Study text: "Essential Mathematics and Statistics for Science", 2nd Edition,

G Currell & A A Dowman, Wiley-Blackwell, 2009

G Currell & A A Dowman, Wiley-Blackwell, 2009

- The spectrophotometric absorbance,
*A*, of a solution can be measured as a function of its concentration,*C*, according to the Beer-Lambert equation:*A*=*εbC*

where*ε*is the absorptivity of the solute and*b*is the path length of light through the solution.

In calibrating the spectrophotometer, the instrument is first ‘zeroed’ so that the value of the absorbance,*A*= 0 when the concentration,*C*= 0. The values of the absorbances,*A*, are then measured for a series of standard solutions of known concentrations,*C*, and plotted on an x-y graph.

When analysing this data, which one of the following statements is true:- Linear regression should be used to calculate the intercept of the 'best-fit' straight line.
- The mathematical calculation of linear regression will assume that the uncertainties in the values of absorbance,
*A*, are more significant than the uncertainties in the concentrations,*C*. - The slope of the ‘best-fit’ line of regression would be given by
*m*= 1 /*εb* - The concentrations,
*C*, should be plotted on the y-axis.

- The pressure,
*P*, for*n*moles of an ideal gas at a fixed temperature,*T*, as a function of volume,*V*, is given by the ideal gas equation:

where*R*is the gas constant.

Which one of the following procedures would be used to calculate the value of*nRT*from the experimental data, so that is possible to calculate the value of*R*(with values of*n*and*T*already known).- Plot
*P*against and use the intercept of the graph to calculate*nRT*. - Plot
*P*against and use the slope of the graph to calculate*nRT*. - Plot
*P*against log(*V*) and use the slope of the graph to calculate*nRT*. - Plot
*P*against log(*V*) and use the intercept of the graph to calculate*nRT*.

- Plot
- In an experiment to measure the energy of photo-electrons, the voltage,
*V*, required to stop the electrons is measured as a function of the wavelength,*λ*, of ultra-violet light releasing the electrons from the material. The relevant equation is

where*h*,*c*and*e*are fundamental constants, and Ø is the work function of the material.

Which of the following procedures would be used to calculate the value of Ø from the experimental data:- Plot
*λ*against*V*and use the intercept of the graph to calculate Ø. - Plot
*λ*against*V*and use the slope of the graph to calculate Ø. - Plot
*V*against and use the intercept of the graph to calculate Ø. - Plot
*V*against and use the slope of the graph to calculate Ø.

- Plot
- The equation
*N*_{t}= N_{0}e^{kt}

describes the growth of a population,*N*, as a function of time,_{t}*t*, where*k*is the growth constant.

This equation can be linearised by:- plotting ln(
*N*) against ln(_{t}*t*) - plotting ln(
*N*) against_{t}*t* - plotting ln(
*N*-_{t}*N*_{0}) against*t* - plotting log(
*N*) against log(_{t}*t*)

- plotting ln(
- The radioactivity,
*A*, is recorded as a function of time,_{t}*t*, starting with an initial activity,*A*_{0}, at time*t*= 0.

Assuming the relationship

ln(*A*) = ln(_{t}*A*) -_{0}*kt*

experimental values of ln(*A*) are plotted against_{t}*t*. A linear regression calculation gives the slope,*m*= - 0.0470, and the intercept,*c*= 3.20. Calculate (to 3 sf) the best estimate value for*A*_{0}from these values.- 0.954
- 1580
- 3.20
- 24.5