1. Introduction

2. Results and Discussion

2.1. Volumetric Properties

The results obtained from the density tests of the DMF + BuOH mixture (Table S1 Supplementary Materials) show that the values increase with increasing DMF content in the mixture and decrease systematically with increasing temperature over the whole range of mixture compositions. To study the properties of real solutions, excess functions are used, which determine the difference between the magnitude of a given molar thermodynamic function in a real solution and its magnitude in an ideal solution. The excess properties were calculated using the following expression:

$$Z^{\mathrm{E}}=Z-Z^{\mathrm{id}}$$

(1)

where $Z^{\mathrm{E}}$ is the excess quantity of the property Z and $Z^{\mathrm{id}}$ is the corresponding ideal value [25].

In this paper will be presented and analyze six excess functions $(V_{\mathrm{m}}^{\mathrm{E}}, E_{p,\mathrm{m}}^{\mathrm{E}}, K_{S,\mathrm{m}}^{\mathrm{E}}, K_{T,\mathrm{m}}^{\mathrm{E}},$ $C_{\mathrm{p},\mathrm{m}}^{\mathrm{E}}, C_{\mathrm{V},\mathrm{m}}^{\mathrm{E}})$ calculated according to Equation (1). In order to calculate excess molar volume $\left(V_{\mathrm{m}}^{\mathrm{E}}\right)$, the molar volume of the mixture was obtained according to Equation (2):

(2)

where: $\rho$ is the density of DMF + BuOH mixture, $x_1$, $x_{2 }$and $M_1$, $M_2$ are the mole fractions and molar masses of the mixture components, respectively, i.e. BuOH (1), DMF (2).

In order to determine the changes taking place in a real solution in relation to an ideal solution in a binary mixture, the values of excess molar volume ($V_{\mathrm{m}}^{\mathrm{E}})$have been executed. $V_{\mathrm{m}}^{\mathrm{E}}$ values of the mixture in the whole composition range and at the temperature range (293.15 K–318.15 K) was calculated according to Equation (3) and presented in Figure 1:

(3)

where: $V_{\mathrm{m}}$ is the molar volume of (DMF+BuOH) mixture, $V_{\mathrm{m}}^{\mathrm{id}}$ is the volume of ideal mixture, $V_1^{*}$,$V_2^{*}$ are the molar volume of pure compounds, i.e. BuOH (1), DMF (2).

The values of excess functions and among others the $V_{\mathrm{m}}^{\mathrm{E}}$ of DMF + BuOH mixture were fitted to the polynomial of Redlich−Kister type:

$$V_{\mathrm{m}}^{\mathrm{E}}=x_1{x}_2{\displaystyle \sum _{j=0}^n{A}_j{(1-2x_1)}^j}$$

(4)

$$\frac{V_{\mathrm{m}}^{\mathrm{E}}}{x_1{x}_2}={\displaystyle \sum _{j=0}^n{A}_j{\left(1-2x_1\right)}^j}$$

(5)

where $A_{\mathrm{j}}$ is the polynomial coefficient calculated by the least-squares method using Equation (5).

As can be seen in Figure 1, the excess molar volume exhibits positive values in a solution with a predominantly BuOH content in the mixture. $V_{\mathrm{m}}^{\mathrm{E}}$ increases when small amounts of DMF are added to pure BuOH and passes through a maximum which appears depending on the temperature among $x_{\mathrm{DMF}}$ ≈ 0.3 and $x_{\mathrm{DMF}}$ ≈ 0.35 values. Above this value of mole fraction, $V_{\mathrm{m}}^{\mathrm{E}}$ values are decreasing and reach the minimum value at $x_{\mathrm{DMF}}$ ≈ 0.9. It is noteworthy that negative values for $V_{\mathrm{m}}^{\mathrm{E}}$are obtained for different mixture compositions depending on the temperature. The mole fraction of DMF, in which we observe the change of sign of the $V_{\mathrm{m}}^{\mathrm{E}}$ function, increases with increasing temperature from the value $x_{\mathrm{DMF}}$ ≈ 0.5 for 293.15 K to the value $x_{\mathrm{DMF}}$ ≈ 0.9 for 318.15 K. One can notice that the volume contraction of the DMF + BuOH mixture increases when the molar fraction of DMF increases above 0.5 and decreases with increasing temperature. In the literature, you can find several reports in which an attempt was made to determine the value of $V_{\mathrm{m}}^{\mathrm{E}}$ for the DMF + BuOH mixture [14,15,16,19,20,21]. However, the research results presented in these articles are divergent. Similar results to ours were obtained by Rao and Reddy [14] at 303 K and Garcia et al. [21] but only at 298K. In the other works, the results diverged from each other and from our results. Moreover, some authors obtained negative values of $V_{\mathrm{m}}^{\mathrm{E}}$ in the whole range of compositions [15,19] which was completely inconsistent with the results obtained by other researchers, including ours. Such ambiguous data and conclusions prompted us to study the properties of this mixture with more accuracy using the three different test methods mentioned earlier.

The sign and magnitude of the excess functions may be attributed to the result of an appropriate combination of the following three major effects. The mutual dissociation of the components due to addition of the second component, the formation of hydrogen bonds between unlike molecules and steric hindrance, as well as geometry of molecular structure, which could be the reason which resist the closeness of the molecules. For DMF + PrOH according to the results obtained by us in our previous work [26], the values were negative over the entire range of the mixture composition. Excess molar volumes are more negative in systems with lower alcohols, which may be attributed to strong interactions between unlike molecules and different molecular sizes [27]. Such properties cause volume contraction in these mixtures. For the DMF + BuOH system, the small increase in the size of the alcohol molecule (extra –CH₂ group in 1-butanol compared to DMF + PrOH [26]) gives completely different results and appears in volume expansion when DMF is added to BuOH. These values are positive in the BuOH rich region. Mostly for the DMF + BuOH system, the results obtained in their absolute value are 1.5 to 2 times smaller than for the DMF + PrOH mixture, however, in some compositions, they are several or ten times smaller than for the DMF + PrOH mixture.

Alcohols are strongly hydrogen-bonded in the pure state. Their molecules are self- and cross-associated [28,29]. The degree of association decreases also with increasing alcohol chain length. Thus, the addition of DMF to pure BuOH breaks the hydrogen bonds between molecules in the structure of the alcohol network, which produces a positive contribution to $V_{\mathrm{m}}^{\mathrm{E}}$. The results obtained on the basis of the dielectric study prove that in a solution with a predominant content of BuOH, the largest changes in the structure of hydrogen bonds occur in the DMF + BuOH solution [30]. At the same time, an increase in the length of the alcohol chain causes a steric hindrance that also contributes to the increase in the real volume of the DMF + BuOH mixture. There is also not much difference in the molecular size of these two compounds, which also makes mutual accommodation difficult. The presence of hydrogen bonds between the components of the mixture tested was also confirmed by other researchers [4,20,30,31]. Prajapati [30] based on the analysis of parameters obtained from the dielectric study provides confirmation of the formation of hydrogen bonds between DMF and BuOH molecules and weak dipole-dipole interactions between the components of the mixture. In contrast to the DMF + PrOH mixture, it is known that the interactions between unlike molecules are very weak in DMF + BuOH solution [32]. Therefore, we observe an increase in $V_{\mathrm{m}}^{\mathrm{E}}$ with increasing DMF content in the mixture to $x_{\mathrm{DMF}}$ ≈ 0.3. The maximum observed at this mole fraction of DMF is related to the fact that in this range of the mixture composition there are probably the weakest interactions between the components of the mixture, which also contributes positively to the $V_{\mathrm{m}}^{\mathrm{E}}$ [32]. Despite the fact that there is probably still the possibility to form some amount of hydrogen bonds between DMF and BuOH molecules, the excess of BuOH amount is a competitive factor in the formation of hydrogen bonds between unlike molecules. According to the researchers [32], the binding energy –O–H···O=C decreases when another alcohol molecule approaches with its oxygen atom favorablely orientated for “change”. As a consequence, instead of hydrogen bonding between DMF and BuOH molecules, the bonding appears between alcohol molecules rather more. When the amount of BuOH prevails in the DMF + BuOH system, there exists a greater tendency to create hydrogen bonds between BuOH molecules than between unlike species. As a result of the specific arrangement of neighbouring molecules around the DMF and BuOH molecules that form the bond, this interaction is weakened, which is confirmed by the negative ${\epsilon }_0^{\mathrm{E}}$ values in the entire concentration range of the solution obtained by Prajapati [30]. This parameter assumes the largest negative deviation for a mixture with the composition $x_{\mathrm{DMF}}$ ≈ 0.3, hence we observe a maximum on the relationship

= f($x_{\mathrm{DMF}})$. Dissociation of bonding of pure components presumably became the main reason for volume expansion of the tested system. When $x_{\mathrm{DMF}}$ > 0.3 $V_{\mathrm{m}}^{\mathrm{E}}$ values decrease and become negative when $x_{\mathrm{DMF}}$ > 0.5 (293.15 K). In a solution with a predominant DMF content in the mixture, they can observe small volume contraction. The volume contraction value is similar to that observed for the DMF + PrOH system in the same concentration area of the mixture. When DMF starts to prevail in the solution, dipole-dipole interactions are most likely to begin to prevail in DMF and the creation of hydrogen bonding between DMF and BuOH molecules.

The analysis of the excess partial molar volume of both components of the mixture may help to explain the observed changes in this range of the mixture composition. The volume of the solution, in the case of a binary mixture, is the sum of the partial molar volumes of both components:

$$V={\mathrm{n}}_{\mathrm{BuOH}}·V_{\mathrm{m},\mathrm{BuOH}}+{\mathrm{n}}_{\mathrm{DMF}}·V_{\mathrm{m},\mathrm{DMF}}$$

(6)

where:${\mathrm{n}}_{\mathrm{BuOH}}$, ${\mathrm{n}}_{\mathrm{DMF}}$ – mole fraction of BuOH i DMF respectively, $V_{\mathrm{m},\mathrm{BuOH}}$, $V_{\mathrm{m},\mathrm{DMF}}$ – partial molar volume of BuOH i DMF respectively.$V_{\mathrm{m},\mathrm{BuOH}}$ and $V_{\mathrm{m},\mathrm{DMF}}$ we can calculate using the following Equations (7) i (8):

$$V_{\mathrm{m},\mathrm{DMF}}=V_{\mathrm{m}}+x_{\mathrm{BuOH}}\left(\frac{\partial V_{\mathrm{m}}}{\partial x_{\mathrm{DMF}}}\right)$$

(7)

$$V_{\mathrm{m},\mathrm{BuOH}}=V_{\mathrm{m}}+x_{\mathrm{DMF}}\left(\frac{\partial V_{\mathrm{m}}}{\partial x_{\mathrm{BuOH}}}\right)$$

(8)

where: $V_{\mathrm{m}}$ – molar volume of the real solution (DMF + BuOH),$x_{\mathrm{DMF}}$, $x_{\mathrm{BuOH}}$ – mole fraction of DMF and BuOH.

Analysis of changes in the partial molar volume of the components of a mixture can be represented by the excess partial molar volume ($V_{\mathrm{DMF}}^{\mathrm{E}}$, $V_{\mathrm{BuOH}}^{\mathrm{E}}$). These values can be calculated using Equations (9) and (10):

$$V_{\mathrm{m},\mathrm{DMF}}^{\mathrm{E}}=V_{\mathrm{m},\mathrm{DMF}}-V_{\mathrm{DMF}}^{*}$$

(9)

$$V_{\mathrm{m},\mathrm{BuOH}}^{\mathrm{E}}=V_{\mathrm{m},\mathrm{BuOH}}-V_{\mathrm{BuOH}}^{*}$$

(10)

where: $V_{\mathrm{m},\mathrm{DMF}}^{\mathrm{E}}$,$V_{\mathrm{m},\mathrm{BuOH}}^{\mathrm{E}}$ – excess partial molar volume of DMF and BuOH respectively,$V_{\mathrm{DMF}}^{*}$, $V_{\mathrm{BuOH}}^{*}$ – molar volume of pure DMF and BuOH.

The results for partial and excess partial molar volume of DMF and BuOH $V_{\mathrm{m},\mathrm{DMF}}^{\mathrm{E}}$ and $V_{\mathrm{m},\mathrm{BuOH}}^{\mathrm{E}}$ are presented in Tables S4-S7 in Supplementary Materials and presented in Figure 2 at T = 293.15 K.

The values of $V_{\mathrm{m},\mathrm{DMF}}^{\mathrm{E}}$ and $V_{\mathrm{m},\mathrm{BuOH}}^{\mathrm{E}}$ express the difference between the value of the partial molar volume of DMF or BuOH in the solution and the molar volume of each of the components in their pure form. Based on the analysis of Figure 2, one can notice that for DMF the values of $V_{\mathrm{m},\mathrm{DMF}}^{\mathrm{E}}$ are negative when $x_{\mathrm{DMF}}$ > 0.4 at 293.15 K. This means that DMF contributes negatively to the real volume of the mixture in this composition range of the solution. Although $V_{\mathrm{m},\mathrm{BuOH}}^{\mathrm{E}}$ assumes positive values in the same concentration range, however the amount of DMF prevails over the amount of BuOH when $x_{\mathrm{DMF}}$ > 0.4. Having been aware that DMF is a weakly associated liquid in contrast to BuOH [30], the results obtained allow us to conclude that the negative contribution to the real volume of the mixture is made by the DMF molecules ($V_{\mathrm{m},\mathrm{DMF}}^{\mathrm{E}}<0).$ This is presumably one of the main factors of the slight contraction of the volume of the mixture causing that we observe $V_{\mathrm{m}}^{\mathrm{E}}$ < 0 when $x_{\mathrm{DMF}}$ > 0.5. Analysis of data on the partial molar volume of each component in the DMF + BuOH mixture (Tables S4 and S5 Supplementary Materials) allows one to observe that the relationships $V_{\mathrm{m},\mathrm{DMF}}=f\left(x_{\mathrm{DMF} }\right)$ and $V_{\mathrm{m},\mathrm{BuOH}}=f\left(x_{\mathrm{DMF} }\right)$ shows only little change with increasing DMF content in solution. This proves that chemical entities, such as various types of associations or complexes built of both molecules of the solution components, are most likely not formed in the system. Otherwise, we would observe characteristic changes in the course of both functions, along with an increase in the content of one of the components of the mixture, as in the case of an aqueous solution of N,N-dimethylformamide [33], which also confirms the conclusions drawn earlier. The partial molar volume of DMF decreases slightly as the DMF content of the mixture increases, with a higher temperature. Garcia et al. [19] presented a similar course of this dependence at a temperature of 298.15 K. In addition, $V_{\mathrm{m},\mathrm{DMF}}$ takes the highest value at the highest temperature. Determined by Zegers and Somsen [22] value of $V_{\mathrm{m},\mathrm{DMF}}$ in pure BuOH is equal to 7.779·10⁻⁵ m³·mol⁻¹ and is close to the one obtained by us (7.749·10⁻⁵ m³·mol⁻¹). The course of the relationship $V_{\mathrm{m},\mathrm{BuOH}}=f\left(x_{\mathrm{DMF} }\right)$ is opposite to that observed for DMF. $V_{\mathrm{m},\mathrm{BuOH}}$ values increase slightly with increasing DMF content. The influence of temperature on the values of $V_{\mathrm{m},\mathrm{BuOH}}$ is analogous to the case $V_{\mathrm{m},\mathrm{DMF}}$. Zegers and Somsen [22] determined also $V_{\mathrm{m},\mathrm{BuOH}}$ in pure DMF, which is equal to 9.197·10⁻⁵ m³·mol⁻¹. The value obtained by them is close to the one determined by us (9.208·10⁻⁵ m³·mol⁻¹).

Using the density values of the mixture at six temperatures, the coefficient of thermal expansion was calculated (α_p) using Equation (11):

$${\alpha }_p=\frac{1}{V_{\mathrm{m}}}{\left(\frac{\partial V_{\mathrm{m}}}{\partial T}\right)}_{\mathrm{p}}=-\frac{\partial \ln \rho }{\partial T}=-\frac{\partial \rho }{\rho \partial T}$$

(11)

$V_{\mathrm{m}}$was calculated with Equation (12) [34]:

(12)

The values of molar isobaric expansion ($E_{p,\mathrm{m}}^{}$) were calculated using Equation (13) [25]:

$$E_{p,\mathrm{m}}^{}={\alpha }_p\cdot V_{\mathrm{m}}$$

(13)

The obtained results of$E_{p,\mathrm{m}}^{}$ as a function of the DMF molar fraction are presented in Figure 3.

The isobaric molar expansion reaches the highest value for pure BuOH and decreases with increasing DMF in the mixture. The $E_{p,\mathrm{m}}$ values increase with increasing temperature over the whole composition range of the mixture which seems to be logical due to the increase in thermal movements at higher temperatures. Such a behaviour of the system in the BuOH rich region is most likely related to the breaking of hydrogen bonds in BuOH after adding DMF to the solution. This causes greater changes in volume expansion with an increase in temperature. With a high BuOH content in the mixture, a slight maximum is visible at the two lowest temperatures (293.15 K, 298.15 K). For higher temperatures on the $E_{p,\mathrm{m}}=f\left(x_{\mathrm{DMF}}\right)$ relationship, we observe only a change in the slope of the function. A greater effect of temperature on the value of this function is observed when BuOH prevails in the mixture. As the DMF content increases, the temperature differentiates the $E_{p,\mathrm{m}}$ values of the mixture increasingly less. Small changes in the value of the isobaric molar expansion in the DMF rich region show that the structure of this solvent remains only slightly associated by dipol-dipol interactions.

Using the calculated values of $E_{p,\mathrm{m}}$ and Equations (14) and (15) [25] excess molar isobaric expansion was determined ($E_{p,\mathrm{m}}^{\mathrm{E}}$) and presented in Figure 4:

$$E_{p,\mathrm{m}}^{\mathrm{id}}={\alpha }_p^{\mathrm{id}}·V_{\mathrm{m}}^{\mathrm{id}}=\left({\phi }_{\mathrm{BuOH}}·{\alpha }_{p,\mathrm{BuOH}}^{*}+{\phi }_{\mathrm{DMF}}·{\alpha }_{p,\mathrm{DMF}}^{*}\right)·(x_{\mathrm{BuOH}}·V_{\mathrm{BuOH}}^{*}+x_{\mathrm{DMF}}·V_{\mathrm{DMF}}^{*})$$

(14)

$$E_{p,\mathrm{m}}^{\mathrm{E}}=E_{p,\mathrm{m}}-E_{p,\mathrm{m}}^{\mathrm{id}}$$

(15)

where:${\phi }_{\mathrm{BuOH}},{\phi }_{\mathrm{DMF}}$ are volume fraction of BuOH and DMF and ${\alpha }_{p,\mathrm{BuOH}}^{*},{\alpha }_{p,\mathrm{DMF}}^{*}$ are the coefficient of thermal expansion of pure BuOH and DMF respectively.

Excess isobaric molar expansion in the entire concentration range takes positive values. A real solution has a greater ability to thermally expand relative to the ideal solution. In the range of 0.4 ≤ $x_{\mathrm{DMF}}$ ≥ 0.5 (depending on the measurement temperature) a maximum appears, which proves the occurrence of characteristic changes in the interactions between molecules in this composition range. The value of $E_{p,\mathrm{m}}^{\mathrm{E}}$ is the lowest at T = 318.15 K, which means that at the highest temperature the volumetric expansion of a real solution is the smallest and increases with decreasing temperature. It should be expected that at the lowest temperatures, the interactions between molecules will be stronger in relation to the intermolecular interactions weakening with increasing temperature.

3. Experimental

4. Conclusions

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