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2.1 Define mass, force, pressure, energy, period, power, work.

Mass: The quantitative measure of inertia, a fundamental property of all matter. Mass is defined in terms of inertia as the ratio of the force acting on a body to the acceleration produced, in accordance with Newton’s laws of motion. The weight of a body is proportional to its gravitational mass, as defined by Newton’s law of gravitation, i.e. W=mg . Unlike weight, mass is a scalar quantity (a quantity having only magnitude and no direction). Two masses are equal if they have identical weight when measured in the same place in a gravitational field. The SI unit of mass is the kilogram, kg. At speeds close to the speed of light, inertial mass varies with speed; as expressed in the relativity theory by the equation E=mc², where E is energy and c is the speed of light. This equation points to the startling modern discovery of the equivalence of mass and energy. The conversion of mass to energy occurs during nuclear fission and fusion.

Force: The physical agency (a vector quantity) that changes the velocity of an object. Quite often an object is acted on by several forces simultaneously and a change in velocity occurs only if the net force is not equal to zero. The second of Newton’s laws of motion defines the net force acting on an object of constant mass as the product of that mass and the acceleration that is induced in it; F=ma , measured in Newtons (N). When a force is exerted over a given distance, work is done; for work (or energy) equals force ^x distance; W=F ^xd , measured in Newton-metres (Nm) or Joules (J). A force continually applied at a certain speed gives rate of working, power, measured in joules per second (J/s) or Watts (W). Although contact forces, or forces in which two bodies interact directly by contact between their surfaces, are common, they are not considered fundamental forces in the physical world. There are four fundamental forces in nature: the gravitational force, the electromagnetic force, which includes both electric and magnetic forces and is responsable for the structure of matter, the strong or "nuclear" force, which is the attractive force that holds together the constituents of the atomic nucleus, and the weak force, important in the interactions between subnuclear particles but so weak that it plays no direct part in ordinary observable behavior.

Pressure: The force per unit area experienced on a surface. Atmospheric pressure is the force produced by the weight of all the air above the spot, so the greater the altitude the less is the pressure. The pressure of a solid on a solid is in proportion to the size of the area of contact. The pressure in a liquid is the product of its density and the depth multiplied by the acceleration of free fall. The SI unit of pressure is the pascal (Pa), which is defined as 1 newton per square metre. A unit that is used in some barometers, as well as for the measure of blood pressure, is the millimetre of mercury (mm Hg); atmospheric pressure is given as 760 mm Hg, this being the height of a column of mercury that the atmosphere can support under standard conditions.

Energy: The capacity of objects or systems to do work. The concept of energy emerged during the mid-19th century, when it was realized that moving bodies could be made to move against resisting forces, thus doing work. This ability came to be known as kinetic energy. Raised bodies exhibit a potential for doing work when they fall, a property known as gravitational potential energy. Energy can take many forms, with the important characteristic that the total energy of a system remeins constant (i.e. Energy is conserved). Other energy forms include chemical energy, thermal energy (heat), electrical energy, and nuclear energy. The standard unit for all forms of energy is the joule (J). Power, the rate at which energy is delivered or converted, is measured in joules per second or watts (W). Energy can be transferred from one body to another by work processes (involving movement), by heating (using their temperature differences), by electromagnetic radiation (such as light and microwaves) and by electricity (flow of electrical energy). Energy can also be converted from one form to another. Although energy is conserved in practice when it is converted from one form to another, some of the energy is converted into an unwanted form, usually heat.

Period: The time interval between recurrent events. The period of an oscillation or wave is the time taken to complete one cycle, usually measured in hertz. Note that the period T is the reciprocal of the frequency f: T=1/f. In astronomy, periods are ascribed to rotational spins of bodies, orbital revolutions, solar magnetic phenomena, and pulsations within stellar atmospheres.

Power: Measured in joules/second, or watts (W), power is defined as the rate of doing work, or the rate at which energy is delivered or converted: P = <W/<t If a 70 kg person runs up a staircase 3.0 m high in 3.5 seconds, we may find his power in climbing the steps. Since in this case, the work done is the change in gravitational potential energy, mgh, so the power is: P= mgh/t = (70 kg)(9.8 m/s)(3.0m)/3.5s = 590 W.

Work: The process of a transfer of energy. As with all forms of energy, the SI unit of work is the joule. Work is the scalar product of the force and displacement vectors. The work done by a force is the product of that force and the distance over which it is applied in the direction in which it acts. One joule of work is done when one newton is applied over one metre.

Ohm’s law states that the electric current I flowing in a conductor is directly proportional to the potential difference (voltage V), and inversely proportional to the resistance R: V = IR , when the resistance R is a constant. Materials whose resistance is constant over a wide range of voltages are said to obey Ohm’s law. In these materials, current is proportional to the applied potential difference V and inversely proportional to the resistance R. Applying this to a wire of known diameter and conductivity, the current is inversely proportional to length.

Ohm’s law is not a law of nature in the sense that conservation of momentum or the universal law of gravitation is a law of nature; rather, it is an experimental observation about the behavior of some materials under a limited range of circumstances.

Capacitance is the property of an electrical system of conductors and insulators that enables it to store charge. The greater the potential, the greater the amount of charge that can be stored. For a charge of magnitude Q and a potential difference of V, the ratio Q/V is the capacitance which is a constant for that capacitor, measured in farads. The farad is a very large unit; in practice, capacitance values range between 10^-12 and 10^-6 farad.

A typical capacitor consists of two parallel conductors consisting of metal plates or electrodes separated by an insulator or dielectric. A capacitor presents an extremely high resistance to a direct current (d.c.): no current will flow in a d.c. circuit containing a capacitor. However, alternating current (a.c.) is allowed to pass much more easily, and the higher the frequency of the a.c. signal, the less opposition the capacitor presents. Because of this ability to separate a.c. and d.c. signals, capacitors are frequently employed in both filters and power supplies. Several types of capacitor exist. Ceramic capacitors are used in radio-frequency circuits, while capacitors made from polyester and polycarbonate plastics have general application.

For a viscous fluid flowing through a horizontal pipe of uniform cross section, the fluid pressure decreases with distance along the direction of flow. Viscosity is that property of a fluid that indicates its internal friction. The more viscous a fluid (gas or liquid), the greater the force required to cause one layer of fluid to slide past another. In the situation of a fluid moving through a horizontal pipe, the wall of the pipe exerts a resistive force, or drag, on the adjacent layers of fluid. These layers, in turn, slow down the next adjacent layers, and so on. As a result, the rate of flow is slowest near the pipe walls and fastest in the center of the pipe. Therefore, for a given rate of flow the pressure difference between two points along the length of the pipe depends on the radius of the pipe. The pressure difference between the two points is also related to a quantity known as the coefficient of viscosity or simply the viscosity of the liquid. The exact relation is given by Poiseuille’s equation: P₁ - P₂ = 8QgL/ oR⁴ , where Q is the flow rate in m3/s, g ("eta") is the coefficient of viscosity, R is the radius of the pipe, and L is the separation between the test points. If R and L are given in meters and the pressure is given in pascals, the unit of the coefficient of viscosity is the pascal-second (Pa^.s). This equation is often used experimentally to determine the coefficient of viscosity of a liquid. Note that this equation allows us to observe that a viscous fluid will not flow through a pipe unless there is a pressure difference between the ends.

Laplace’s law is a relation tying the pressure differential between the two sides of an elastic membrane or a liquid film to the surface tension of the membrane or film. If we consider a spherical membrane filled with a fluid, the membrane wall exerts a force per unit of length called surface tension c; this force per unit of length depends on the thickness of the wall and is thus related to the membrane in its entirety, not simply to the two surfaces. The same applies in the case of a liquid film. The lungs and the heart may be approximately described by means of this elastic membrane model. Given <P = P_i - P_o as the difference between the external and internal pressures, Laplace’s law gives us the relation between <P and the surface tension c for a spherical membrane of radius r: P_i - P_o = 2c/r

When a liquid is in a state of equilibrium with its own vapour, the pressure of the gaseous phase is called vapour pressure. We can understand this as the pressure necessary to prevent the liquid from further evaporation; this pressure must be therefore equal to the pressure difference P_i - P_o across the liquid-vapour interface. Example: If the surface tension of water is 7.28 ^x 10^-2 Nm-¹ at 20^Û C, and the vapour pressure of water at 20^Û C is 2.33 x 10³ Pa, what’s the radius of the smallest drop of water that can possibly form without evaporating? Here we know that the pressure difference cannot exceed 2.33 ^x 10³ Pa. Using LaPlace’s law to solve for r, r = 2c/P_i - P_o = 2(7.28 ^x 10^-2 Nm^-1)/2.33 ^x 10³ Pa = 6.25 ^x 10^-5 m .

For every bubble of soap, there exist two spherical surfaces, each one subject to its own surface tension. This entails a change in LaPlace’s equation by a factor of 2 for a spherical bubble: P_i - P_o = 4c/r .

For a cylindrical membrane of radius r, LaPlace’s equation becomes: P_i - P_o = c/r .

Bernoulli’s equation relates a fluid’s pressure, velocity, and height as it moves along a pipe or other tube of flow, expressing conservation of energy in a moving fluid. For the steady, nonviscous flow of an incompressible fluid:

P₁ + qgh₁ + ½qv₁ ² = P₂ + qgh₂ + 1/2qv₂² = constant.

In practice, we can also apply the above equation to compressible fluids of negligible viscosity in laminar flow.

If the tube of flow (pipe) under consideration is horizontal, and thus h₁ = h₂, the fluid flows more rapidly in a constricted region of the pipe. Considering a horizontal pipe for which the area A₁ is greater than A₂, Bernoulli’s adjusted equation shows that the pressure is lower in the constricted region:

P₂ = P₁ + qv₂2(A₂² - A₁²)/2A₁²

This equation holds strictly for incompressible, nonviscous fluids. If the fluid is incompressible but viscous, such as water or blood, the above equation becomes an inequality to show that some of the work done is dissipated by the internal frictional forces in the viscous liquid:

P₂ \ P₁ + qv₂2(A₂² - A₁²)/2A₁²

The result for which Robert Boyle is best known is the observed relationship between the pressure and the volume of an enclosed gas at a constant temperature. Boyle’s law states that the pressure exerted by a gas at constant temperature is inversely proportional to the volume in which it is enclosed. Boyle’s law is usually written:

PV = constant

Alternatively, Boyle’s law may be written:

P₁V₁ = P₂V₂

Here, the subscripts 1 and 2 refer to different physical states of the same sample of gas with the temperature held constant.

Charles’s law states that under constant pressure, the volume of a gas is proportional to the temperature (expressed in kelvins), written:

V₁/V₂ = T₁/T₂ or V/T = constant

Boyle’s law and Charles’s law combined gives us the ideal gas law, which relates pressure, volume, temperature, and quantity of gas, the latter expressed in moles, n. The ideal gas law depends on the universal gas constant R, where R=0.0821 L-atm/mol-k or R=8.314 J/mol-K, given in terms of L-atm or joules depending on the units involved in the rest of the equation, written:

PV = nRT

Note: An equation such as the one above that links the pressure, volume, and temperature of a sample of matter is called an equation of state. Quantities that describe the condition of state of a system are called state variables.

While studying the properties of air, John Dalton observed that the total pressure of a mixture of gases equals the sum of the pressures that each would exert if it were present alone. The pressure exerted by a particular component of a mixture of gases is called the partial pressure of that gas. Dalton’s law of partial pressures can be written as follows:

P_total = P₁ + P₂ + P₃ + ...

The saturated vapour pressure is the pressure at the point of equilibrium between a liquid and the vapour in contact with it, when as many molecules are passing from liquid to vapour as are passing from vapour to liquid. The saturated vapour pressure increases as temperature increases.

Molar mass of the gas: Diffusion is faster for light molecules than for heavy ones. The ratio of the rates of diffusion for two gases is inversely proportional to the square root of the ratio of their molar masses, as states Graham’s law of effusion: r₁/r₂ = H(M₂/M₁) .

Density of the gas: The diffusion of gases is much slower than molecular speeds of effusion (one particle at a time through a pinhole) because of molecular collisions. The diffusion of a molecule from one point to another consists of many short, straight-line segments as collisions buffet it around in random directions. The higher the density of a gas, the smaller the mean free path; that is the smaller the average distance traveled by a molecule between collisions. The more molecules there are in a given volume, the shorter the average distance traveled between collisions.

Temperature: Temperature determines how effective attractive forces between gas molecules are. As a gas is cooled, the average kinetic energy decreases, while intermolecular attractions remain constant. In a sense, cooling a gas deprives molecules of the energy they need to overcome their mutual attractive influence; rates of diffusion are thus slightly decreased at cooler temperatures. As temperature increases, a gas behaves more like an ideal gas (molecules in continuous random motion; attractive and repulsive forces between gas molecules negligable) and intermolecular attractions are minimized.

Pressure: The pressure of a gas can also affect the level of attractive forces between molecules. When molecules are crowded together at high pressures, the attractive forces between molecules come into play at short distances, and their effect is to lessen the impact of a given molecule with the wall of the container and reduce the rate of diffusion. At lower pressures, attractive forces are minimized and the gas behaves more like an ideal gas.

The solubility of a gas in any solvent is increased as the pressure of the gas over the solvent increases. (By contrast, the solubilities of solids and liquids are not appreciably affected by pressure). The relationship between pressure and solubility is expressed in terms of a simple equation known as Henry’s law: C_g = kP_g , where C_g is the solubility of the gas in the solution phase - usually expressed as molarity or moles/liter - P_g is the partial pressure of the gas over the solution, and k is a proportionality constant known as the Henry’s law constant. The Henry’s law constant is different for each solute-solvent pair. It also varies with temperature. Note that bottlers use the effect of pressure on solubility to produce carbonated beverages such as champagne, beer, and soft drinks; these are bottled under a carbon dioxide pressure slightly greater than 1 atm. When the bottles are opened to the air, the partial pressure of CO₂ above the solution is decreased, the solubility of CO₂ decreases, and CO₂ bubbles out of the solution.

The principle put forward by the French industrial chemist Le Chatelier can be stated as follows: If a system at equilibrium is disturbed by a change in temperature, pressure, or the concentration of one of the components, the system will shift its equilibrium position so as to counteract the effect of the disturbance. Le Chatelier’s principle can be used to make qualitative predictions about the response of a system at equilibrium to various changes in external conditions:

1. Change in Reactant or Product Concentrations: Le Chatelier’s principle states that the shift will be in the direction that minimizes or reduces the effect of the change. Therefore, if a chemical system is at equilibrium and we add a substance (either reactant or product), the reaction will shift so as to reestablish equilibrium by consuming part of the added substance. Conversely, removal of a substance will result in the reaction moving in the direction that forms more of the substance.

2. Effects of Volume and Pressure Changes (gaseous mixtures): If a system is at equilibrium and its volume is decreased, thereby increasing its total pressure, Le Chatelier’s principle indicates that the system will respond by shifting its equilibrium position to reduce the pressure. This occurs when a system reduces the total number of gas molecules; fewer molecules of gas exert a lower pressure. Thus, reducing the volume of a gaseous equilium mixture causes the system to shift in the direction that reduces the number of moles of gas. Conversely, increasing the volume causes a shift in the direction that produces more gas molecules.

3. Effect of Temperature Changes: Le Chatelier’s principle predicts that when heat is added to the system, the equilibrium shifts in the direction that absorbs heat. When heat is added to an endothermic reaction whereby Reactants + heat J products , heat is absorbed as reactants are converted to products and thus the equilibrium shifts to the right. If heat is added to an exothermic reaction, whereby Reactants J products + heat , the opposite occurs; heat is absorbed as products are converted to reactants and thus the equilibrium shifts to the left. Cooling a reaction has the opposite effect of heating it: As we remove heat from the system, the equilibrium shifts to the side that produces heat. Thus, cooling an endothermic reaction shifts the equilibrium to the left and cooling an exothermic reaction shifts the equilibrium to the right.

Heat or thermal energy is never strictly lost, but can be transferred by (1) evaporation, (2) conduction, (3) convection, and (4) thermal radiation.

Heat loss by evaporation: When a liquid evaporates, molecules escape from its surface and move about freely as a gas. Within the liquid, the vibrating molecules collide with other, some gaining kinetic energy and others losing it. At the surface, some of the faster, upward-moving molecules have enough kinetic energy to overcome the attractions from other molecules and escape from the liquid. With these faster molecules gone, the average KE of those left behind is reduced; i.e the temperature of the liquid falls: hence the cooling effect of evaporation. The rate of evaporation (and therefore the rate at which heat is lost from a liquid) is increased if:

Heat loss by conduction: Conduction is a process in which thermal energy is transferred without any net movement of the material itself. In gases, fast-moving molecules pass on kinetic energy to slower-moving ones when they collide with them. In non-metal solids and liquids, the molecules are coupled to each other by the forces between them; thus the molecules with the most vibrational energy pass on some of this to those with less energy. In metals, which contain free electrons in thermal equilibrium with the surrounding atoms, the free electrons travel at higher speeds, and transfer energy quickly from one part of the metal to another. That is why metals are such good conductors of heat. (They also conduct some heat by the transfer of vibrational energy). A substance’s thermal conductivity can be measured. With x and A the width and area of the substance under consideration, respectively, its thermal conductivity k is a function of the rate of flow of heat, the substance’s own dimensions, and the temperature gradient: DQ/Dt = -kA(DT/x) . The minus sign indicates that the heat flow is in the direction of decreasing temperature. Good conductors have high k values; good insulators have low k values. Note that rate of flow of heat is the same as power; its unit is the watt. The following are some examples of various substances’ thermal conductivities:

Heat loss by convection: Convection is a more rapid process of heat transfer, accomplished through the mass motion or flow of some fluid, such as air or water. Room heaters (including radiators and refrigerators) lose most of their heat by convection, which works thus: A hot surface heats the air next to it. The hot air rises, to be replaced by cooler air which then heats up, and so on. If air is blown across it by forced convection, the surface will lose heat more quickly. Generally, natural convection is caused by an upthrust: While a region of cold air simply floats in the cold air around it because it displaces its own weight, heated air expands. It weighs the same as before, but now displaces more cold air, so the increased upthrust pushes it upwards.

Heat loss by thermal radiation: Radiation is the most rapid kind of transfer of thermal energy, a process that requires neither contact nor mass flow. Vibrating and spinning molecules in one object give off electromagnetic radiation, similar to light and capable of passing through empty space (a vacuum), whose energy can be absorbed by molecules in another object so that they speed up. This radiation is called thermal radiation. From most warm or hot objects, thermal radiation is mainly infrared. The best absorbers and emitters of thermal radiation are black bodies; these reflect no radiation. The sun is effectively a black body radiator. A hot object also loses energy by radiation. The warmth felt next to a fire is due to this radiation; if an object is hot enough, some of the radiation is visible and can be seen. The rate at which an object radiates energy is proportional to its surface area A and to the fourth power of its absolute temperature T. The total energy radiated from an object per unit time (that is, radiated power) is found experimentally to be:

P = reAT⁴ (The Stefan-Boltzmann law),

where r is the Stefan-Boltzmann constant which has the value r = 5.67 ^x 10^-8 Wm^-2K^-4 and e is the emissivity, a dimensionless number between 0 and 1 that describes the nature of the emitting surface. The emissivity is larger for dark, rough surfaces and smaller for smooth, shiny ones. All objects radiate energy no matter their temperatures; they also absorb radiation from surrounding objects, however, and eventually come to thermal equilibrium with their environment. If an object is at one temperature T and its surroundings are at a different temperature T_s, the net energy lost or gained per second by the object is given by:

P_net = reA(T⁴ - T_s⁴) .

Refractive index: A value by which the capacity of a material to cause refraction is measured. It is the ratio of the speeds of a wave in two different media (one of them usually being air or a vacuum), and is chiefly used of light. Between air and glass the refractive index is about 1.6, and between air and water at 25°C it is about 1.3: that is, light travels 1.3 times as fast in air as in water. Light of different wavelengths is refracted by different amounts. This is the effect known as dispersion; it is responsible for the ability of a prism to separate the lines of atomic spectra.

Focal length: The distance between a lens or mirror and its focus is called its focal length. The focal ratio of a lens or mirror is found by dividing its focal length by its diameter (aperture).

Wavelength: The distance between equivalent points on adjoining waves in a series, between the crests on water waves for example, or the points of maximum compression in sound waves. Sound waves range in wavelength from several metres to one or two centimetres. Electromagnetic waves have the widest range of all, with wavelengths form several kilometres for some radio waves down to less than 10^-10 metres for gamma radiation.

Dioptre: A unit of measurement of the refractive power of lenses equal to the reciprocal of the focal length in meters.

Note: From the equation C = n l, we can find the frequency n by dividing the speed of light (C = 3 ^x 10⁸ m/s) by the wavelength l.

Concave (diverging) lens: Concave lenses always generate virtual images and have, formally, negative focal lengths. In the case of beams of light parallel to the lens’s principal axis, a concave lens spreads the parallel light rays so that they appear to come from a focal point on the other side of the lens. The concave lens and the convex mirror are diverging systems; they both defocus light. Convex mirrors are often used where you want to see what's happening over a large area; for example, many motor vehicle drivers have added convex mirrors to allow them to observe other vehicles behind them. The concave lens is normally used for the eyeglasses to correct nearsidedness.

A decibel is a unit for expressing the ratio of two amounts of electric or acoustic signal power equal to 10 times the common logarithm of this ratio. This unit is useful for expressing the relative intensity of sounds on a scale from zero - for the least perceptible - to about 130 for the average pain threshold for the human ear.