Determining the Density and Percent Air Space in Sand Using a TI-82 Calculator

Description:
When sand fills a container, a portion of its volume is due to the air spaces between the individual particles. As water is added to the container, it will seep into the gaps and openings, displacing the air space. The differences between the predicted maximum volume and the actual measured volume of the mixture can be determined, and these values used to calculate the density of sand and the percent of air space between the granuals. The experimental data can be analyzed with the help of a graphing calculator.

Procedure:

1. Use a funnel to pour sand into a dry, 10 mL graduated cylinder to a height of 5-6 mL. Record the exact volume. Place a beaker(or piece of weighing paper) on an elctronic balance and press the 'tare' button to re-zero the balance. Transfer the sand sample to the beaker and record the mass of the sample. Label the beaker as "sample 1". Repeat this step to obtain 3 more samples of the dry sand, labeled as samples 2, 3 and 4.
2. Add "sample 1" of the dry sand to a 50 mL cylinder, using the funnel to avoid spilling it.
3. Measure 7-8 mL of water in the 10 mL cylinder, record the exact volume, then add it to the 50 mL cylinder. Stir the mixture to allow the water to fill the spaces in between the sand, then record the volume of the mixture.
4. Repeat steps 2 and 3 for the other three samples of sand. (Add "sample 2" of the dry sand to the 50 mL cylinder, add another 7-8 mL of water, measure the mixture's volume, etc.)
5. Calculate the total mass of sand and total volumes of sand and water used in each trial. Pour the wet sand and water into a paper towel, then place under the heat lamp to dry.

Enter your data and totals into the chart below:

 - Enter in L1 Enter in L2 Enter in L3 Enter in L4 - Total mL of Sand Total g of Sand Total mL of Water Mixture Volume, mL Trial 1 1. 1. 1. 1. Trial 2 2. 2. 2. 2. Trial 3 3. 3. 3. 3. Trial 4 4. 4. 4. 4.

Analysis of Data Using a TI-82 Calculator

Preliminary Calculator set-up

You must first clear the calculator's memory of any formulas or data that might interfere with your analysis.

1. Turn calculator ON (lower left)
2. Press Y= (upper left) and CLEAR if there is an equation listed after the Y1=. Press the 2nd and QUIT keys to exit this mode.
3. Press the STAT and ENTER keys to check(and edit) any data that is stored in the calculator as a "list". (Note: There are 6 lists labeled L1 to L6. Use the right arrow key to view the last 3 lists.) To clear data from the statistics menu, press the STAT and 4 keys to select "ClrList", then press the following six keys to clear lists L1 and L2: 2nd 1 , 2nd 2 ENTER The data in lists L3 to L6 can be cleared in a similar way, but only L1 and L2 will be used to plot a graph.

TI-82 graphing

1. Turn calculator ON (if it OFF)
2. Push the STAT and ENTER keys to add new data to the lists.
3. Type in your experimental data using either the down arrow or ENTER after each entry. Place your data for the "Total mL of Sand" in L1, "Total g of Sand" in L2, "Total mL of Water" in L3, and "Mixture Volume" in L4.
4. Press the 2nd button, and then the STAT PLOT (under the Y=)
5. Enter number 1 or, just push the ENTER button.
6. Use the arrow and ENTER keys to select the graphing options to match those shown in the copy of the calculator display at the right. This set up will plot a graph of the total mass of the sand versus its total volume, which equals the density of the dry sand.
7. Press GRAPH (upper right) and then ZOOM 9 to expand the graph to fill the screen window.
8. Draw a sketch of the displayed graph.

TI-82 Regression Line

1. Push the STAT button.
2. Move the cursor to CALC.
3. Press number 5 to select the function to find a linear regression line, y = ax + b.
4. Press 2nd and the L1 key (under the number 1 key) and then the (,) comma followed by 2nd and the L2 key (under the number 2 key). This selects the data in lists L1 and L2 as the x and y values.
5. Press ENTER to perform the calculation.
6. Copy on paper the a(slope), b(intercept), and r(correlation coefficient) values. The slope is equal to the density of the dry sand. The correlation coefficient(r) indicates how well the data fits the straight line. A value of 1 indicates a perfect match(100% correlation).

Paste the formula in the Y= in order to graph the regression line

1. Press Y= (Press CLEAR if there is an equation listed after the Y=).
2. Press VARS and push number 5 to select "Statistics".
3. Move the cursor to EQ and push number 7 to select "RegEQ" and paste the slope and intercept values into the Y= equation, then ENTER to signal the end of the formula.
4. Press GRAPH to see the regression line(best-fitting straight line) drawn for your data. Sketch a diagram of this graph and note any data points that do not touch the line.

TI-82 Programs to Perform a Defined Set of Operations

1. Push the PRGM button to access the calculator's program menu.
2. Select the AIRSPACE program and press ENTER twice to calculate the average percent of air space in sand, based on the 4 trials you performed. This program determines the volume differences between the separate sand and water as compared to the mixture. The decrease in volume of the mixture is due to the air space being displaced by the water. The ratio of air space to sand volume is expressed as a percentage value. Record the % air space displayed on the calculator's screen. The programming steps for AIRSPACE are listed below:
: For ( N , 1 , 4 , 1 )
: L1(N) + L3(N) --> L5(N)
: ( ( L5(N) - L4(N) ) / L1(N) ) * 100 --> L6(N)
: End
: ( L6(1) + L6(2) + L6(3) + L6(4) ) / 4 --> A
: Disp "AVERAGE PERCENT"
: Disp "AIR SPACE IS", A

3. Write your own program to determine the density of only the solid sand particles by:
• Pressing PGRM and select the option for a NEW program.
• Type in the name ASAND by pressing the keys having the appropriate white letters at the upper right corner above each key. Press ENTER when done typing in the name.
• Create a program loop to perform the same calculation for each of the 4 trials by pushing PRGM 4 ALPHA N , 1 , 4 , 1 ) ENTER This will start the calculations with the variables beginning with N equal to 1 and ending with those with N = 4, by increments of 1.
• The actual calculation divides the mass of sand by the volume of solid sand only. The volume of solid sand is equal to (100% - % air space)(Total mL of Sand). The AIRSPACE program already determined the % air space and stored the values in list L6. Type in the following keystrokes to make the string: ( 2nd L2 (ALPHA N) — ( (100 - 2nd L6 (ALPHA N) ) x 2nd L1(ALPHA N) x .01) STO 2nd L5 (ALPHA N) ENTER
• PRGM 7 completes the programming. The actual lines on your calculator should look like:
: For ( N , 1 , 4 , 1 )
: ( L2(N) / ( ( 100 - L6(N) ) * L1(N) * .01 ) --> L5(N)
: END
• Press 2nd and QUIT to exit the program editting mode.
4. Press PRGM and select your ASAND program by number, then press ENTER to run it. When the message "Done" appears, the new density values for solid sand have been calculated and stored as list L5.
5. Press the keys STAT ENTER and the right arrow buttons to view the densities. Record the 4 values displayed in L5 on your paper.

TI-82 Shut Down Procedure

Press 2nd QUIT then CLEAR, followed by 2nd OFF to clean up the calculator's display and shut it off. Return your calculator to your teacher for final check out.

Teacher Notes

An analysis of the results from 35 lab groups showed that the average percent of air space in fine-grain sand was 45% with a standard deviation of 5%. The density of the sand with air space was 1.56 g/mL, while the density of only solid sand was 2.84 g/mL. It was also obvious that there was a wider range of calculated values for Trial 1, and precision improved with each subsequent trial. This emphasized the fact that measurement errors become less significant as the sample size increases.

The AIRSPACE program can be entered into one calculator, then shared with other calculators by using the LINK mode. To enter the original programming, complete the following steps:

1. Press PGRM and select the option for a NEW program.
2. Type in the name AIRSPACE by pressing the keys having the appropriate white letters at the upper right corner above each key. Press ENTER when done typing in the name.
3. Create a program loop to perform the same calculation for each of the 4 trials by pushing PRGM 4 ALPHA N , 1 , 4 , 1 ) ENTER This will start the calculations with the variables beginning with N equal to 1 and ending with those with N = 4, by increments of 1.
4. The maximum possible volume of the mixture is determined by adding the total volume of sand(L1) and the total volume of water(L3), then it is stored as L5 in the calculator. Type in the following keystrokes to make the string: ( 2nd L1 (ALPHA N) + 2nd L3 (ALPHA N) STO 2nd L5 (ALPHA N) ENTER
5. The actual volume of the mixture stored in L4 is subtracted from the maximum possible volume of the mixture in L5 to determine the volume of air space. This volume is divided by the total volume of sand in L1, then multiplied by 100 to obtain a % of air space, which is stored in L6 of the calculator. Type in the following keystrokes to make the string: ( ( 2nd L5 (ALPHA N) - 2nd L4 (ALPHA N) ) / 2nd L1 (ALPHA N) ) * 100 STO 2nd L6 (ALPHA N) ENTER
6. Press PRGM 7 ENTER to signal the end of the calculation loop.
7. Add the values for the % of air space in each trial and calculate the average by typing the following keystrokes: ( 2nd L6 ( 1 ) + 2nd L6 ( 2 ) + 2nd L6 ( 3 ) + 2nd L6 ( 4 ) ) / 4 STO ALPHA A ENTER The average value is stored as variable "A".
8. To display a message and the value of "A" on the calculator screen, press PRGM the right arrow button to select I/O and 3. The prompt Disp appears on the screen for you to type in your message of 16 characters or less. Press 2nd ALPHA "AVERAGE PERCENT" ENTER
9. Again press PRGM right arrow button and 3. Press 2nd ALPHA "AIR SPACE IS" ALPHA , ALPHA A ENTER
10. Press 2nd and QUIT to exit the program editting mode.