INTRODUCTION:

Contemporary measuring techniques allow researchers to collect increasingly extensive data within shorter timeframes. Regression analysis, a statistical method, can be used to identify the parameter values that best represent experimental data by establishing a specific relationship between two or more variables. This method is applied to determine the optimal curve or line for the data points by minimizing the sum of the squares of the discrepancies (or residuals) between the actual values and those predicted by a theoretical model. It improves precision, provides optimal parameters, and addresses value complexity. It helps to smooth out and filter random measurement errors in the data, leading to a more reliable estimate of surface tension.

The method provides the most precise and unbiased estimate of the unknown parameters of the model being fitted for predicting the surface tension of n-pentane, n-heptane, and their mixtures within the studied temperature range. Regression analysis is a statistical method used to identify the parameter values that best represent extensive experimental data by establishing a specific relationship between variables. It determines the optimal curve by minimizing the sum of the squares of the discrepancies (residuals) between actual and predicted values. This process improves precision, filters random measurement errors, and provides a reliable, unbiased estimate of unknown parameters, such as the surface tension of n-pentane and n-heptane mixtures.

Theoretical Expressions Related to Least Square Method:

Description of the Problem:

Often in real world one expects to find linear relationships between variables. For example, Ferguson equation^1-5 has the form , where and are constants for a substance. (Here, , is the difference between the critical temperature and the observed temperature, is the surface tension of the substance). Thus, they assemble data of the form () for unfortunately, it is unlikely that we will observe a perfect linear correlation. Two factors contribute to this. The first relates to experimental errors; the second suggests that the underlying relationship may not be strictly linear but rather approximately linear. To determine the line that provides the "best fit" for the data, the method of least squares^6-18 is a procedure that requires some understanding of calculus and linear algebra. Thus for ‘best-fit’, the general problem is given as: for function, find values of coefficients such that the linear combination is the best fit approximation to the data.

Variance and standard deviation:

Given a sequence of data, we define the mean (or the expected value) to be.

(1)

We denote this by writing a line above ; thus

(2)

The variance of is denoted by , is

(3)

The standard deviation is the square root of the variance

(4)

Equation of the least squares regression line

(5)

Where A is the y-intercept, and B is the gradient /slope. The equation of the gradient/slope is given by

(6)A

where, r is the correlations coefficients, is the standard deviation of the x-values and is the standard deviation of the y-values.

The equation of the y-intercept is given by

(7)

Where, is the mean of the x-values and, is the mean of the y-values.

The method of least squares:

Given data , we may define the error associated to saying by

(8)

This is just N times the variance of the data set. The choice between using the variance or N times the variance to calculate our error is inconsequential. It's important to remember that the error is a function of two variables. Our goal is to identify the values of A and B that will minimize this error. In multivariable calculus, we learn that we need to determine the values of (A, B) such that.

(9)

Least Square Fitting:

The Least Squares Method is utilized to derive a generalized linear equation between two variables. The dependent and independent variables are represented as x and y coordinates within a 2D Cartesian coordinate system. Initially, known values are plotted, resulting in what is referred to as a scatter plot. The Least Squares technique is a well-established mathematical approach for data fitting, evaluation, regression analysis, and predictive modeling. In the context of regression analysis, this method is commonly employed to approximate sets of equations when there are more equations than unknowns.

In the least square fitting, we have the following relations between A, B, Δ, and can be given by¹:

(10)

(11)

For the least squares estimates of the constants A and B, we were omitted i=1 to N from the summation signs ∑ and also omit the subscripts i then,is also written as and the two equations (1) and (2), also called normal equations, and easily solved.

(12)

(13)

The Equation (3) and (4) give the best estimation for the constants A and B. where i have introduced the convenient abbreviation for the denominator,

(14)

The results (3) and (4) give the best estimates for the constants A and B of the straight line

y=A+ BT, based on the N measured points (T₁, σ₁), (T_N, ). The resulting line is called the least-squares fit to the data, or the line of regression of σ on T. then equation becomes,

(15)

calculate the uncertainty in the measurements of and finally we get the value of , as

(16)

(17)

In most cases, when analyzing experimental data, some "scatter" can be attributed to errors. These errors may result from imprecision in instruments, overlooked restrictions or factors, human mistakes, or various other sources. Often, we seek to determine the correlation between the variables in our data, which we achieve through regression analysis. A challenge arises when selecting a "best-fit" line because many different lines may appear to be the "best." Personal preferences regarding the fit of our line can also influence our choices, making it difficult to identify an objective best-fit line. The least squares method is a technique that helps select a line that minimizes the error between all data points and the line that best represents the data. In this analysis, we will explore experimental data on temperature, the square of temperature, surface tension, and the product of surface tension and temperature for n-pentane, n-heptane, n-decane, are given in Table 1, Table 3 and Table 5(see Ref.1) and Table 2, Table 4 and Table 6 from Ref.2. For binary mixture of n-pentane+ n-heptane at composition and are given in Table 7 and Table 8 respectively¹.

Table 1: -pentane surface tension values and comparison with literature¹ (T vs σ).

Obs.	*T/K*	σ/(mNm^-1)	T_i²	T_iσ_i
1	293.15	15.94	85936.92	4672.811
2	298.15	15.30	88893.42	4561.695
3	305.15	14.36	93116.52	4381.954
4	318.15	12.60	101219.4	4008.69
5	323.15	11.95	104425.9	3861.643
N = 5	= 1537.75	= 70.15	= 473592.2	= 21486.79

Table 2: n-pentane surface tension values and comparison with literature²(T vs σ).

Obs.	*T/K*	σ/(mNm^-1)	T_i²	T_iσ_i
1	293.15	16.11	85936.92	4722.647
2	298.15	15.52	88893.42	4627.288
3	305.15	14.56	93116.52	4442.984
4	318.15	12.89	101219.4	4100.954
5	323.15	12.19	104425.9	3939.199
N = 5	= 1537.75	= 71.27	= 473592.2	= 21833.07

Fig.1 The plot illustrates the relationship between surface tension and temperature for n-pentane. The lines represent the best fit lines (refer to Table 11 for coefficients A and B), while the points are derived^1,2 from Table 1 and Table 2.

Fig.2 The plot illustrates the relationship between surface tension and temperature for n-heptane.

The lines represent the best fit lines (refer to Table 11 for coefficients A and B), while the points are derived^1,2 from Table 3 and Table 4. The surface tension values for n-heptane at different temperatures were obtained and are presented in Tables 3 and 4. The measured surface tension values of n-pentane were compared with those in Ref. 2. Our studies on the pure components indicate that an increase in temperature results in a decrease in surface tension.

Table 5: n-decane surface tension values and comparison with literature¹. (T vs σ)

Obs.	*T/K*	σ/(mNm^-1)	T_i²	T_iσ_i
1	293.15	24.12	85936.92	7070.778
2	303.15	23.16	91899.92	7020.954
3	313.15	22.22	98062.92	6958.193
4	323.15	21.17	104425.9	6841.086
5	333.15	20.15	110988.9	6712.973
6	343.15	19.23	117751.9	6598.775
N = 6	= 1908.9	= 130.05	= 609066.5	= 41202.76

Table 6: n-decane surface tension values and comparison with literature²(T vs σ).

Obs.	*T/K*	σ/(mNm^-1)	T_i²	T_iσ_i
1	293.15	23.83	85936.92	6985.765
2	303.15	22.91	91899.92	6945.167
3	313.15	21.91	98062.92	6861.117
4	323.15	21.07	104425.9	6808.771
5	333.15	20.15	110988.9	6712.973
6	343.15	19.23	117751.9	6598.775
N = 6	= 1908.9	= 129.1	= 609066.5	= 40912.57

Fig. 3: The plot illustrates the relationship between surface tension and temperature for n-decane. The lines represent the best fit lines (refer to Table 11 for coefficients A and B), while the points are derived^1,2 from Table 5 and Table 6.

The surface tension values for n-decane at different temperatures were obtained and are presented in Tables 5 and 6. The measured surface tension values of n-pentane were compared with those in Ref [2]. Our studies on the pure components indicate that an increase in temperature leads to a decrease in surface tension.

Table 7: Surface tension values of [n-pentane (x₁) +n-heptane (x₂)] Binary Mixtures² at x = 0.165.

Obs.	*T/K*	σ/(mNm^-1)	T_i²	T_iσ_i
1	293.15	18.60	85936.92	5452.59
2	298.15	17.90	88893.42	5336.885
3	305.15	17.13	93116.52	5227.22
4	318.15	15.66	101219.4	4982.229
5	323.15	14.88	104425.9	4808.472
N =5	= 1537.75	= 84.17	= 473592.2	= 25807.4.97

Table 8: Surface tension values of [n-pentane (x₁) +n-heptane (x₂)] Binary Mixtures² at x = 0.524.

Obs.	*T/K*	σ/(mNm^-1)	T_i²	T_iσ_i
1	293.15	19.62	85936.92	5751.603
2	298.15	18.98	88893.42	5658.887
3	305.15	18.16	93116.52	5541.524
4	318.15	16.91	101219.4	5379.917
5	323.15	16.33	104425.9	5277.04
N = 5	= 1537.75	= 90	= 473592.2	= 27608.97

Fig.4 The plot illustrates the relationship between surface tension and temperature for binary mixture of n-pentane and n-heptane at =0.165 and =0.52. The lines represent the best fit lines (refer to Table 11 for coefficients A and B), while the points are derived¹ from Table 7 and Table 8.

Tables 7 and 8 present the surface tension data for binary mixtures of n-pentane and n-heptane at various temperatures. The composition dependence of the surface tension in these mixtures can be represented in terms of excess surface tension. Our studies on the pure components indicate that an increase in temperature leads to a decrease in surface tension. This trend suggests that the interactions between the components in the mixture are affected by thermal energy, which may change their molecular arrangements. Further investigation into the excess surface tension could offer deeper insights into the molecular interactions occurring in these mixtures.

Calculation:

Table 9: Calculated values of A, B Δ, and by equations (12), (13), (14), (15), (16) and (17).

Table	Substaces	From Eq-12 A	From Eq-13 B	From Eq-14 Δ	From Eq-15	From Eq-16	From Eq-17
1.	n-decane	50.71374	-0.09177	10500	61.21450	466.2212	1.463306
2.	n-decane	53.06277	-0.09866	10500	59.10691	450.1694	1.869125
3.	n-pentane	55.13650	-0.13366	3286	57.04694	684.8585	2.225275
4.	n-pentane	54.50786	-0.13089	3286	57.04694	684.8585	2.225275
5.	n-heptane	49.2268	-0.0993	5000	57.04694	547.4721	1.803983
6.	n-heptane	50.03773	-0.1015	5000	57.04694	547.4721	1.803983
7	n-pentane+ n-heptane at x₁ = 0.165	51.00594	0.10732	3286	57.04694	684.8585	2.225274
8.	n-pentane+ n-heptane at x₁ = 0.542	53.84483	0.12034	3286	57.04694	684.8585	2.225274

Table 10: Statistical summary

Constant		Estimate	Std. Error	t-value	P(>\|t\|)	R-Squared
A	n-pentane	55.1365	0.1816	303.6495	0.0000	0.9999
B	n-pentane	-0.1337	0.0006	-226.5403	0.0000	0.9999
A	n-pentane	52.2764	1.6406	31.8634	0.0001	0.9945
B	n-pentane	-0.1241	0.0053	-23.3304	0.0002	0.9927 *
A	n-heptane	49.2268	0.1391	353.9666	0.0000	0.9999
B	n-heptane	-0.0993	0.0005	-216.6904	0.0000	0.9999 *
A	n-heptane	50.0377	0.0303	1648.7998	0.0000	1.0000
B	n-heptane	-0.1015	0.0001	-1015.0000	0.0000	1.0000 *
A	n-decane	51.9600	0.3786	137.2591	0.0000	0.9994
B	n-decane	-0.0950	0.0012	-79.9803	0.0000	0.9992
A	n-decane	52.2764	1.6406	31.8634	0.0001	0.9945
B	n-decane	-0.1241	0.0053	-23.3304	0.0002	0.9927
A	Binary Mixture of (n-pentane + n-heptane) at x₁=0.165	51.0059	0.9350	54.5539	0.0000	0.9976
B	Binary Mixture of (n-pentane + n-heptane) at x₁=0.165	-0.1073	0.0030	-35.3263	0.0000	0.9968 *
A	Binary Mixture of (n-pentane + n-heptane) at x₁=0.52	53.6187	1.0891	49.2301	0.0000	0.9975
B	Binary Mixture of (n-pentane + n-heptane) at x₁=0.52	-0.1213	0.0035	-34.2623	0.0001	0.9966 *

* Adjusted R-Squared

RESULTS AND DISCUSSION:

The surface tension of n-decane, n-pentane, n-heptane, and binary mixtures at various temperatures indicates that an increase in temperature results in a decrease in surface tension. The surface tension values of n-decane for temperatures ranging from T = 293.15 to 343.15 K were compared with data from references^1,2. Similarly, the surface tension values for n-pentane and for n-heptane as well as for n-decane at these temperatures (i.e., T = 283.15 to 323.15 K), are presented^1,2 in Tables 1 through 6 and plotted in Figures 1 through 4. These comparisons can also confirm the consistency of the values. The results show that the temperature increase causes the surface tension decrease. Table 7 shows the values of surface tension data of (n-pentane + n-heptane) binary mixtures at temperature T = (293.15 to 323.15K) at . Fig 1-4 represents the plots of surface tension versus temperature for n-Pentane, n-Heptane and n-decane. The lines represent the best fit lines (see Table 11 for A and B values), while the points are derived^1,2 from Table 1 and Table 2. Table 11 provides a statistical summary that includes the estimates for A and B, standard errors, t-values for each case of the considered substances, P(>|t|) and R-squared (R²) is defined as an estimate of the relationship between the movements of a dependent variable based on the movements of an independent variable. The values indicate how well the independent variable(s) in a statistical model explain the variance in the dependent variable. R-squared is applicable only in a basic linear regression model with a single explanatory variable. In multivariate regression, which involves multiple independent variables, the R-squared value must be adjusted. The adjusted R-squared helps assess the explanatory power of regression models that include different numbers of predictors.

CONCLUSIONS:

A type of mathematical analysis is employed to establish the line of best fit for various data sets, which results in a visual representation known as the least squares fit. This method identifies the optimal line that illustrates the relationship between known values and estimates, thereby indicating a more accurate model fit. It is commonly utilized across multiple fields, including economics, biology, and engineering, to predict outcomes and identify patterns. By minimizing the differences between actual values and model predictions, researchers can derive meaningful insights from their data. A smaller standard error in the estimate suggests a superior model fit.

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Received on 25.01.2026 Revised on 19.02.2026

Accepted on 13.03.2026 Published on 27.05.2026

Available online from May 30, 2026

Asian J. Research Chem.2026; 19(3):253-258.

DOI: 10.52711/0974-4150.2026.00039

This work is licensed under a Creative Commons Attribution-Non Commercial-Share Alike 4.0 International License. Creative Commons License.

Obs.	*T/K*	σ/(mNm^-1)	T_i²	T_iσ_i
1	283.15	21.11	80173.92	5977.297
2	293.15	20.12	85936.92	5898.178
3	303.15	19.13	91899.92	5799.26
4	313.15	18.11	98062.92	5671.147
5	323.15	17.15	104425.9	5542.023
N= 5	= 1515.75	= 95.62	= 460499.6	= 28887.9