Question 13: Predicting density - Partnering with two other students in the class, find a mineral sample set. Use the descriptions in the table below to identify which mineral is which. Which of the mineral samples in your set is the densest? Least dense? In the table below, order your samples by density (best you can together predict). Discuss with your fellow students and note your hypothesis below. Remember: density = mass / volume (mass divided by volume). How confident are you in our answers? Why? Predicted Densities Corundum Galena Pyrite Quartz Description brownish opaque silver and metallic gold and metallic clear (transparent) 6- barrel-shaped prism sided prism that comes to a point Predicted relative density (1 = densest and 4 = least dense) Question 14: Measuring mass - Find a scale and have each person in your group choose one mineral sample to weigh. Follow the measurement procedure and weigh the sample 3 times. Average your answers. Show work and give final answer below, adding uncertainty based on your range of answers. Then note below the masses of the other minerals measured by your partners and any challenges you faced. Mass Measurements Corundum Galena Pyrite Quartz Mass (with unit and uncertainty) 3. 3. 3. AVE: AVE: AVE AVE: Question 15: Measuring volume - Since density is mass over volume, now that we have mass, we just need volume to confirm density. But how do we measure the volume of an irregularly shaped object? We can use a graduated cylinder and water levels. 1. First fill the cylinder halfway with water and record its volume. Do this process three times (have your group mates share the load so you each are giving a second and third measurement for each other). Average your answers and determine uncertainty by taking the high and low values, subtracting them and dividing that difference by 2. Give final answer below, adding additional uncertainty if necessary. 2. Then add the mineral sample and record the new volume (with uncertainty). Do it 3 times, as noted above. 3. Subtract Ending volume from Starting volume to get Volume of the mineral (carry over the highest uncertainty). 4. Repeat for the volumes of the other minerals. and record any challenges you faced. Use the figure on the next page to understand best how to read a volume measurement. Graduated Cylinder Volume Measurements Corundum Galena Pyrite Quartz STARTING VOLUME 1. (WATER ONLY) with w ! uncertainty AVE: AVE: AVE FINAL: ENDING VOLUME 1. 1. 1 . 1. (WATER + MINERAL) N 2. 2. with uncertainty 3. 3. 3. 3 AVE AVE AVE AVE: VOLUME OF THE MINERAL (subtract two averages and add uncertainties)Make reading mL here 25 20 20 20 15 Figure 2. Reading the meniscus of liquid in a graduated cylinder. Source: LibreTexts Chemistry CC BY-NC-SA 4.0 Carrying through precision and uncertainty for Addition and Subtraction: Round to the least precise measurement and add the uncertainties When we add or subtract measured numbers, we round the answer to the precision of the least precise of any of our starting measurements. Then we add any uncertainty together. Examples: 12.6 cm +/- 0.2 cm + 3 cm +/- 0.3 cm = 15.6 cm +/- 0.5 cm > 16 cm +/- 0.5 cm (3 cm was the least precise number so we round to the one's place; uncertainties add) 12.6 cm +/- 0.2 cm - 3 cm +/- 0.3 cm = 9.6 cm +/- 0.5 cm > 10 cm +/-0.5 cm (3 cm was the least precise number so we round to the one's place; uncertainties add) Multiplication or Division by a constant is the same as addition or subtraction (Example: 6 cm x 2 is the same as 6 cm + 6 cm). Therefore, in these cases (both), we add uncertainties for our final answer, which means multiplying by the constant. Division is the opposite. We divide our original uncertainty by the constant. 12 cm +/- 0.2 cm x 3 = 36 cm +/-0.6 cm (precision stays the same; uncertainty is multiplied by 3) 12 cm +/- 0.2 cm + 2 = 6 cm +/- 0.1 cm (precision stays the same; uncertainty is divided by 2) Question 16: 12.7 cm +/- 0.05 cm + 6.43689 cm +/- 0.000005 cm = Question 17: 13.7 cm +/-0.05 cm - 3 cm +/- 0.5 cm = Question 18: 12.27 cm +/- 0.05 cm x 1.5 = (*Note that only one number here has units, so only one was measured. Your answer should match the precision of the starting number. Also, be sure to multiply your uncertainty by the multiplication factor as well - just like you would if it were an addition problem with one measurement being added to another measurement half its value.) Question 19: 12.7 cm +/- 0.06 cm + 3 = (*Note that only one number here has units, so only one was measured. Your answer should match the precision of the starting number. Also, be sure to divide your uncertainty by the division factor as well..) Question 20: Go back to your volume measurements for Q15. When you subtracted your two volume measurements was your final volume only as precise as the least precise measurement? Did you add your uncertainties? If not, go back now and round correctly or add the right uncertainty. Advanced Rule: Multiplication and Division: complete three calculations. When performing division or multiplication between two measured numbers, rounding is more complicated, as is the uncertainty. To simplify the process, 1. We take our measurements and complete the calculation to get a preliminary answer with no uncertainty. Examples: a. Multiplication: 2 cm +/-0.5 cm x 1.2 cm +/-0.05 cm = 2.4 cm? b. Division: 2 cm +/- 0.5 cm + 1.2 s +/- 0.05 s = 1.67 cm/s 2. Then we return to our original measurements, and for each one we add and subtract their uncertainties to get: a. The highest possible number it could be b. The original measurement we made C. The lowest possible number it could be d. Examples: 2 cm +/- 0.5 cm > 1.5 cm to 2.5 cm; 1.2 cm +/-0.05 cm -> 1.15 cm to 1.25 cm3. Then we complete the calculation two more times to get the highest and lowest possible results. a. If we're multiplying two numbers, i. We multiply the two highest possible numbers to get the highest possible result. il. We multiply the two lowest possible numbers to get the lowest possible result. ill. Example: 2.5 cm x 1.25 cm = 3.125 cm and 1.5 cm x 1.15 cm = 1.725 cm b. If we're dividing two numbers, 1. We divide the highest possible number by the lowest to get the highest possible result. il. We divide the lowest possible number by the highest to get the lowest possible result. lil. Example: 2.5 cm + 1.15 s = 2.1739 cm/s and 1.5 cm + 1.25 s = 1.2 cm/s 4. We now have three answers. We take the highest answer and subtract from it the lowest answer to get our range of answers. Then we divide that by two to get our uncertainty. Examples: a. 3.125 cm' - 1.725 cm? = 1.4 cm (range) -> 1.4 cm? + 2 = 0.7 cm? > +/-0.7 cm? (uncertainty) b. 2.1739 cm/s - 1.2 cm/s = 0.9739 cm/s (range) -> 0.9739 cm/s + 2 = 0.48695 cm/s -> +/- 0.5 cm/s (uncertainty) 5. We look again at our preliminary answer and round it if necessary so that it as precise or one place less precise than the precision of our uncertainty. Examples: 2.4 cm? is as precise as the +/-0.7 cm uncertainty, so we keep it at one decimal place to match. Final answer: 2.4 cm2 +/-0.7 cm? b. 1.67 cm/s is more precise than the 0.5 cm/s uncertainty, so we round it to one decimal place to match. Final answer: 1.7 cm/s +/- 0.5 cm/s Question 21: 12.12 cm +/- 0.01 cm x 1.5 cm +/- 0.05 cm = Show ALL work, with units. Question 22: 12 cm +/- 0.1 cm + 1.55 s +/- 0.005 s = Show ALL work, with units. Question 23: Calculating density -- Which of the three mineral samples in your set is the densest? For each sample, take its mass and divide it by its density. Use the 3-calculation process described above to ensure you have the correct precision and uncertainty for your final answer. Label samples from densest #1 to least dense as #4. Final Densities Corundum Galena Pyrite Quartz Mass Volume Mass/Volume = Density Density ranking (1 = densest; 4 = least) Question 24: How do your results compare to your original hypothesis about which was denser? If not, why do you think you didn't get it correct originally? How do they compare to the actual densities of these minerals? If they don't match exactly, why not? (Corundum is 3.98 g/ml; Galena is 7.6 g/ml; Pyrite is about 5 g/ml; Quartz is 2.65 g/mL.)