Volume of an irregular shape

Mass can be determined by using a balance. If the object has a regular shape, such as a cube or a cylinder, volume can be calculated from length measurements. However, most objects have irregular shapes, and the volume must be determined indirectly. One way to measure the volume of an irregularly shaped item that does not dissolve in or react with water is by water displacement. An item that is entirely submerged in water will displace a volume of water equal to its volume. 

FIGURE 15.14 Using a balance to find the volume of an irregularly shaped object. 

Problem The volume of an irregularly shaped solid can be determined from the volume of water it displaces. A graduated cylinder contains 19.9 mL of water. When a small piece of galena, an ore of lead, is added, it sinks and the volume increases to 24.5 mL. What is the volume of the piece of galena in cm and in L ... 

In fact, the derived SI unit for volume is the cubic meter. It is easy to visualize a cubic meter imagine a large cube whose sides are each 1 m in length. The volume of an irregularly shaped solid can be determined using the water displacement method, a method used in the MiniLab in this section. 

The volume of an irregularly shaped solid can be determined by measuring the amount of water it displaces. 

The Monte Carlo statistical sampling methods have been known for more than half a century and applied to numerous fields of physics. A simple example for their applications is the determination of the volume of an irregularly shaped body. Let the body be in a unit-volume cube. N points are chosen randomly in the cube, from which falls into the investigated volume. If the number of points is sufficiently large, the expectation value of the sought for volume is NJN. 

To determine the volume of an irregularly shaped glass vessel, the vessel is weighed empty (121.3 g) and when filled with carbon tetrachloride (283.2 g). What is the volume capacity of the vessel, in milliliters, given that the density of carbon tetrachloride is 1.59 g/mL ... 

The density of an irregularly shaped solid is usually determined by measuring the mass and then measuring the volume of liquid that it displaces. The volume of liquid in a graduated cylinder is measured before the object is submerged and then measured again with the object submerged. The difference in the volume equals the volume of the object. 

However, in the typical case of an irregular-shaped particle, it is not easy to calculate its volume and thus dsph is taken equal to the mean nominal diameter measured by sieve analysis dp. In the present book, dp is considered to be equal to the average sieve diameter. 

To illustrate this method, we will consider the determination of the density of an irregularly shaped solid. In this determination we make three measurements. First, we measure the mass of the object on a balance. Next, we must obtain the volume of the solid. The easiest method for doing this is to partially fill a graduated cylinder with a liquid and record the volume. Then we add the solid and record the volume again. The difference in the measured... 

Fig. 1.5.2. Compression of an irregularly shaped object by application of an external pressure that diminishes the volume.

Raymond and DeVries (1977) have recently postulated that the functional saturation can be realized even with the simpler theory previously developed by Kuhn (1956). Reasoning that l differential surface-to-volume ratio, an irregularly shaped surface will lead to larger surface area for the same spherical volume, these investigators showed that l = [2pac] 112, where p is related to polymer size, a is an entrapment coefficient of the polymer in ice, and c is the concentration of polymers in solution. If this value of l is used for the evaluation ofK in Eq. (27), then even when the Flory-Huggins factor is not incorporated (x = i), ATioweringrxcin can be seen. Qualitative... 

The equivalent volume diameter (Dg) of an irregularly shaped particle is the diameter of a sphere having the same volume as the particle in consideration. The equivalent volume diameter is used to describe the dynamic particle behaviour of non-spherical particles in combination with the shape factor (see definition). 

For a spherical particle, the diameter is taken as the size. However, the size of an irregularly shaped particle is a rather uncertain quantity. We therefore need to define what the particle size represents. One simple definition of the size of an irregularly shaped particle is the diameter of the sphere having the same volume as the particle. This is not much help because in many cases the volume of the particle is ill-defined or difficult to measure. Usually, the particle size is defined in a fairly arbitrary manner in terms of a number generated by one of the measuring techniques described later. A particle size measured by one technique may therefore be quite different from that measured by another technique, even when the measuring instruments are operating properly. 

How can you find the volume of a solid that has an irregular shape ... 

D By placing an irregularly shaped solid into a graduated cylinder with a known volume of water, the water is displaced by a certain volume. If the density of the solid is known, then the mass of the object can be calculated. If the mass of the solid is known, then the density of the object can be calculated. 

Porosity is greatly determined by the particle shape. Ball shape allows for the minimum void volume in an irregular filling. However, the layer porosity characteristics obtained for the simplest case of ball-shaped evenly-sized particles cannot be applied for determining the type of the filling, or the shape of the channels formed by the particles and, consequently, for estimating the layer resistance... 

FIGURE 4 (A) An irregularly shaped particle with an infinite number of diameters. (B) The equivalent volume, surface, and projected area diameter of (A). 

In the above, <7C is the equivalent volume diameter, i.e. the diameter of a sphere having the same volume as an irregular particle. As Hinds (1982) points out The equivalent volume diameter can be thought of as the diameter of the sphere that would result if an irregular particle were melted to form a droplet <7C is calculated from microscopic measurement of the actual particles being considered, while x is the dynamic shape factor which is included to allow for the effects of shape on terminal velocity. For example, talc dust is characterized by a dynamic shape factor (/ ) of 1.88, sand particles by 1.57, etc. Spheres have a dynamic shape factor of 1.0 while cubes have a dynamic shape factor of 1.08. 

For larger particles or nanoparticle aggregates, SMPS measurements can be coupled with an aerodynamic particle sizer (APS). For spherical particles, it is easy to relate the measured diameters from the SMPS and APS because no corrections need to be made for shape and volume, but for irregularly shaped particles the APS reports an aerodynamic diameter. Da, by comparing the settling velocity to a spherical particle with a density of 1 g cm to compute the particle size. A volume equivalent diameter, D g, which is defined as the volume of a sphere with the same volume as a particle with an irregular shape, is used to relate the aerodynamic diameter from the APS with mobility diameter, D , from the SMPS (46) ... 

Propose a method for measuring the volume of water in an irregularly shaped swimming pool. You have 1 gallon of water that contains a radioisotope with a long half-life, and a Geiger-MiiUer counter. 

---