Density of Matter Version 6.3

Density of Matter Version 6.3 Michael J. Vitarelli Jr. Department of Chemistry and Chemical Biology Rutgers University, 610 Taylor Road, Piscataway, NJ 08854 I. INTRODUCTION Physical properties of matter can be classified into two categories: extensive properties and intensive properties. Extensive properties are additive and depend on the size, amount, or volume of matter. Intensive properties are independent of the amount of a substance or its volume and are not additive. Temperature is an example of an intensive property. If a substance, at uniform temperature, is cut in half both halves will have the same temperature. Mass, however, is an extensive property. If a slab of matter is cut in half, then each half has half the original amount of mass. Each new slab does not have the same mass as the original slab. If the ratio of two extensive properties such as mass and volume is taken, then the resultant is an intensive property, in this case density: density = mass volume If a substance, with uniform density, is cut in half, then the density of each half is the same. The primary purpose of this lab is for the student to determine the density of various objects. As shown in Equation 1, to obtain the density of a substance we need both its mass and its volume. The mass is usually easy to determine. If it is a solid simply put it on a balance and weigh it. If it is a liquid, put it in a flask then weigh the flask with the liquid in it. Then subtract the mass of the flask. Similarly for a gas except the flask must be sealed. However, the volume of a substance may not be as easily determined. There are three standard methods for measuring the volume of an object. The first is to calculate the volume based on its dimensions. If it is a rectangular solid then the volume is the product of the length, the width, and the height: (1) V = L W H. (2) If it is a right circular cylinder then the volume is the square of the base radii times π times height: If it is a cylindrical tube then the volume is: If it is a sphere then the volume is: V cyl = πr 2 h. (3) V tube = π(r 2 outer r 2 inner)h. (4) V sph = 4 3 πr3. (5) However, with the sphere the radius may not be easy to measure. Furthermore as the geometry of the object becomes more complicated this method may lose tractability. Fortunately, there is

a second method for measuring the volume of an object. This entails immersing the object in a liquid (usually water) and measuring the change in volume of the liquid. The change in volume of the liquid is equal to the volume of the solid. As an example, let s say we fill a graduated cylinder to a height of 5.62 ml. Then we gently place an object in the cylinder such that it is completely submerged. After this we find that the height of the water has risen to 6.81 ml. The difference between these two measurements, 6.81 ml - 5.62 ml = 1.21 ml, is the volume of our object. A final method in obtaining the volume of an object is by exploiting Archimedes principle. If an object is submerged or at least partially submerged in a fluid it will displace some of that fluid. This displacement of fluid will cause a force to be exerted back onto the object; this force is known as buoyancy. The magnitude of this force is equal to the weight of the displaced fluid. Thus objects in water are lighter then in air. Consider an object partially submerged, floating in a liquid. Gravity acts downward on this object while the force of the displaced liquid acts upwards. If in equilibrium, these forces must balance: F g =F buoy. Recall that force is mass times acceleration. Thus, the force of gravity is equal to the mass of the object times the acceleration due to gravity, g. While, the buoyancy force is equal to the mass of the displaced liquid times the acceleration due to gravity; therefore: m obj g=m liq g. Using equation 1 and cancelling g on both sides we obtain a rather useful equation for floating objects: V obj d obj = V liq d liq (6) If the density of the object is less then the density of the liquid, then the object will, to some extent, float in the liquid. If we are given the density of the object and that of the liquid we can use equation 6 to determine what percentage of the object is submerged. An iceberg may look small, but much of it is hidden under the surface of the ocean. During this experiment and in all future experiments error will occur. This error could be from the equipment being used or from the experimenter. The accuracy of a given experiment refers to how close the measured value is from the true or accepted value. Using equation 7 we can calculate the percent error in this particular measurement because we happen to know the right (or at least accepted) value: % error = Actual value Theoretical value Theoretical value 100% (7) Lastly, we come to the reproducibility of a given measurement. An experiment may be performed several times, possibly by one or many individuals. It is unlikely that each experiment will produce exactly the same value. How close these values are to each other is known as precision. Suppose we throw ten darts at a dart board, if they are all clustered near each other then we say these throws have high precision. If these darts are all in the center of the of the board, the desired result, then these throws also have high accuracy. 2

3 II. PRE-LAB QUESTIONS 1.(2 point) Severus Snape knows that density of his powder is 3.00 g/cm 3. He also knows he needs 3.00 cm 3 of this powder. What mass in grams does he require? 2.(2 points) Calculate the density of rock that weights 14.5 kg and has a volume of 141 cm 3. Express your answer in g/cm 3. 3.(2 points) Trevor, Neville Longbottom s pet toad, has a mass of 13.6 kg. Trevor jumps into a large cylinder containing water. The water level rises 2.50 L. What is the density of the toad? Express your answer in kg/l. 4.(2 points) The philosopher s stone weighs 43.2 g is placed in a graduated cylinder containing 63 ml of water. After the stone is added to the cylinder the water rises to 74 ml. What is the density of the stone? Express your answer in g/ml. 5.(2 points) Harry s spell book is a rectangular solid and has dimensions 4.0 cm by 13 cm by 9.2 cm and a mass of 4.2 kg. What is its density? Express your answer in g/cm 3. 6.(2 points) Harry s Golden Snitch, the smallest ball in Quidditch, is a sphere with radius of 3.10 cm, and a density of 6.12 g/cm 3. What is its mass? Express your answer in grams. 7.(2 points) Mr. Finnigan has a particular proclivity for pyrotechnics. He will use right circular cylinders with a height of 12.0 cm and a radius of 11.6 cm. If the density of his powder is 8.41 g/cm 3, how many grams of powder does he need to stop the Snatchers from getting to Hogwarts if he needs 14 of these cylinders? Express your answer in kg. 8.(2 points) Hermione Granger uses her Time-Turner and is suddenly whisked away to a cold April night in 1912 in the middle of the Atlantic ocean. She is on the Titanic enjoying the party. In the distance she sees a large white object approaching the ship. She thinks its pretty and not much danger to the ship so she carries on with her festivities. Later the large white object (an iceberg) hits the ship, and the ship begins to sink. As the ship is sinking she thinks, hm, it wasnt that big of an ice-berg (and of course this ship cant sink...). What percentage of the iceberg was under water? Let the density of the water be 1.0 g/cm 3, and the density of the ice be 0.92 g/cm 3. Express your answer as a percentage (not a decimal). 9.(2 points) Albus Dumbledore is a strict headmaster. He, like your professor in this course, requires you to answer all questions with the proper number of significant figures. To test your knowledge, what is the answer to the following problem, reported to the correct number of significant figures? A. 0.15 B. 0.152 C. 0.1518 D. 0.15183 E. 0.151835 9.072 8.7497 21.75 0.09764

4 10.(2 points) Severus Snape thinks it s important to understand the fundamentals. He would like to know the SI units for volume. What are they? A. L B. ml C. m 3 D. gal E. ft 3 11. (2 points) Ollivander sells sheets of aluminum in a variety of thicknesses: 1.0 mm at 0.55 $/cm 2, 2.0 mm at 0.70 $/cm 2, and 3.0 mm at 0.85 $/cm 2 and various thicknesses in between all following the same linear trend. Harry needs 10.0 cm 2 of 2.2 mm thick aluminum. How much does this cost him? Express your answer in Dollars. 12. (2 points) A ring weighing 9.45 g is placed in a graduated cylinder containing 25.3 ml of water. After the ring is added to the cylinder the water rises to 27.4 ml. What metal is the ring made out of? Assume the ring is a single metal. A. Titanium B. Copper C. Iron D. Lead E. Aluminum III. PROCEDURE The student will obtain an unknown metal with a number printed on it. Record this number on the data sheet. The students first task is to determine the metals mass on an electronic balance. Simply zero the balance then place the metal on the balance and record the mass in the data sheet. Next, the student will measure the volume of the unknown metal using each of the three methods listed below: Determining The Volume Via Formula The student will measure all necessary dimensions of their object then, using equations 2-5, the student will calculate the volume of their object. Record this calculated volume in the data sheet. Determining The Volume Via Displacement Obtain and fill a 100 ml graduated cylinder with 50 ml of water. Record this volume in the data sheet. Next place the object into the graduated cylinder. Simply dropping the object into the cylinder could cause water lose due to splashing, or may even break the cylinder. To avoid this tilt the cylinder and slide the object into the cylinder. Record this new volume in the data sheet. The difference between the second volume and the first volume is the volume of the object. Record this difference in the data sheet.

5 Determining The Volume Via Archimedes Principle Again, the general principle is that an object is lighter in water then in air. First hang the object from the support arm of the beam balance and record the mass in the data sheet. Next, fill a 400 ml beaker with 350 ml of water. With the object still hanging from the support arm, suspend this object in the water. Make sure the object is completely submerged, yet does not touch the sides or bottom of the beaker. Record the mass the balance indicates in the date sheet. The difference between these two masses is the mass of the displaced water. Measure the temperature of the water and record this temperature in the data sheet. Using the density of water from Table I, and equation 1, the student can calculate the volume of the water that has been displaced. This volume is equal to the volume of the object! TABLE I: Density of Water Temp [ C] Density [g/ml] 18 0.9986 19 0.9984 20 0.9982 21 0.9980 22 0.9978 23 0.9975 24 0.9973 25 0.9970 26 0.9968 27 0.9965 28 0.9962 TABLE II: Density of Solids @ 20 C Solid Density [g/cm 3 ] Aluminium 2.70 Brass 8.70 Carbon (diamond) 3.51 Carbon (graphite) 2.27 Copper 8.96 GOLD! 19.3 Iron 7.87 Magnesium 1.74 Tin 7.27 Titanium 4.11 Zinc 7.14

6 IV. CHEMICAL HAZARD AWARENESS FORM: DENSITY OF MATTER This form is to be turned in prior to beginning your experiment. Name: Section: Experiment: List below any chemicals used in this experiment. Also list their hazards and any handling precautions for these chemicals.

7 V. DATA SHEET: DENSITY OF MATTER Name: Section: Instructor: Unknown metal number: Mass of unknown metal: [g] Laboratory temperature: [ C] Via Formula Object volume via formula: [cm 3 ] Object density via formula: [g/cm 3 ] Via Displacement Initial graduated cylinder volume: [ml] Final graduated cylinder volume: [ml] Object volume via displacement: [cm 3 ] Object density via displacement: [g/cm 3 ] Via Archimedes Object mass on beam balance in air: [g] Object mass on beam balance in water: [g] Mass of displaced water: [g] Object volume via Archimedes Principle: [cm 3 ] Object density via Archimedes Principle: [g/cm 3 ]

8 VI. RESULTS: DENSITY OF MATTER Name: Section: Instructor: Results: Unknown number: Density of unknown: Via Formula Density of unknown: Via Displacement Density of unknown: Via Archimedes Principle Average density of unknown: Average of the three above densities Identity of unknown metal: From Table II Density of unknown metal: From Table II % Error Between average density and that from Table II Conclusions and Comments:

9 VII. POST-LAB QUESTIONS: DENSITY OF MATTER Name: Section: Instructor: 1. Calculate the density of a plate that has a mass of 7.42 g and has a volume of 4.2 cm 3. Express the answer in g/cm 3. 2. A graduated cylinder is filled with 24.81 ml of water. Five marbles with a total mass of 36.18 g are gently placed into the cylinder. The water level rises to 27.49 ml. What is the density of one of these marbles? Express the answer in g/cm 3. 3. A metal right circular cylinder has a height of 4.00 cm, a base diameter of 2.50 cm and a mass of 77.2 g. What is this cylinder most likely made out of? 4. If a spherical steel ball has a radius of 2.00 cm, what volume of the ball is submerged in a large beaker of mercury? Steel has a density of 7.75 g/cm 3, while mercury has a density of 13.53 g/ml. Express the answer in cm 3.