The assignment should be done with MS Excel Part 1: Mathematically modelling the density of mercury liquid between -20.0 and 50.0 C (1.00 atm pressure).

The file HgDensity contains two columns of data: the temperature in C and the density in g cm-3 of pure liquid mercury.

1. a) Plot the data on a density against temperature graph. [Select both data columns, go to Insert, Chart section and select Scatter this will give a plot with just points, no connecting lines]

2. b) Carry out a linear regression. One way to do this is to add a trendline. [Click on the data, right-click to open a pop-up menu, select Add trendline, from Format trendline box, select Linear, Display Equation on chart and Display r-squared value on chart. The equation may not have many figures displayed. Youll need to right-click on the equation, select Format trendline label, Scientific format and 7 decimal places]

3. c) The data looks linear, but is a linear trendline really the best description? To determine this, use the coefficients in the linear equation from the trendline and a cell formula to calculate the density for the first temperature. This is the density predicted by the linear fit. Copy the formula down the column for all temperatures.

4. d) Subtract the calculated density from the experimental one for each temperature. These differences are called the residuals to the fit because they are what is left over when you remove the experimental data. Plot the residuals against temperature and examine the graph. If the linear equation is an adequate description, the residuals will appear randomly distributed around zero and there will be no obvious pattern suggesting underlying structure to the data.

5. e) The pattern you should be seeing in the linear fit residuals resembles a parabola. This suggests that the linear model is not reproducing all of the information in the original data. The shape (parabola) suggests that the information not captured in the linear model is quadratic in nature. Consequently, we should try to fit the original data to a

quadratic equation. Go back to the original data set (not the linear residuals!) and add a trendline that is polynomial, order 2 (this is a quadratic equation) instead of linear. Again, display the equation on the chart, with parameters in scientific notation and at least 7 decimal places.

6. f) Once again, use the coefficients from the quadratic equation of the trendline and a cell formula to calculate the density predicted by the quadratic equation for the first temperature. Copy the formula down the column for all temperatures.

7. g) Calculate and plot the residuals to the quadratic fit (that is, the difference between the experimental and the densities predicted by the quadratic fit. Does the data look more randomly distributed? You may see some odd patterns due to the relatively low numeric precision of the data, but over all it should look a lot more random. If you use your imagination, you might see a slight cubic structure to the residual plot, but the residuals are all less than the number of significant figures in the density data. This means that the quadratic equation for the density has captured all of the information in the original data set

8. h) The advantage of a mathematical model, such as the quadratic equation youve determined is that it is a very concise description of the data you only need the equation and the associated constants (two for linear, three for quadratic, etc). It is also helpful if you want to calculate intermediate values. Calculate the density of mercury at 11.7C using the two equations you fitted (linear and quadratic). The original data had 7 significant figures. What number of significant figures would you have to drop the data to in order for the linear fit to become an equivalent description to the quadratic fit?

9. i) Would it be appropriate to use this equation to calculate the density of pure mercury at 85 C? Why or why not?

For Part 1, please submit the spreadsheet showing your work, with labelled charts. Also, please provide a brief report of your results: the two equations you obtained, your observations and the answers to any questions above.