Skip to main content
Chemistry LibreTexts

\(^1H\) NNR Solvent Shifts

  • Page ID
    22643
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

    \( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)

    ( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)

    \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

    \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)

    \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

    \( \newcommand{\Span}{\mathrm{span}}\)

    \( \newcommand{\id}{\mathrm{id}}\)

    \( \newcommand{\Span}{\mathrm{span}}\)

    \( \newcommand{\kernel}{\mathrm{null}\,}\)

    \( \newcommand{\range}{\mathrm{range}\,}\)

    \( \newcommand{\RealPart}{\mathrm{Re}}\)

    \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

    \( \newcommand{\Argument}{\mathrm{Arg}}\)

    \( \newcommand{\norm}[1]{\| #1 \|}\)

    \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

    \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)

    \( \newcommand{\vectorA}[1]{\vec{#1}}      % arrow\)

    \( \newcommand{\vectorAt}[1]{\vec{\text{#1}}}      % arrow\)

    \( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vectorC}[1]{\textbf{#1}} \)

    \( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)

    \( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)

    \( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)

    \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

    \(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)

    The following table lists Proton NMR Chemical Shifts of common laboratory solvents and impurities.

    proton mult, J (Hz) CDCl3 CD2Cl2 THF-d8 toluene-d8 C6D6 C6D5Cl (CD3)2CO (CD3)2SO CD3CN TFE-d3 CD3OD D2O
    solvent residual signals - - 7.26 5.32 1.72 2.08 7.16 6.96 2.05 2.5 1.94 5.02 3.31 4.79
    - - - - 3.58 6.97 - 6.99 - - - 3.88 - -
    - - - - - 7.01 - 7.14 - - - - - -
    - - - - - 7.09 - - - - - - - -
    water OH s 1.56 1.52 2.46 0.43 0.4 1.03 2.84 3.33 2.13 3.66 4.87 -
    acetic acid CH3 s 2.10 2.06 1.89 1.57 1.52 1.76 1.96 1.91 1.96 2.06 1.99 2.08
    acetone CH3 s 2.17 2.12 2.05 1.57 1.55 1.77 2.09 2.09 2.08 2.19 2.15 2.22
    acetonitrile CH3 s 2.10 1.97 1.95 0.69 0.58 1.21 2.05 2.07 1.96 1.95 2.03 2.06
    allyl acetate CHCH2 ddt 5.93 5.92 5.9 5.67 5.68 5.77 5.92 5.91 5.93 5.93 5.94 5.99
    CHCH2(1) ddt 5.32 5.31 5.27 5.05 5.06 5.15 5.29 5.29 5.29 5.32 5.30 5.37
    CHCH2(2) ddt 5.24 5.22 5.15 4.94 4.94 5.04 5.18 5.20 5.21 5.25 5.21 5.30
    CH2 ddd 4.57 4.55 4.50 4.34 4.38 4.44 4.53 4.52 4.53 4.58 4.56 4.62
    CH3 s 2.09 2.05 1.98 1.63 1.63 1.80 2.02 2.03 2.02 2.07 2.05 2.13
    benzaldehyde CHO s 10.03 10.01 9.98 9.57 9.64 9.77 10.05 10.02 10.01 9.88 10 9.96
    CH(2,6) m 7.88–7.91 7.87–7.89 7.86–7.88 7.45–7.47 7.49–7.53 7.59–7.61 7.92–7.94 7.91–7.93 7.89–7.91 7.90–7.92 7.90–7.93 7.97–7.99
    CH(3,5) m 7.51–7.57 7.53–7.57 7.51–7.55 6.95–6.99 6.93–6.99 7.15–7.19 7.59–7.63 7.61–7.67 7.57–7.61 7.56–7.59 7.56–7.60 7.57–7.66
    CH(4) m 7.61–7.65 7.63–7.67 7.60–7.64 7.03–7.07 7.01–7.07 7.24–7.28 7.69–7.73 7.69–7.75 7.67–7.71 7.68–7.72 7.66–7.70 7.76–7.80
    benzene CH s 7.36 7.35 7.31 7.12 7.15 7.20 7.36 7.37 7.37 7.36 7.33 -
    tert-butyl alcohol CH3 s 1.28 1.24 1.15 1.03 1.05 1.12 1.18 1.11 1.16 1.28 1.40 1.24
    OH s - - 3.16 0.58 0.63 1.30 - 4.19 2.18 2.20 - -
    tert-butyl methyl ether CCH3 s 1.19 - - - 1.07 - 1.13 1.11 1.14 - 1.15 1.21
    OCH3 s 3.22 - - - 3.04 - 3.13 3.08 3.13 - 3.20 3.22
    BHA ArH s 6.76 6.73 6.68 6.83 6.93 6.83 6.72 6.62 6.73 6.87 6.71 -
    OH s 4.76 4.76 5.64 4.45 4.53 4.62 5.65 6.52 4.98 - 4.85 -
    ArOCH3 s 3.77 3.73 3.68 3.48 3.48 3.61 3.72 3.66 3.72 3.79 3.72 -
    ArC(CH3)3 s 1.44 1.42 1.4 1.34 1.41 1.37 1.41 1.36 1.4 1.44 1.41 -
    BHT ArH s 6.98 6.97 6.92 6.99 7.05 6.97 6.96 6.87 6.97 7.06 6.92 -
    OH s 5.01 5.00 5.81 4.72 4.79 5.50 - 6.65 5.20 - - -
    ArCH3 s 2.27 2.25 2.21 2.23 2.24 2.20 2.22 2.18 2.22 2.24 2.21 -
    ArC(CH3)3 s 1.43 1.42 1.4 1.36 1.38 1.37 1.41 1.36 1.39 1.43 1.40 -
    chloroform CH s 7.26 7.32 7.89 6.10 6.15 6.74 8.02 8.32 7.58 7.33 7.90 -
    18-crown-6 CH2 s 3.67 3.59 3.57 3.36 3.39 3.41 3.59 3.51 3.51 3.64 3.64 3.80
    cyclohexane CH2 s 1.43 1.44 1.44 1.40 1.40 1.37 1.43 1.40 1.44 1.47 1.45 -
    cyclohexanone CH2(2,6) t 2.33 2.29 2.24 1.95 1.98 2.08 2.27 2.25 2.27 2.38 2.34 2.40
    CH2(3,5) m 1.84–1.86 1.81–1.87 1.77–1.82 1.33–1.39 1.28–1.37 1.48–1.53 1.79–1.83 1.74–1.78 1.79–1.84 1.87–1.92 1.85–1.87 1.85–1.90
    CH2(4) m 1.71–1.73 1.69–1.72 1.68–1.71 1.16–1.20 1.08–1.16 1.33–1.37 1.70–1.74 1.64–1.66 1.67–1.72 1.75–1.78 1.74–1.76 1.70–1.75
    diallyl carbonate CHCH2 ddt 5.94 5.95 5.92 5.63 5.65 5.75 5.96 5.93 5.96 5.92 5.94 5.99
    CHCH2(1) ddt 5.37 5.35 5.31 5.09 5.09 5.17 5.35 5.33 5.34 5.35 5.34 5.40
    CHCH2(2) ddt 5.27 5.26 5.19 4.92 4.92 5.03 5.23 5.25 5.25 5.28 5.25 5.32
    CH2 ddd 4.64 4.61 4.58 4.34 4.38 4.46 4.62 4.61 4.61 4.62 4.61 4.69
    1,2-dichloroethane CH2 s 3.73 3.76 3.77 2.91 2.90 3.26 3.87 3.90 3.81 3.71 3.78 -
    dichloromethane CH2 s 5.30 5.33 5.51 4.32 4.27 4.77 5.63 5.76 5.44 5.24 5.49 -
    diethyl ether CH3 t, 7 1.21 1.15 1.12 1.1 1.11 1.1 1.11 1.09 1.12 1.2 1.18 1.17
    CH2 q, 7 3.48 3.43 3.38 3.25 3.26 3.31 3.41 3.38 3.42 3.58 3.49 3.56
    diglyme CH2 m 3.65 3.57 3.43 3.43 3.46 3.49 3.56 3.51 3.53 3.67 3.61 3.67
    CH2 m 3.57 3.50 3.53 3.31 3.34 3.37 3.47 3.38 3.45 3.62 3.58 3.61
    OCH3 s 3.39 3.33 3.28 3.12 3.11 3.16 3.28 3.24 3.29 3.41 3.35 3.37
    1,2-dimethoxyethane CH3 s 3.4 3.34 3.28 3.12 3.12 3.17 3.28 3.24 3.28 3.4 3.35 3.37
    CH2 s 3.55 3.49 3.43 3.31 3.33 3.37 3.46 3.43 3.45 3.61 3.52 3.6
    dimethylacetamide CH3CO s 2.09 2.02 1.94 1.59 1.6 1.74 1.97 1.96 1.97 2.09 2.07 2.08
    NCH3 s 3.02 2.97 2.95 2.56 2.57 2.65 3.00 2.94 2.96 3.05 3.31 3.06
    NCH3 s 2.94 2.87 2.82 2.11 2.05 2.42 2.83 2.78 2.83 2.94 2.92 2.90
    dimethyl carbonate CH3 s 3.79 3.75 3.69 3.31 3.30 3.48 3.72 3.69 3.72 3.77 3.74 3.69
    dimethyl malonate CH3 s 3.75 3.72 3.65 3.24 3.23 3.41 3.68 3.65 3.68 3.76 3.72 3.78
    CH2 s 3.4 3.37 3.35 2.92 2.97 3.15 3.42 3.53 3.38 3.41 3.44 3.60
    dimethylformamide CH s 8.02 7.96 7.91 7.57 7.63 7.73 7.96 7.95 7.92 7.86 7.97 7.92
    CH3 s 2.88 2.82 2.76 1.96 1.86 2.30 2.78 2.73 2.77 2.88 2.86 2.85
    dimethyl sulfoxide CH3 s 2.62 2.55 2.45 1.64 1.68 2.03 2.52 2.54 2.50 2.63 2.65 2.71
    1,4-dioxane CH2 s 3.71 3.65 3.56 3.33 3.35 3.45 3.59 3.57 3.60 3.76 3.66 3.75
    ethane CH3 s 0.87 0.85 0.85 0.81 0.80 0.79 0.83 0.82 0.85 0.85 0.85 0.82
    ethanol CH3 t, 7 1.25 1.19 1.10 0.97 0.96 1.06 1.12 1.06 1.12 1.22 1.19 1.17
    CH2 q, 7 3.72 3.66 3.51 3.36 3.34 3.51 3.57 3.44 3.54 3.71 3.60 3.65
    OH s 1.32 1.33 3.30 0.83 0.50 1.39 3.39 4.63 2.47 - - -
    ethyl acetate CH3CO s 2.05 2.00 1.94 1.69 1.65 1.78 1.97 1.99 1.97 2.03 2.01 2.07
    CH2CH3 q, 7 4.12 4.08 4.04 3.87 3.89 3.96 4.05 4.03 4.06 4.14 4.09 4.14
    CH2CH3 t, 7 1.26 1.23 1.19 0.94 0.92 1.04 1.20 1.17 1.20 1.26 1.24 1.24
    ethyl methyl ketone CH3CO s 2.14 2.09 2.03 1.59 1.58 1.78 2.07 2.07 2.06 2.16 2.12 2.19
    CH2CH3 q, 7 2.46 2.43 2.39 1.82 1.81 2.06 2.45 2.43 2.43 2.49 2.5 3.18
    CH2CH3 t, 7 1.06 1.00 0.96 0.84 0.85 0.89 0.96 0.91 0.96 1.05 1.01 1.26
    ethylene CH2 s 5.40 5.40 5.36 5.25 5.25 5.29 5.38 5.41 5.41 5.40 5.39 5.44
    ethylene glycol CH2 s 3.76 3.66 3.48 3.36 3.41 3.58 3.28 3.34 3.51 3.72 3.59 3.65
    furan CH(2,5) dd 7.45 7.46 7.48 7.10 7.13 7.24 7.56 7.67 7.52 7.44 7.49 7.57
    CH(3,4) dd 6.40 6.41 6.37 6.07 6.08 6.19 6.43 6.47 6.44 6.42 6.40 6.51
    H grease CH3 m 0.84–0.87 0.84–0.90 0.85–0.91 0.89–0.96 0.90–0.98 0.86–0.92 0.90 0.82–0.88 - 0.88–0.94 0.86–0.93 -
    CH2 br s 1.25 1.27 1.29 1.33 1.32 1.30 1.29 1.24 - 1.33 1.29 -
    hexamethylbenzene CH3 s 2.24 2.20 2.18 2.10 2.13 2.10 2.17 2.14 2.19 2.24 2.19 -
    hexamethyldisiloxane CH3 s 0.07 0.07 0.07 0.10 0.12 0.10 0.07 0.06 0.07 0.08 0.07 0.28
    n-hexane CH3 t,7 0.88 0.89 0.89 0.88 0.89 0.85 0.88 0.86 0.89 0.91 0.90 -
    CH2 m 1.26 1.27 1.29 1.22 1.24 1.19 1.28 1.25 1.28 1.31 1.29 -
    HMPA CH3 d, 9.50 2.65 2.60 2.58 2.42 2.40 2.47 2.59 2.53 2.57 2.63 2.64 2.61
    hydrogen H2 s 4.62 4.59 4.55 4.50 4.47 4.49 4.54 4.61 4.57 4.53 4.56 -
    imidazole CH(2) s 7.67 7.63 7.48 7.30 7.33 7.53 7.62 7.63 7.57 7.61 7.67 7.78
    CH(4,5) s 7.1 7.07 6.94 6.86 6.90 7.01 7.04 7.01 7.01 7.03 7.05 7.14
    methane CH4 s 0.22 0.21 0.19 0.17 0.16 0.15 0.17 0.20 0.20 0.18 0.20 0.18
    methanol CH3 s 3.49 3.42 3.27 3.03 3.07 3.25 3.31 3.16 3.28 3.44 3.34 3.34
    OH s 1.09 1.09 3.02 - - 1.30 3.12 4.01 2.16 - - -
    nitromethane CH3 s 4.33 4.31 4.31 3.01 2.94 3.59 4.43 4.42 4.31 4.28 4.34 4.40
    n-pentane CH3 t, 7 0.88 0.89 0.89 0.87 0.87 0.84 0.88 0.86 0.89 0.90 0.90 -
    CH2 m 1.27 1.30 1.31 1.25 1.23 1.23 1.27 1.27 1.29 1.33 1.29 -
    propane CH3 t, 7.3 0.90 0.90 0.90 0.89 0.86 0.84 0.88 0.87 0.90 0.90 0.91 0.88
    CH2 sept, 7.3 1.32 1.32 1.33 1.32 1.26 1.26 1.31 1.29 1.33 1.33 1.34 1.30
    2-propanol CH3 d, 6 1.22 1.17 1.08 0.95 0.95 1.04 1.10 1.04 1.09 1.20 1.15 1.17
    CH sept, 6 4.04 3.97 3.82 3.65 3.67 3.82 3.90 3.78 3.87 4.05 3.92 4.02
    propylene CH3 dt, 6.4, 1.5 1.73 1.71 1.69 1.55 1.55 1.58 1.68 1.68 1.70 1.70 1.70 1.70
    CH2(1) dm, 10 4.94 4.93 4.89 4.92 4.95 4.91 4.90 4.94 4.93 4.93 4.91 4.95
    CH2(2) dm, 17 5.03 5.03 4.99 4.98 5.01 4.98 5.00 5.03 5.04 5.03 5.01 5.06
    CH m 5.83 5.84 5.79 5.70 5.72 5.72 5.81 5.80 5.85 5.87 5.82 5.90
    pump oil CH3 m 0.83–0.89 0.84–0.89 0.86–0.90 0.88–0.96 0.91–0.97 0.88–0.91 0.87 0.74 0.85 0.99 0.86–0.91 -
    CH2 br s 1.26 1.27 1.29 1.30 1.37 1.31 1.29 1.15 1.27 1.41 1.29 -
    pyridine CH(2,6) m 8.62 8.59 8.54 8.47 8.53 8.51 8.58 8.58 8.57 8.45 8.53 8.52
    CH(3,5) m 7.29 7.28 7.25 6.67 6.66 6.90 7.35 7.39 7.33 7.40 7.44 7.45
    CH(4) m 7.68 7.68 7.65 6.99 6.98 7.25 7.76 7.79 7.73 7.82 7.85 7.87
    pyrrole NH br t 8.40 8.69 9.96 7.71 7.80 8.61 10.02 10.75 9.27 - - -
    CH(2,5) m 6.83 6.79 6.66 6.43 6.48 6.62 6.77 6.73 6.75 6.84 6.72 6.93
    CH(3,4) m 6.26 6.19 6.02 6.27 6.37 6.27 6.07 6.01 6.10 6.24 6.08 6.26
    pyrrolidine CH2(2,5) m 2.87 2.82 2.75 2.54 2.54 2.64 - 2.67 2.75 3.11 2.80 3.07
    CH2(3,4) m 1.68 1.67 1.59 1.36 1.33 1.43 - 1.55 1.61 1.93 1.72 1.87
    silicone grease CH3 s 0.07 0.09 0.11 0.26 0.29 0.14 0.13 -0.06 0.08 0.16 0.1 -
    tetrahydrofuran CH2(2,5) m 3.76 3.69 3.62 3.54 3.57 3.59 3.63 3.60 3.64 3.78 3.71 3.74
    CH2(3,4) m 1.85 1.82 1.79 1.43 1.40 1.55 1.79 1.76 1.8 1.91 1.87 1.88
    toluene CH3 s 2.36 2.34 2.31 2.11 2.11 2.16 2.32 2.30 2.33 2.33 2.32 -
    CH(2,4,6) m 7.17 7.15 7.10 6.96–7.01 7.02 7.01–7.08 7.10–7.20 7.18 7.10–7.30 7.10–7.30 7.16 -
    CH(3,5) m 7.25 7.24 7.19 7.09 7.13 7.10–7.17 7.10–7.20 7.25 7.10–7.30 7.10–7.30 7.16 -
    triethylamine CH3 t, 7 1.03 0.99 0.97 0.95 0.96 0.93 0.96 0.93 0.96 1.31 1.05 0.99
    CH2 q, 7 2.53 2.48 2.46 2.39 2.4 2.39 2.45 2.43 2.45 3.12 2.58 2.57
    1. Gottlieb, H. E.; Kotlyar, V.; Nudelman, A. J. Org. Chem. 1997, 62, 7512-7515.
    2. Fulmer, G. R.; Miller, A. J. M.; Sherden, N. H.; Gottlieb, H. E.; Nudelman, A.; Stoltz, B. M.; Bercaw, J. E.; Goldberg, K. I. Organometallics 2010, 29, 2176-2179.

    Contributors and Attributions


    \(^1H\) NNR Solvent Shifts is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by LibreTexts.

    • Was this article helpful?