Chapter 4, Problem 4.35P

(a)

Interpretation Introduction

Interpretation:

Using spreadsheet and using the method of least squares the slope, intercept and their uncertainties has to be found out and also a calibration curve has to be constructed.

Concept Introduction:

Linear Calibration Curve:

The equation of linear calibration line is:

$y(\pm s_y)=[m(\pm u_m)]x+[b(\pm u_b)]$

Where,

$y$ is the corrected absorbance.

$\text{corrected absorbance = observed absorbance - blank absorbance}$

Standard deviation of y: $s_y$

The standard deviation of y is:

${\sigma }_y\approx s_y=\sqrt{\frac{{\displaystyle \sum (d_i{}^2)}}{n-2}}$

Standard uncertainty of slope and intercept:

The standard uncertainty of slope is:

${\text{u}}_m^2=\frac{s_y^{2}n}{D}$

The standard uncertainty of intercept is:

${\text{u}}_b^2=\frac{s_y^2{\displaystyle \sum (x_i{}^2)}}{D}$

(b)

Interpretation Introduction

Interpretation:

Using spreadsheet the average corrected signal for the unknown with given replicate measurements has to be found out.

Concept Introduction:

Linear Calibration Curve:

The equation of linear calibration line is:

$y(\pm s_y)=[m(\pm u_m)]x+[b(\pm u_b)]$

Where,

$y$ is the corrected absorbance.

$\text{corrected absorbance = observed absorbance - blank absorbance}$

To Find: The average corrected signal for the unknown using spreadsheet.

(c)

Interpretation Introduction

Interpretation:

The concentration of the unknown with its standard uncertainty $(u_x)$ at the $95\%$ confidence interval has to be found out.

Concept Introduction:

Uncertainty with a Calibration Curve:

Propagation for the equation $y=mx+b$ (but not $y=mx)$ gives,

The standard uncertainty in x is given as:

$u_x=\frac{s_y}{\left|m\right|}\sqrt{\frac{1}{k}+\frac{1}{n}+\frac{{(y-\overline{y})}^2}{m^2{\displaystyle \sum {(x_i-\overline{x})}^2}}}$

Where,

$u_x$ is standard uncertainty for $x$

$y$ is the corrected absorbance of unknown

$x_i$ is the mass of sample in standards

$\overline{y}$ is the average of $y$ values

$\overline{x}$ is the average of $x$ values

k is the number of replicate measurements

To Find: The concentration of the unknown with its standard uncertainty $(u_x)$ at the $95\%$ confidence interval