Question

In the trans form of nitrous acid, all the atoms are in the same plane, but the HO is directed away from the other \(\mathrm{O}\). The \(\mathrm{O}-\mathrm{H}\) distance is \(0.098 \mathrm{~nm} ;\) the distance from the central \(\mathrm{N}\) to the hydroxy1 \(\mathrm{O}\) is \(0.146 \mathrm{~nm} ;\) and the distance to the other \(\mathrm{O}\) is \(0.120 \mathrm{~nm}\). If the \(\mathrm{H}-\mathrm{O}-\mathrm{N}\) bond angle is \(105^{\circ}\) and the \(\mathrm{O}-\mathrm{N}-\mathrm{O}\) bond angle is \(118^{\circ}\), how far is the \(\mathrm{H}\) from the other \(\mathrm{O}\) ?

Short answer

The distance between the hydrogen atom and the other oxygen atom in trans nitrous acid is approximately 0.202 nm.

Step-by-step solution

Remember the law of cosines

The law of cosines is a useful rule to find the third side of a triangle when we know two sides and the angle between them. The formula for the law of cosines is: \(c^2 = a^2 + b^2 - 2ab \cdot \cos(\gamma)\) In our case, \(c\) represents the distance between the hydrogen and the other oxygen atom, which we want to find. The side \(a\) will be the N-O distance, with length 0.120 nm. The side \(b\) will be the H-O distance, with length 0.098 nm. The angle \(\gamma\) will be the sum of the bond angles H-O-N and O-N-O, which is 105° + 118°.

Calculate the sum of the bond angles

We need the value of the angle between sides a and b, this is equal to the sum of angles H-O-N and O-N-O. \(\gamma = 105^{\circ} + 118^{\circ} = 223^{\circ}\)

Convert the given angle to radians

To apply the law of cosines, we need the angle in radians, therefore we will convert the angle \(\gamma\) in degrees to radians: \(\gamma_{rad} = \gamma \cdot \frac{\pi}{180} = 223 \cdot \frac{\pi}{180} \approx 3.89\ \text{radians}\)

Apply the law of cosines

Now, we can apply the law of cosines using the lengths of sides a, b and the angle \(\gamma_{rad}\) in radians: \(c^2 = (0.120)^2 + (0.098)^2 - 2(0.120)(0.098) \cos(3.89)\)

Calculate the third side's length

Finally, we can calculate the value of \(c\): \(c = \sqrt{(0.120)^2 + (0.098)^2 - 2(0.120)(0.098) \cos(3.89)}\) \(c \approx 0.202\ \text{nm}\) So, the distance between the hydrogen atom and the other oxygen atom in trans nitrous acid is approximately 0.202 nm.

Key concepts

Trans Form of Nitrous Acid

Nitrous acid (HNO2) can exist in two forms, which are isomers of each other: the cis and trans forms. In the trans form of nitrous acid, a pivotal concept is that all of the atoms are co-planar, aligning on a single geometric plane. This alignment distinguishes the trans form from the cis form, where the atoms are not all in the same plane. The trans isomer is characterized by the orientation of the hydroxyl (OH) group away from the other oxygen atom, as stated in the exercise. Understanding the spatial distribution is crucial for grasping the physical and chemical properties of the molecule, such as reactivity and intermolecular interactions.

By visualizing the molecule in a planar structure, we can better predict and calculate bond angles and distances using geometric principles. This planar structure also influences how the molecule will interact with other substances, such as how it might form hydrogen bonds in an aqueous solution.

Molecular Geometry

Molecular geometry is the three-dimensional arrangement of atoms within a molecule. The geometry determines many physical and chemical properties, including reactivity, polarity, phase of matter, color, magnetism, biological activity, and intermolecular interactions. Nitrous acid, for instance, has a bent geometry. Aside from the overall molecular geometry, individual bond angles significantly impact the molecule's shape.

Understanding molecular geometry also involves recognizing how electron pairs, whether they are in bonds or lone pairs, repel each other and shape the molecule. The Valence Shell Electron Pair Repulsion (VSEPR) theory provides a fundamental method for predicting the geometry of molecules based on the number of electron pairs surrounding a central atom.

Bond Angles Calculation

The calculation of bond angles is a crucial aspect of chemistry that helps us understand molecular geometries. Bond angles describe the angle between two bonds originating from the same atom. The trans form of nitrous acid has two specific bond angles given in the exercise, namely the H-O-N and O-N-O angles.

To accurately calculate the distance or angle not directly provided, chemists often turn to trigonometric methods and rules, such as the law of cosines. In the case of nitrous acid, knowing the bond angles and the distances between certain pairs of atoms enables us to calculate other distances within the molecule, thereby providing a complete geometric description of the molecule.

Cosine Rule in Chemistry

The law of cosines, also known as the cosine rule, has significant applications in chemistry, particularly when it comes to determining distances between atoms in a molecule when direct measurement is not feasible. This rule applies the fundamental principles of trigonometry to the molecular geometry, enabling chemists to calculate unknown aspects of molecular structure.

In the context of nitrous acid, the law of cosines is employed to calculate the unknown distance between the hydrogen atom and the non-hydroxyl oxygen atom. The exercise involves adding the bond angles together, converting them to radians (as trigonometric functions are radian-based), and applying the law of cosines formula. As a result, the law of cosines provides an essential tool for chemists in the exploration and confirmation of molecular structure, contributing to the broader understanding of chemical behavior and properties.