Problem 73 - Chemical Principles

Question

(a) Calculate the surface areas of the isomers 2,2 dimethylpropane and pentane. Assume that 2,2-dimethylpropane is spherical with a radius of \(254 \mathrm{pm}\) and that pentane can be approximated by a rectangular prism with dimensions \(295 \mathrm{pm} \times\) \(692 \mathrm{pm} \times 766 \mathrm{pm}\). (b) Which has the larger surface area? (c) Which do you expect to have the higher boiling point?

Step-by-Step Solution

Answer

The surface area of 2,2-dimethylpropane is larger than that of pentane, thus it can be expected to have a higher boiling point due to the presence of stronger London dispersion forces.

Step 1: Calculate the Surface Area of 2,2-Dimethylpropane

Use the formula for the surface area of a sphere, which is given by the equation \( 4\pi r^2 \), where \( r \) is the radius of the sphere. For 2,2-dimethylpropane, the radius is given as \( 254 \) pm. Substituting this value into the formula: \( 4\pi (254 \mathrm{pm})^2 \).

Step 2: Calculate the Surface Area of Pentane

The surface area of a rectangular prism is found using the formula \( 2lw + 2lh + 2wh \), where \( l \) is the length, \( w \) is the width, and \( h \) is the height. For pentane, these dimensions are \( 295 \) pm, \( 692 \) pm, and \( 766 \) pm respectively. Substituting these into the formula yields: \( 2(295 \mathrm{pm})(692 \mathrm{pm}) + 2(295 \mathrm{pm})(766 \mathrm{pm}) + 2(692 \mathrm{pm})(766 \mathrm{pm}) \).

Step 3: Perform the Calculations

Calculate the numerical surface areas for both molecules using the given formulas and dimensions: For 2,2-dimethylpropane: \( 4\pi(254 \mathrm{pm})^2 \) and for pentane: \( 2(295 \mathrm{pm} \times 692 \mathrm{pm}) + 2(295 \mathrm{pm} \times 766 \mathrm{pm}) + 2(692 \mathrm{pm} \times 766 \mathrm{pm}) \).

Step 4: Compare the Surface Areas

After computing the numerical values, compare them to determine which molecule has the larger surface area.

Step 5: Predict the Higher Boiling Point

The molecule with the larger surface area tends to have a higher boiling point due to stronger London dispersion forces. Based on the calculated surface areas, predict which molecule has a higher boiling point.

Key Concepts

Surface Area of a Sphere

When visualizing a sphere, like a perfectly round ball, one can understand the surface area as the amount of material needed to cover it. In the context of chemistry, molecules that are spherical, such as 2,2-dimethylpropane, can be represented by a sphere's surface area.
The formula for calculating the surface area of a sphere is given by \( 4\pi r^2 \) where \( r \) represents the radius of the sphere. This calculation tells us how much space the molecule's surface covers, which is especially relevant when considering properties like reactivity and physical behavior.

Real-World Application

In practice, knowing the surface area of spherical molecules helps chemists understand how these molecules might interact with other substances, as a larger surface area can lead to higher reactivity or different physical properties.

Surface Area of a Rectangular Prism

A rectangular prism can be likened to a box with six faces, each being a rectangle. We encounter this shape commonly in everyday objects like shoeboxes or bricks. The surface area of a rectangular prism tells us how much area all of these faces cover combined.
The formula for calculating this is \( 2lw + 2lh + 2wh \), where \( l \) is length, \( w \) is width, and \( h \) is height. For molecules approximated by this shape, like pentane in this case, calculating the surface area provides insight into the exposure each molecule has to its environment.

Understanding Molecule Behavior

This becomes significant when predicting how a substance might act, especially in relation to forces between molecules. The more surface available, the more area there is for these forces to act upon.

Intermolecular Forces

Intermolecular forces are the forces of attraction or repulsion between neighboring particles (atoms, molecules, or ions). They are weaker than the strong covalent bonds that hold compounds together but are pivotal in determining the physical properties of a substance, including its phase, melting point, and boiling point.
There are several types of intermolecular forces, with London dispersion forces, dipole-dipole interactions, and hydrogen bonding being the most significant.

The Role in Phase Changes

During phase changes, such as transitioning from liquid to gas, these forces must be overcome. So, the strength and type of intermolecular forces present help determine at what temperature these phase changes occur.

Boiling Point

Boiling point, the temperature at which a substance transitions from a liquid to a gas state, is greatly influenced by intermolecular forces. Stronger intermolecular forces will require more energy to overcome, leading to a higher boiling point.
The boiling point can tell us about the volatility of a liquid – a lower boiling point indicates higher volatility.

Comparing Substances

When comparing substances, if we assume similar molar masses, the one with the higher boiling point usually has stronger intermolecular forces between its molecules. In the case of molecules with different sizes and shapes, the surface area in contact with other molecules greatly affects these forces.

London Dispersion Forces

London dispersion forces, a type of van der Waals force, arise from attractions between instantaneously induced dipoles in molecules. These are the weakest of the intermolecular forces but are present in all molecules, regardless of their polarity.
The larger the molecule, or the larger the surface area it has, the more pronounced these London dispersion forces become.

Influence on Physical Properties

They influence physical properties such as boiling point because molecules with larger surface areas create stronger London dispersion forces, requiring more heat energy to reach the boiling point. Hence, understanding the relationship between molecular size, surface area, and London dispersion forces is fundamental in predicting how a substance will behave under various conditions.