PHYSICAL CHEMISTRY

R. K. MAZITOV

MAGNETIC RELAXATION OF PROTONS AND DEUTERONS IN AQUEOUS SOLUTIONS OF $\mathrm{Cr}^{3+}$ IONS

(Presented by Academician A. E. Arbuzov, 26 XII 1963)

Hausser and Laukien ($^{1}$) found that the longitudinal time $T_1$ of nuclear magnetic relaxation of protons in “green” aqueous solutions of $\mathrm{CrCl}_3$ becomes shorter upon heating in the temperature interval $0$–$20^\circ\mathrm{C}$. Gutovsky et al. ($^{2}$) obtained analogous results for “violet” solutions of $\mathrm{CrCl}_3$ and suggested that changes in the structure of the solvation shells of $\mathrm{Cr}^{3+}$ ions may be responsible for this decrease. A different explanation of this fact was given by Bloembergen and Morgan ($^{3}$). They believe that the shortening of $T_1$ is caused by a decrease in the residence time $\tau_{\mathrm{H}}$ of protons in the first hydration shell of the $\mathrm{Cr}^{3+}$ ion.

It is known ($^{4}$) that the rates of chemical exchange of protons and deuterons differ substantially. This difference should be reflected in the nuclear relaxation times if the latter are determined by the rates of hydrogen exchange. It therefore seemed of interest to carry out a comparative study of the magnetic relaxation of protons and deuterons in solutions of $\mathrm{Cr}^{3+}$ ions as a function of temperature and of the isotopic composition of the solvent (water).

The times $T_{1\mathrm{D}}$, $T_{2\mathrm{D}}$, $T_{1\mathrm{H}}$, $T_{2\mathrm{H}}$ were determined by the spin-echo method. Proton resonance was observed at a frequency of 28.7 MHz, and deuteron resonance at 4.4 MHz, in the same constant magnetic field of strength 6730 oersteds. The transverse relaxation times were measured using $90$–$180^\circ$, and the longitudinal times using, respectively, $90$–$180^\circ$—$90$–$180^\circ$ sequences of radio-frequency pulses. The accuracy of the measurements was: $T_{1\mathrm{D}} \pm 15\%$, $T_{2\mathrm{D}} \pm 10\%$, $T_{1\mathrm{H}} \pm 10\%$, $T_{2\mathrm{H}} \pm 7\%$, temperatures $\pm 2^\circ\mathrm{C}$. Chromium nitrate of “chemically pure” grade and heavy water containing 99.5% deuterium were used for preparation of the samples. After preparation, the solutions were kept for several days at room temperature. It turned out that the influence of the temperature “prehistory” of the solution on $T_{1\mathrm{H}}$ and $T_{2\mathrm{H}}$, which had been observed in ($^{2}$) for “violet” solutions of $\mathrm{CrCl}_3$, is also present in “violet” solutions of $\mathrm{Cr(NO_3)_3}$. The values of the times $T_{1\mathrm{H}}$ and $T_{2\mathrm{H}}$ of samples kept for several hours at $90^\circ\mathrm{C}$ were 15% higher than $T_{1\mathrm{H}}$ and $T_{2\mathrm{H}}$ of samples not subjected to heating.

The results of measuring the temperature dependence of the relaxation times are given in Fig. 1. Heating of the solution causes an increase in $T_{1\mathrm{D}}$ in the temperature interval $0$–$100^\circ\mathrm{C}$, whereas $T_{1\mathrm{H}}$ first decreases, passes through a minimum, and only then begins to increase. An increase in temperature is accompanied by a slight increase in $T_{2\mathrm{D}}$, while $T_{2\mathrm{H}}$ becomes much shorter. Apparently, such a difference in the relaxation characteristics of protons and deuterons can be explained only by taking account of the mechanism of hydrogen exchange. To interpret the results we use the formula ($^{3}$)

$$ (T_1)^{-1} = (T_{1W})^{-1} + (Nn/N_{\mathrm{n}})(T_{1c} + \tau_{\mathrm{n}})^{-1}, \tag{1} $$

where $T_1$ is the measured nuclear relaxation time, $T_{1W}$ is the nuclear relaxation time in the pure solvent, $T_{1c}$ is the relaxation time of nuclei in the first hydration sphere of a paramagnetic ion, $\tau_{\mathrm{n}}$ is the residence time of nuclei in the first hydration sphere, $N$ is the concentration of paramagnetic ions, $N_{\mathrm{n}}$ is the con-

centration of nuclei in the solution, \(n\) is the number of nuclei in the first hydration sphere. Under the experimental conditions, \((N = 0.1 M)\), the first term on the right-hand side of (1) is considerably smaller than the second. Denoting \((Nn/N_{\mathrm{I}})=Pb\), we write:

\[ (T_1)^{-1} \simeq Pb\,(T_{1c}+\tau_{\mathrm{я}})^{-1}. \tag{2} \]

An analogous relation also holds for \(T_2\)

\[ (T_2)^{-1} \simeq Pb\,(T_{2c}+\tau_{\mathrm{я}})^{-1}. \tag{3} \]

It may be assumed that \(Pb\) does not depend on the isotopic composition of the solvent.

Fig. 1. Temperature dependence of the relaxation times of protons and deuterons in an aqueous solution \((50\%\, \mathrm{H_2O}+50\%\, \mathrm{D_2O})\) containing \(0.1\,M\) chromium nitrate:
\(1 — NT_{2\mathrm{H}}\); \(2 — NT_{1\mathrm{H}}\), \(3 — NT_{2\mathrm{D}}\), \(4 — NT_{1\mathrm{D}}\)

The relaxation times \(T_{1c}\) and \(T_{2c}\) entering equations (1)—(3) are:

\[ (T_{1c})^{-1}=(T_{1g.g})^{-1}+(T_{1\,\mathrm{обм}})^{-1}, \tag{4} \]

\[ (T_{2c})^{-1}=(T_{2\mathrm{д.д}})^{-1}+(T_{2\mathrm{обм}})^{-1}. \tag{5} \]

In (4) and (5) the first terms on the right-hand sides denote the contribution of the dipole interaction, and the second terms that of the exchange interaction, to the relaxation times.

Let us first consider the behavior of the longitudinal times. Usually \((T_{1\mathrm{обм}})^{-1} \ll (T_{1\mathrm{д.д}})^{-1}\). Then \((T_{1c})^{-1} \simeq (T_{1\mathrm{д.д}})^{-1} \sim \gamma_I^2\). Here \(\gamma\) is the gyromagnetic ratio of the nucleus whose relaxation is being considered. Since \((\gamma_{\mathrm{H}}/\gamma_{\mathrm{D}})^2 = 6.53\), it may be expected that the time \(T_{1\mathrm{D}\,\mathrm{д.д}}\) should be 42.5 times longer than \(T_{1\mathrm{H}\,\mathrm{д.д}}\). The residence time of a deuteron, \(\tau_{\mathrm{D}}\), in the first hydration shell of a \(\mathrm{Cr}^{3+}\) ion should also be longer than \(\tau_{\mathrm{H}}\), but only by a factor of 5—10 \((^4)\). Consequently, the assumed excess in (3) of \(\tau_{\mathrm{H}}\) over \(T_{1c\mathrm{H}}\) at low temperatures should decrease or disappear entirely when the relaxation of deuterons is considered. This is indeed observed experimentally. The dependence of \(T_{1\mathrm{D}}\) on temperature bears no traces whatever of the influence of \(\tau_{\mathrm{D}}\) and is determined mainly by the temperature dependence of the correlation time of the magnetic dipole interaction between the paramagnetic ion and the deuteron. To account for the possible shortening of the times \(T_{1\mathrm{D}}\) due to quadrupole relaxation of deuterons, the temperature dependence of \(T_{1w\mathrm{D}}\) in pure water was measured. It turned out that quadrupole relaxation contributes no more than \(\sim 10\%\) to the measured \(T_{1\mathrm{D}}\). Taking this contribution into account, it was found that \((T_{1\mathrm{D}\,\mathrm{д.д}}/T_{1\mathrm{H}\,\mathrm{д.д}})=36\) at \(90^\circ\mathrm{C}\), which is close to the theoretically expected value 42.5. The times \(T_{1\mathrm{H}}\) were also measured in solutions of \(\mathrm{Cr}^{3+}\) ions in \(100\%\,\mathrm{H_2O}\) and in a mixture of \(5\%\,\mathrm{H_2O}+95\%\,\mathrm{D_2O}\). The form of the curves obtained is similar to that shown in Fig. 1 for \(T_{1\mathrm{H}}\) in a solution containing \(50\%\,\mathrm{H_2O}\) and \(50\%\,\mathrm{D_2O}\), with the only difference that in the solution containing \(95\%\,\mathrm{D_2O}\) the minimum is shifted toward higher temperatures, and in the \(100\%\,\mathrm{H_2O}\) solution toward lower temperatures. The values of \(T_{1\mathrm{H}}\) in the range \(20—0^\circ\mathrm{C}\) increase as heavy water is added. At temperatures above \(\sim 60^\circ\mathrm{C}\), the values of \(T_{1\mathrm{H}}\) for all three solutions do not differ from one another within the limits of experimental error. Consequently, the correlation times of the dipole interaction cannot depend appreciably on the isotopic composition of the solvent. From the slope of the \(T_{1\mathrm{H}}\) curves in the high-temperature region and of \(T_{1\mathrm{D}}\) in the interval \(0—100^\circ\mathrm{C}\), we found the activation energy for reorientation of the hydrated chromium ion to be \(V_c = 3.2\) kcal/mol.

and the dipole correlation time \(\tau_c(300^\circ\mathrm{K}) = 10\cdot 10^{-11}\) sec; the latter is in good agreement with the value \(\tau_c(300^\circ\mathrm{K}) = 8\cdot 10^{-11}\) sec obtained in \(^{(3)}\). In determining these quantities it was assumed that \(\omega_s\tau_c \gg 1\) (\(\omega_s\) is the Larmor precession frequency of the electron spins), and it was taken that \(\tau_c=\tau_c^0=\exp(V_c/RT)\). With the aid of formula (2), from the temperature dependence of \(T_{1\mathrm{H}}\), the values of the residence times of protons \(\tau_{\mathrm{H}}\) in the first hydration sphere of \(\mathrm{Cr}^{3+}\) ions were determined. They proved to be \(3.6\cdot 10^{-6}\) sec, \(5.4\cdot 10^{-6}\) sec, and \(8.7\cdot 10^{-6}\) sec at \(300^\circ\mathrm{K}\) for \(100\%\ \mathrm{H_2O}\), \(50\%\ \mathrm{H_2O}+50\%\ \mathrm{D_2O}\), and \(5\%\ \mathrm{H_2O}+95\%\ \mathrm{D_2O}\), respectively. The temperature dependence of \(\tau_{\mathrm{H}}\) satisfies the condition \(\tau_{\mathrm{H}}=\tau_{\mathrm{H}}^0\exp(V_{\mathrm{H}}/RT)\) in the range \(0\)–\(40^\circ\mathrm{C}\), with an activation energy for proton exchange \(V_{\mathrm{H}}=10\pm 0.2\) kcal/mole for all three solutions. These quantities are in agreement with the values of \(\tau_{\mathrm{H}}\) and \(V_{\mathrm{H}}\) obtained in \(^{(3)}\).

The comparison made above between \(T_{1\mathrm{D}}\) and \(T_{1\mathrm{H}}\) is also valid for transverse relaxation times, since \((T_{2\,\mathrm{ex}})^{-1}\sim \gamma_l^2\), according to \(^{(6)}\), and \((T_{2\,\mathrm{d.d.}})^{-1}=1.16\,(T_{1\,\mathrm{d.d.}})^{-1}\). \(T_{2\mathrm{D}}\) is not affected by hydrogen exchange, and the weak increase of \(T_{2\mathrm{D}}\) upon heating may be attributed to a decrease in the contribution from the dipole mechanism. On the contrary, the strong shortening of \(T_{2\mathrm{H}}\) in the range \(0\)–\(30^\circ\mathrm{C}\) is due to a decrease in \(\tau_{\mathrm{H}}\). The values of \(T_{2\mathrm{H}}\) increase upon addition of \(\mathrm{D_2O}\) in the low-temperature region and prove to be independent of the isotopic composition of the solution at \(t^\circ \gg 65^\circ\mathrm{C}\). Assuming that \(T_{2\mathrm{H}}\) at high temperatures does not depend on \(\tau_{\mathrm{H}}\), and that the time \(T_{2c\mathrm{H}}\) changes upon heating in the same way as \(T_{2c\mathrm{D}}=(1/Pb)T_{2\mathrm{D}}\), constants characterizing hydrogen exchange were found. These constants practically coincided with those calculated from the temperature dependence of the \(T_{1\mathrm{H}}\) times. In the case where \(\tau_s \ll \tau_{\mathrm{H}}\), the exchange contribution to \(T_2\) is expressed by formula (5):

\[ (T_{2\,\mathrm{ex}})^{-1}=(1/3)\,Pb\,S(S+1)(A/\hbar)^2\tau_s, \tag{6} \]

where \(S=3/2\) is the electron spin of the \(\mathrm{Cr}^{3+}\) ion, \(A\) is the exchange-interaction constant, and \(\tau_s\) is the electron relaxation time. Taking \(\tau_s=5\cdot 10^{-10}\) sec \(^{(3)}\), from the experimental data, with the aid of (6), we found*: \((A_{\mathrm{H}}/h)=3\cdot 10^6\) Hz and \((A_{\mathrm{D}}/h)=4.3\cdot 10^5\) Hz. The value of \((A_{\mathrm{H}}/h)\) found by us is greater than \((A_{\mathrm{H}}/h)=2\cdot 10^6\) Hz, reported in \(^{(3)}\). According to the theory \(^{(6)}\), \(A_I\sim \gamma_I|\psi(0)|_I^2\); here \(|\psi(0)|_I^2\) is the density of unpaired electrons of the ion at the nuclei of the hydrogen atoms in the hydration shell. Comparison of the quantities \(A_{\mathrm{H}}\) and \(A_{\mathrm{D}}\) gave: \(|\psi(0)|_{\mathrm{D}}^2=0.93|\psi(0)|_{\mathrm{H}}^2\), i.e., the density of unpaired electrons at deuteron sites is somewhat lower than at proton sites. An analogous conclusion was made earlier for aqueous solutions of \(\mathrm{Mn}^{2+}\) ions \(^{(7)}\).

Fig. 2. Dependence of the relaxation times of protons and deuterons on the isotopic composition of the solvent (water) at a temperature of \(0^\circ\mathrm{C}\). Chromium nitrate concentration \(0.1\,M\): 1—\(NT_{2\mathrm{H}}\), 2—\(NT_{1\mathrm{H}}\), 3—\(NT_{2\mathrm{D}}\), 4—\(NT_{1\mathrm{D}}\).

Figure 2 presents the dependence of the relaxation times of protons and deuterons in aqueous solutions containing \(0.1\,M\) chromium nitrate at \(0^\circ\mathrm{C}\) on the isotopic composition of the solvent. \(T_{1\mathrm{H}}\) and \(T_{2\mathrm{H}}\) lengthen as \(\mathrm{D_2O}\) is added, whereas the times \(T_{1\mathrm{D}}\) and \(T_{2\mathrm{D}}\) do not undergo changes exceeding the experimental errors. A weak (\(\sim 30\%\)) increase—

* Calculations were performed for a temperature of \(90^\circ\mathrm{C}\); there are grounds to believe that in the range \(\sim 20\)–\(100^\circ\mathrm{C}\) the time \(\tau_s\) for a \(0.1\,M\) solution of \(\mathrm{Cr}^{3+}\) should not change substantially with temperature.

$T_{1\mathrm{H}}$ and $T_{2\mathrm{H}}$ upon addition of $\mathrm{D_2O}$ was found at room temperature. These results can be readily understood on the basis of the foregoing.

The author expresses gratitude to A. I. Rivkind for supervising the work and to T. A. Neklyudova for participation in the measurements.

Kazan State University
named after V. I. Ulyanov-Lenin

Received
26 XII 1963

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