Linear Least-Squares Regression

Problem 14.7.

The following data were gathered to determine the relationship between pressure and temperature of a fixed volume of 1 kg of nitrogen. The volume is 10 m³.

Employ the ideal gas law pV = nRT to determine R on the basis of these data. Note that for the law, T must be expressed in kelvins. 

Solution:

Step 1: Rearrange ideal gas law with p in terms of T. 

p = (nR/V)T

Note that the above expression is a linear equation (intercept = 0, slope = nR/V).

Step 2: Apply criteria for best-fit line by minimizing the sum of squares of the residuals.

The sum of squares of the residuals can be written as:

To determine the value for a₁, the above equation is differentiated with respect to the unknown:

Next, setting the derivative equal to zero will result in minimum S_r:

Therefore, 

Step 3: Set up the data given in tabular form and compute the necessary sums. 

a₁ = 20601452.5/693933.5 = 29.688

Thus, the best-fit line is p (N/m²) = 29.688T (K). 

Step 4: Determine R based on the slope of best-fit line. 

First, we need to compute n (no of moles of nitrogen):

n = 1 kg / 28 kg/kmol = 0.035714 kmol

Given volume, V = 10 m³. Since nR/V = slope = 29.688, R = 29.688V/n.

R = 8.3127 J/mol.K

*Try to verify the unit.