1. Recall the trend of electron gain enthalpy

Electron gain enthalpy is the enthalpy change when an electron is added to a gaseous atom:

$X(g) + e^- \rightarrow X^-(g)$

• More negative electron gain enthalpy means greater tendency to accept an electron.
• Halogens have highly negative electron gain enthalpies.
• Noble gases have positive electron gain enthalpies because adding an electron means putting it into a new shell/subshell, which is unfavorable.

2. Compare the elements involved

• $\mathrm{Cl}$: very negative electron gain enthalpy
• $\mathrm{F}$: negative electron gain enthalpy, but less negative than $\mathrm{Cl}$
• $\mathrm{Ne}$: positive electron gain enthalpy
• $\mathrm{Ar}$: positive electron gain enthalpy

Also, among noble gases:

$\mathrm{Ne} > \mathrm{Ar}$

in electron gain enthalpy value (more positive for Ne), because adding an electron to Ne is even less favorable.

And among halogens:

$\mathrm{Cl} < \mathrm{F}$

(where $\mathrm{Cl}$ is more negative than $\mathrm{F}$).

3. Find where the difference is maximum

We need the pair with the largest numerical difference in electron gain enthalpy.

That will happen when one element has the most positive value and the other has the most negative value.

From the given set:

• Most positive: $\mathrm{Ne}$
• Most negative: $\mathrm{Cl}$

Hence, the maximum difference is between:

$\boxed{\mathrm{Ne\ and\ Cl}}$

4. Check options

• A: Ar and Cl → large difference, but less than Ne and Cl
• B: Ne and Cl → maximum difference
• C: Ne and F → large, but less than Ne and Cl because $\mathrm{F}$ is less negative than $\mathrm{Cl}$
• D: Ar and F → not maximum

5. Final answer

$\boxed{\text{B: Ne and Cl}}$