To understand which molecule can absorb infra-red (IR) radiation, we need to consider the property of molecules that makes them IR active. A molecule can absorb IR radiation if it undergoes a change in its net dipole moment during vibration. This change in dipole moment allows the molecule to interact with the electric field component of the IR radiation.

Let's analyze the given options:

• Infra-red Absorption and Dipole Moment: Molecules that absorb IR radiation are said to be IR active. For a molecule to be IR active, its vibrational or rotational motion must cause a change in its overall dipole moment.
• Homonuclear vs. Heteronuclear Diatomics:
  • Homonuclear Diatomic Molecules: These molecules consist of two atoms of the same element, like $\text{N}_2$ ($\text{N}-\text{N}$) or $\text{O}_2$ ($\text{O}-\text{O}$). They are nonpolar because the electron distribution is symmetric, and there is no permanent dipole moment. When these molecules vibrate, the symmetry is maintained (though the bond length changes), and the dipole moment remains zero. Therefore, homonuclear diatomic molecules are generally IR inactive. Even with different isotopes like $^{13}\text{N}^{14}\text{N}$, the molecule is still symmetric in charge distribution, making it effectively nonpolar and IR inactive.
  • Heteronuclear Diatomic Molecules: These molecules consist of two atoms of different elements, like $\text{NO}$ ($\text{N}-\text{O}$) or $\text{CO}$ ($\text{C}-\text{O}$). Due to the difference in electronegativity between the two atoms, there is an unequal sharing of electrons, resulting in a permanent dipole moment. When these molecules vibrate, the bond length changes, and this change affects the magnitude of the dipole moment. Since the dipole moment changes during vibration, heteronuclear diatomic molecules are IR active.

Now let's look at the options provided:

• $^{13}\text{N}_2$: This is a homonuclear diatomic molecule (${^{13}\text{N}}-{^{13}\text{N}}$). It is nonpolar and expected to be IR inactive.
• $^{13}\text{N}^{16}\text{O}$: This is a heteronuclear diatomic molecule (${^{13}\text{N}}-{^{16}\text{O}}$). Nitrogen and Oxygen are different elements, so the molecule is polar and possesses a permanent dipole moment. Its vibration will change the dipole moment, making it IR active.
• $^{16}\text{O}_2$: This is a homonuclear diatomic molecule (${^{16}\text{O}}-{^{16}\text{O}}$). It is nonpolar and expected to be IR inactive.
• $^{13}\text{N}^{14}\text{N}$: This is a homonuclear diatomic molecule (${^{13}\text{N}}-{^{14}\text{N}}$). Despite having different isotopes, the molecule structure is symmetric, making it nonpolar and IR inactive.

Based on the principle that IR absorption requires a change in dipole moment during vibration, the heteronuclear diatomic molecule, which is inherently polar, is the one that can absorb infra-red radiation.

Comparing the options, $^{13}\text{N}^{16}\text{O}$ is the only heteronuclear diatomic molecule listed.