To determine if a molecule or ion is non-linear, we need to analyze its geometry using VSEPR theory. The geometry depends on the number of electron domains (bonding pairs and lone pairs) around the central atom.

VSEPR (Valence Shell Electron Pair Repulsion) theory predicts the geometry of molecules based on minimizing the repulsion between electron pairs in the valence shell of the central atom. The number of electron domains (sum of bonding pairs and lone pairs) determines the electron geometry, which in turn influences the molecular geometry.

Let's analyze the geometry of each option:

A) ${\mathrm{CS}}_2$

• Central atom: Carbon (C)
• Valence electrons of C: 4
• Valence electrons of S: 6
• Total valence electrons: $4+2\times 6=16$
• Lewis structure: S=C=S
• Number of bonding pairs around C: 2 (double bonds count as one electron domain)
• Number of lone pairs around C: 0
• Total electron domains: 2
• Electron geometry: Linear
• Molecular geometry: Linear

B) $\mathrm{HCN}$

• Central atom: Carbon (C)
• Valence electrons of C: 4
• Valence electrons of H: 1
• Valence electrons of N: 5
• Total valence electrons: $1+4+5=10$
• Lewis structure: H-C$\equiv$N
• Number of bonding pairs around C: 2 (triple bond counts as one electron domain)
• Number of lone pairs around C: 0
• Total electron domains: 2
• Electron geometry: Linear
• Molecular geometry: Linear

C) ${\mathrm{SnCl}}_2$

• Central atom: Tin (Sn)
• Valence electrons of Sn: 4 (Group 14)
• Valence electrons of Cl: 7
• Total valence electrons: $4+2\times 7=18$
• Lewis structure: Cl-Sn-Cl with one lone pair on Sn.
• Number of bonding pairs around Sn: 2
• Number of lone pairs around Sn: 1
• Total electron domains: 3
• Electron geometry: Trigonal planar
• Molecular geometry: Bent or V-shaped (due to the lone pair)
• Therefore, ${\mathrm{SnCl}}_2$ is non-linear.

D) ${\mathrm{NO}}_2^{+}$

• Central atom: Nitrogen (N)
• Valence electrons of N: 5
• Valence electrons of O: 6
• Total valence electrons: $5+2\times 6-1$ (for +1 charge) $=16$
• Lewis structure: O=N=O with no lone pairs on N.
• Number of bonding pairs around N: 2 (double bonds count as one electron domain)
• Number of lone pairs around N: 0
• Total electron domains: 2
• Electron geometry: Linear
• Molecular geometry: Linear

Based on the analysis, only ${\mathrm{SnCl}}_2$ has a bent molecular geometry due to the presence of a lone pair on the central tin atom, making it non-linear.

Correct Answer: (C)