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Why does the water peak appear at different chemical shift values (ppm) in different solvents in proton NMR spectra? For example, the water peak in DMSO-d6 appears at nearly 3.33 ppm, but the same moisture peak in $ceCDCl3$ appears at 1.56 ppm.

What is the reason behind it? Why should the same species ($ceH2O$) give rise to two different chemical shift values?

Consider also water in addition to the solvents you mention:

• water-d2 (or, $ceD2O$) is an extensively H-bonding polar solvent (dielectric constant $epsilon = 79$)


• 
  $ceDMSO-d6$) is polar ($epsilon= 47$) but aprotic (no-H-bond donation ability)


• chloroform-d is an apolar aprotic solvent ($epsilon=4.81$)

Now consider the state of a trace amount of $ceH2O$ in each of these solvents, giving rise to an NMR signal:

• in $ceD2O$ the trace $ceH$ can exchange and forms $ceHDO$ which retains an extensive H-bond network. $ceH$ is strongly deshielded by bonded and H-bonded electronegative oxygen and due to the formation of oxonium ion with $ceH$ carrying a significant positive partial charge. The chemical shift is $approx pu4.7 ppm$


• in DMSO-d6, $ceH$ is strongly coordinated (H-bonded) by DMSO oxygen atoms, resulting in substantial shielding. The chemical shift is $approx pu3.33 ppm$
  


• in chloroform-d interactions with the solvent are comparably weak and mainly dipolar or dispersion interactions. $ceH$ experiences deshielding mainly due to directly bonded oxygen. The chemical shift is $approx pu1.56 ppm$
  

The following table shows shifts for residual water $ceH$ in different solvents ([1]; dielectric constants are for non-deuterated solvents):

$beginarrayc
beginarrayc
textsolvent &textshift(ppm)&epsilon\hline
ceC6D6 &0.40 & 2.28\
texttoluene-d8 &0.43 &2.38\
ceC6D5Cl &1.03 &5.69\
ceCD2Cl2 &1.52 &9.08\
ceCDCl3 &1.56 &4.81\
ceCD3CN &2.13 &36.64\
textTHF-d8 &2.46 &7.52\
ce(CD3)2CO &2.84 &21.01\
ce(CD3)2SO &3.33 &47\
textTFE-d3 &3.66&8.55 \
ceCD3OD &4.87 &32.6\
endarray
endarray$

Note that the most deshielding solvents are the protic H-bonding ones (the bottom two).
Next in effectiveness are those with oxygen atoms with lone electron pairs available for H-bonding.
Note also that the top three have an additional effect on water shifts due to the aromatic ring current.

Reference
[1] Fulmer et al. Organometallics 2010, 29, 2176–2179

Why do shifts change anyway? You might consider that different solvents may (de-)stabilize certain conformers, or result in different interactions? An example from a previous life (sadly a long time ago, and I can't recall the exact detail or even if NMR was reported in the paper I was following) is that in certain 3,4,5-trisubstituted pyridines (with non identical C3 and C5 substituents) H-2 and H-6 have similar shifts in d-CDCl3 (so it looks like an integral of 2) but they are clearly separated in d6-DMSO.