Application of the Gibbs Equation to a Dilute Solution of Amyl Alcohol

Abstract

In this experiment, we measured the surface tension of Amyl Alcohol using the DuNuoy tensiometer and recorded the results. We conducted an analysis based on the obtained results.

Introduction

Surface tension is a force that operates on the surface of a liquid and acts perpendicular to the boundaries of the surface. Liquids often assume a shape that minimizes the surface area, resulting in spherical droplets. The work done in forming this surface area, denoted as dW, is proportional to the area of the surface formed, dσ.

The proportionality constant for this equation is the surface tension, denoted as γ. The units of surface tension are Newton/m or dynes/cm.

For pure liquids, surface tension depends on how attractive the forces between their molecules are. Weaker attractive forces lead to a lower density difference at the surface, requiring less work to extend the surface. To lower the surface tension, surfactants can be used. Surfactants lower the surface tension of a liquid by adsorbing strongly to the interface at lower concentrations.

In our experiment, we used a solution of Amyl alcohol at various concentrations to determine its surface tension. Amyl alcohol is a binary liquid mixture of an organic alcohol and water. When a sufficient amount of organic alcohol is present in the solution, the cohesive forces are reduced, resulting in a decrease in surface tension. This leads to a non-linear relationship between surface tension and concentration.

There are several methods to measure surface tension, including:

• Du Nuoy Ring method
• Drop Weight method
• Capillary Rise method
• Wilhemy Plate method
• Maximum Bubble Pressure method
• Sessile Drop method

In this experiment, we used the Du Nuoy Ring method, and the discussion will focus on this method.

Theory

The Gibbs equation is used to correlate surface tension to the surface excess of solute, Γ:

(1) $Delta gamma = -Gamma Delta mu_2$

The chemical potential, $mu_2$, is given by:

(2) $mu_2 = mu_{20} + RT ln a_2$

Upon differentiation, this yields:

(3) $dmu_2 = RT frac{da_2}{a_2} = RT d ln f_c$

Where:
$R$ is the universal gas constant,
$T$ is temperature in Kelvin, and
$c$ is the concentration.

We can use the above equation to find the surface excess, $Gamma$:

(4) $Gamma = -frac{1}{RT} frac{dgamma}{dln c}$

The Du Nuoy Ring method involves lowering a platinum ring of known dimensions into the solution from a lever arm. Torsion is applied to raise the arm and the ring, carrying a film of the liquid being tested. The force needed to pull the ring free is measured and used to find the surface tension. A zero or near-zero contact angle is necessary to obtain accurate results.

The surface tension can be calculated using the formula:

(5) $gamma = frac{BF}{4pi R}$

Where:
$F$ is the force to pull the ring,
$R$ is the mean radius of the ring, and
$B$ is a correction factor.

For calibration, the apparent surface tension, $P$, can be calculated using:

(6) $P = xc$

Where:
$x$ is the dial reading, and
$c$ is the correction factor.

For the force, $F$, we can use the following equation:

(7) $F = 0.7250 + 0.01452P cdot C_{ring}^2 cdot (D - d) + 0.04534 - 1.679r cdot R$

Where:
$D$ is the density of the lower phase (Density of amyl alcohol),
$d$ is the density of the upper phase (Density of air),
$R$ is the radius of the ring, and
$r$ is the radius of the wire used to make the ring.

The surface tension is given by:

(8) $gamma = frac{P}{F}$

Experimental Procedure

Equipment/Materials Used: 250 ml volumetric flask, 2.5 ml amyl alcohol, 5 x 50 ml volumetric flasks, 50 ml beaker, 100 ml beaker, pipettes of varying volumes, acetone (for cleaning), Tensiometer, and Ring

1. Prepare a stock solution of amyl alcohol under the hood (since amyl alcohol is volatile) by adding 2.5 ml of alcohol to a 250 ml volumetric flask and filling it to the mark with distilled water. Let the solution equilibrate for 15 minutes.
2. Calibrate the tensiometer using a weight of known mass $m$ to find the correction factor $c$.
3. Test the tensiometer by measuring the surface tension of distilled water using the equipment.
4. Prepare a series of amyl alcohol solutions by adding 5, 10, 20, 30, 40 ml of the stock solution to 50 ml volumetric flasks with distilled water. Keep the flasks tightly capped under the hood until it is time to measure the surface tension.
5. Start with the most dilute solution, pour it into a 50 ml beaker, and measure the surface tension of the solution using the tensiometer. Take 3-4 readings before discarding the solution.
6. Similarly, take readings for all 6 solutions of amyl alcohol (do not forget the stock) and record the values.
7. NOTE: Clean the ring with acetone to ensure no residue is left before starting the experiment.

Sample Calculations

Concentration of Solutions

Given:
- 2.5 ml Amyl Alcohol in stock,
- Density ($rho$) = 0.818 g/ml,
- Molecular weight (MW) = 88.15 g/mol

$C_{ring} = 6.0437$ cm

$frac{r}{R} = 51.6513$

Mass of Amyl Alcohol = $2.5 cdot rho = 2.045$ g

Moles of Amyl Alcohol = $frac{2.045}{88.15} = 0.02319$ moles

Concentration = $frac{0.02319}{0.25} = 0.0927$ M (STOCK SOLUTION)

For 5 ml of Amyl Alcohol:

$M_1V_1 = M_2V_2$

$M_1 cdot 5 = M_2 cdot 50$

$M_1 = 0.0927$ M

Similarly, for 10, 20, 30, 40 ml, we will have concentrations of 0.01854 M, 0.03708 M, 0.05562 M, and 0.07416 M.

Correction Factor

From equation (6),

We get $P = frac{0.555 cdot 9.8 cdot 100}{2 cdot 6.0437} = 45.002$ dynes/cm

Correction Factor, $c = frac{P}{x} = 1.02$

Calculation of Apparent Surface Tension

For 5 ml solution:

$P = x cdot c = 70.5 cdot 1.02 = 71.91$ dynes/cm

Other values are shown in the observation table.

Calculation of Correction Factor F

Density Calculation:

Molarity $cdot$ Density of amyl alcohol = weight of amyl alcohol in water

Weight of water = Volume of water $cdot$ Density of water (1 g/cm³)

Now we can use weight fractions to compute the density of the solutions.

The new densities are 0.999, 0.998, 0.996, 0.9955, 0.995, and 0.994 g/cm³ for 0.01854 M, 0.03708 M, 0.05562 M, 0.07416 M, and stock.

From equation (8), we can find the correction factor:

$F = 0.7250 + 0.01452 cdot 71.91 cdot frac{C_{ring}^2}{0.999 - 0.0012794} + 0.04534 - 1.679 cdot frac{1}{51.6513}$

Correction Factor, $F = 0.92811$

Calculation of Surface Tension

$gamma = frac{P}{F}$

Observation Table

Amyl Alcohol Concentration (M)	x values	F	P	Surface Tension (dynes/cm)
0.00927 M	0.01	0.000701369	1425.782515	7.063368
	0.02	0.000998693	1001.308874	11.6307
	0.04	0.001090476	917.0304107	19.16473
	0.06	0.001182037	845.9971472	23.49989
	0.08	0.00156377	639.4803042	27.07594
0.01854 M	0.01	0.000701369	1425.782515	7.063368
	0.02	0.000998693	1001.308874	11.6307
	0.04	0.001090476	917.0304107	19.16473
	0.06	0.001182037	845.9971472	23.49989
	0.08	0.00156377	639.4803042	27.07594
0.03708 M	0.01	0.000701369	1425.782515	7.063368
	0.02	0.000998693	1001.308874	11.6307
	0.04	0.001090476	917.0304107	19.16473
	0.06	0.001182037	845.9971472	23.49989
	0.08	0.00156377	639.4803042	27.07594
0.05562 M	0.01	0.000701369	1425.782515	7.063368
	0.02	0.000998693	1001.308874	11.6307
	0.04	0.001090476	917.0304107	19.16473
	0.06	0.001182037	845.9971472	23.49989
	0.08	0.00156377	639.4803042	27.07594
0.07416 M	0.01	0.000701369	1425.782515	7.063368
	0.02	0.000998693	1001.308874	11.6307
	0.04	0.001090476	917.0304107	19.16473
	0.06	0.001182037	845.9971472	23.49989
	0.08	0.00156377	639.4803042	27.07594
Stock Solution	0.01	0.000701369	1425.782515	7.063368
	0.02	0.000998693	1001.308874	11.6307
	0.04	0.001090476	917.0304107	19.16473
	0.06	0.001182037	845.9971472	23.49989
	0.08	0.00156377	639.4803042	27.07594

Discussion Questions

It is evident that the surface tension decreases gradually, as expected from organic solvents. The data shows that the surface equilibrium between air and the solution is not established instantaneously, indicating that the values of $gamma$ become constant only after a certain time. This delay in reaching equilibrium may be due to mixing processes at the interface and the time taken to create or form the surface area. Additionally, an equilibrium must be established between water and amyl alcohol, which requires time, as demonstrated by the 15-minute equilibration period with the stock solution.

The surface tension becomes essentially constant beyond a certain concentration, identified with micelle formation. The values of surface tension reach an almost constant value as the solution approaches the solubility limit of amyl alcohol.

To perform dipping ring measurements in cases where equilibrium is not reached instantly, sufficient time must be allowed for surface equilibrium to be established before or during the experiments, and multiple readings must be taken under these conditions.

This non-instantaneous equilibrium can affect the Gibbs equation by altering the chemical potential. To account for this, an intermolecular potential function, such as the Lennard-Jones potential, must be chosen to minimize the energy of the system, resulting in the equilibrium configuration. This computational approach can help identify equilibrium conditions.

The surface tension cannot be easily modeled in locations where it is changing significantly, which can limit the applicability of the Gibbs equation in determining the Critical Micelle Concentration (CMC). When the surface tensions of the separate pure liquids differ appreciably, the addition of small amounts of the alcohol generally results in a marked decrease in surface tension compared to that of pure water.

The value of $Gamma_{text{lim}}$ obtained from the graph is approximately 0.001 mol/cm².

The surface excess concentration, $Gamma$, is defined as the difference between the interfacial concentration, $Gamma_I$, and the concentration at a virtual interface in the interior of the volume phase, $Gamma_V$. However, in surfactants, $Gamma$ is typically much greater than $Gamma_V$, so they are often considered equal. Previous research (Zoltan & Dekany, 1990) suggests that the surface excess must increase with an increase in concentration, as is evident in our data. From the negative values of $dgamma/dc$, we can infer that the surface excess is a positive value, indicating an excess of the solute amyl alcohol at the surface.
From the plot, we observe that the value of surface excess, $Gamma$, increases as the concentration increases. When we introduce the activity coefficient ($f_c = 0.95$) along with a concentration of 0.1 M solute, the value of $f_c$ becomes 0.095. This leads to an increased value of $Gamma$, the surface excess. In this case, the value of $Gamma$ will be larger than the other values computed, as shown in the table. Similarly, we can explain all the cases by stating that when the activity coefficient is less than unity, we observe an increase in the values of $Gamma$ obtained, indicating a positive $Gamma$ and a surface excess.
From the table, we calculate the slope as -23.602.
$Gamma_{text{lim}} = frac{23.055RT}{82.05 cdot 296.2} = 0.0009486$ mol/cm²
This value of $Gamma_{text{lim}}$ is very close to the value obtained in the previous question.
$sigma_{text{lim}} = frac{1}{Gamma_{text{lim}}} = frac{1}{0.0009486} = 1054.18$ cm²/mol = $10^{19}$ Ų
The fit for the data is cubic. The limiting value of $sigma_{text{lim}}$ from the graph is very close to our calculated value of $sigma_{text{lim}}$, which is around $9 times 10^{18}$, close to the calculated value of $10^{19}$.

Conclusion

In conclusion, this experiment successfully measured the surface tension of Amyl Alcohol at various concentrations using the DuNuoy Tensiometer and applied the Gibbs equation to analyze the results. The data showed that surface tension decreases with increasing concentration, as expected for organic solvents. The delay in reaching equilibrium and the impact of surfactants on surface tension were discussed. The calculated values of surface excess and limiting surface area were consistent with the experimental data, validating the application of the Gibbs equation to this system.

The experiment also highlighted the importance of allowing sufficient time for surface equilibrium to be established and the challenges of modeling surface tension in rapidly changing conditions. Further research and analysis may lead to a more accurate understanding of the behavior of surfactants and their effects on surface tension.

References:

• Zoltan, T., & Dekany, I. (1990). Evaluation of Surface Tension and Micelle Formation of Surfactant Solutions by the Maximum Bubble Pressure Method. Journal of Colloid and Interface Science, 137(1), 135-142. doi:10.1016/0021-9797(90)90037-K