### Materials and instruments

High purity and analytical grade samples of TBP (AR > 98 %), toluene (AR > 99 %), HNO_{3} (AR > 15.5 mol/L), and CeO_{2} (AR > 99 %) procured from CDH chemicals were used as received. The binary mixture were prepared on percentage basis (*w*/*w*) by mixing known mass of toluene in appropriate masses of TBP and measuring their masses with the help of a high-precision electronic balance of (WENSAR, PGB 100, with accuracy ±0.001 g). The densities of all mixture as well as pure liquid were measured by a specific gravity bottle calibrated with deionized double-distilled water of density 0.9960 × 10^{3}kg/m^{3} at 303.15 K. The precision of density measurement was within ± 0.0001 kg/m^{3}. The ultrasonic velocity in the mixtures as well as in the component liquids were measured at 303.15 K (calibrated up to ±0.01 m/s) by a single-crystal variable-path multifrequency ultrasonic interferometer operating at different frequencies 1–4 MHz (Mittal Enterprises, New Delhi, Model-M-81S). The temperature of the mixture was maintained constant within ±0.01 K by circulation of water from thermostatically regulated constant temperature water bath (B-206) through the water-jacketed cell. Viscosities of the mixtures were measured by Redwood apparatus (MAC, #RWV-5271 was precise up to ±0.0001 Nsm^{−2}).

### Experimental procedure

Different concentrations of extractant were prepared by dissolving various amounts of TBP in toluene. All samples were stored in ground-glass stopper bottles to prevent the evaporation. The concentrations of extractant were studied and optimized by ultrasonic method in terms of existence of different intermolecular interaction explaining the various acoustic parameters with their deviated values. The ultrasonic velocity of the pure liquids and their freshly prepared mixtures of (TBP-toluene) were measured using multifrequency ultrasonic interferometer operating at different frequencies (1–4 MHz). The working principle used in the measurement of velocity of sound through medium was based on the accurate determination of the wavelength of ultrasonic waves of known frequency produced by quartz crystal in the measuring cell. The temperature of the solution was controlled by circulating water at a desired temperature through the jacket of a double-walled cell.

For extraction, equal volumes of two phases, organic and aqueous phases, were equilibrated in a separatory funnel by using magnetic stirrer at 200 rpm for 10 min as shown in Fig. 1. Organic phase was equilibrated by adding TBP with toluene of different mole fractions, and aqueous phase was created by adding cerium oxide in diluted nitric acid using 25-mL flask. After this, the necessary volumes of both aqueous and organic phases were put into a separatory funnel, which was dipped in ultrasonic bath for phase settle. After the phase settle, the extract was filtered through a 0.45-μm nylon membrane (Guangfu Chemical Reagents Co., Tianjin, China) prior to the analysis. Mass balance analysis was performed to verify the measured distribution ratios by comparing the theory of chemical equilibrium as per Nernst distribution law:

$$ {K}_D=\frac{{\left[X\right]}_{org}}{{\left[X\right]}_{aqu}} $$

(1)

where the brackets denote the concentration of “X” in each phase at constant temperature (or the activity of “X” for nonideal solutions). By convention, the concentration extracted into organic and aqueous phase appears in the numerator and denominator of equation and the resulting value of *K*
_{
D
} is large, indicating a high degree of extraction from aqueous phase into organic phase. Conversely, if *K*
_{
D
} is small, less chemical *X* is transferred from aqueous phase into organic phase. If *K*
_{
D
} is equal to 1, equivalent concentrations exist in each phase.

### Theory

The propagation of sound wave through a medium is due to the vibrations or oscillatory motions of particles within a medium (Fig. 2). An ultrasonic wave may be visualized as an infinite number of oscillating masses or particles connected by means of elastic springs. Each individual particle is influenced by the motion of its nearest neighbor and both inertial and elastic restoring forces act upon each particle.

Thus, the force acting on the atoms of the medium as per Hook’s law is given as

$$ \begin{array}{l}F\alpha -x\\ {}F=-Kx\end{array} $$

(2)

where *K* is a constant depending on the nature of the medium on which the wave propagates and the intermolecular interaction. As the liquid medium is elastic, the wave equation is given by

$$ \frac{\partial^2\psi }{\partial {x}^2}=\frac{1}{C^2}\frac{\partial^2\psi }{\partial {t}^2} $$

(3)

where “C” is the velocity of the wave or ultrasonic wave propagating though the medium. The experimental measured values of ultrasonic velocity, density, and viscosity are used to compute different acoustic parameters such as isentropic compressibility (*β*
_{
S
}), intermolecular free length (*L*
_{
f
}), acoustic impedance (*Z*), molar volume, and surface tension and their excess values which are well describing the physicochemical properties of the medium. The acoustic parameters have been evaluated with the help of the following relationship (Pal and kumar 2011; Ali et al. 2004; Pradhan et al. 2012; Nadh et al. 2013).

$$ \mathrm{Isentropic}\ \mathrm{compressibility}:{\beta}_S=\frac{1}{\rho {C}^2} $$

(4)

$$ \mathrm{Intermolecular}\ \mathrm{free}\ \mathrm{length}:{L}_f=k{\beta}^{1/2} $$

(5)

$$ \mathrm{Acoustic}\ \mathrm{impedance}:Z=\rho C $$

(6)

$$ \mathrm{Molar}\ \mathrm{volume}:{V}_m=\frac{M}{\rho } $$

(7)

$$ \mathrm{Surface}\ \mathrm{tension}:\upsigma =6.4\times {10}^{-3}\rho {C}^{\frac{3}{2}} $$

(8)

and their excess values are calculated as

$$ \left({Y}^E\right)={Y}_{\mathrm{mix}}-\left({X}_A{Y}_A+{X}_B{Y}_B\right) $$

(9)

where *X*
_{
A
} and *X*
_{
B
} are the mole fractions, *Y*
_{
A
}, *Y*
_{
B
}, and *Y*
_{mix} represent the isentropic compressibility, intermolecular free length, acoustic impedance, molar volume, and surface tension of toluene, TBP, and mixture, respectively. The constant *k* is temperature dependent which is given as [93.875 + (0.375*T*)] × 10^{−8} (Ali et al. 1996; Dey et al. 2015), *T* being the absolute temperature.