {}\text{CO}_2}\).