VIEWS: 16 PAGES: 4 POSTED ON: 2/27/2011
From http://theory.uwinnipeg.ca/physics/fluids/node8.html and http://library.thinkquest.org/12596/ideal.html (up to non-ideal behavior) Variation of Pressure with Depth One might guess that the deeper you go into a liquid or gas, the greater the pressure on you from the surrounding fluid will be. The reason for the increased pressure is that the deeper into a fluid you go, the more fluid, and thus the more weight, you have over top of you. We can calculate the variation of pressure with depth by considering a volume of fluid of height h and cross-sectional area A . Variation of Pressure with Depth If this volume of fluid is to be in equilibrium, the net force acting on the volume must be zero. There are three external forces acting on this volume of fluid. These forces are: 1. The force PTA due to the pressure on top of the volume of fluid. If the fluid is open to the air, PT = PO = 1.01 x 105 Pa, which is atmospheric pressure. 2. The weight of the volume of fluid, w = Mg . Remembering the definition of density, = M/V , and that the volume of the fluid may be calculated as V = Ah , we can write the weight of the fluid as w = ghA . 3. The force pushing up on the bottom of the volume of fluid, PBA , due to the fluid below the volume under consideration. If we take the up direction to be positive and add the forces we get PBA - ghA - PTA = 0, which gives PB = PT + gh. 1 This provides the general formula relating the pressures at two different points in a fluid separated by a depth h . Note: Only the density of the fluid and the difference in depth affects the pressure. The shape and size of the container are irrelevant. Thus the water pressure 6 inches below the surface of the ocean is the same as it is 6 inches below the the surface of a glass of salt water. Idea: Pascal's Principle states that any pressure applied to an enclosed fluid is transmitted undiminished to every point of the fluid. Thus, Pascal's Principle a pressure of P1 = F1/A1 applied downward to the surface on the left of the container gets transmitted as an equal pressure upward of P2 = P1 on the surface on the other side of the container. The force on the other side is therefore: F2 = P2A2 = F1 . Thus if A1 < A2 , the transmitted force, F2 , is greater than the applied force, F1 . This is the principle behind the hydraulic press. For example, the transmitted force F2 is used to balance the weight of a car in a hydraulic lift. Ideal Gas Law: Pressure x Volume = Moles x Ideal Gas Constant x Temperature Substituting in variables, the formula is: PV=nRT Explanation and Discussion: 2 The Ideal Gas Law may be the largest and most complex of the gas laws. This is in part because of the number of variables in the equation, and in part to the abstraction of an "ideal" gas that the law is built on. The Ideal Gas Law is also designed as a sort of umbrella for Boyle's, Charles', and Avogadro's laws. First, we'll go over the parts of the equation, PV=nRT. P is pressure. Pressure can be in either atmospheres (atm) or kilopascals (kPa). V is volume in liters (L). n is the number of moles of the gas. Because moles of a substance are determined by mass divided by molecular mass, it can create an interesting variant we will discuss later. R is the Ideal Gas Constant. Depending on whether atmosphers or kilospascals were used, the value is either 0.0821 L-atm/mol-K or 8.31 L-kPa/mol-K, respectively. Temperature is in absolute degrees Kelvin. An interesting aspect of the Ideal Gas Law is its flexibility. It contains elements that allow you to solve for other quantities, such as density or molecular mass. To solve for molecular mass: PV=nRT - start with the equation PV=mass/mol. mass x RT - change moles to mass(m) in grams divided by molecular mass in grams mol. mass x PV = mRT - multiply by molecular mass molecular mass = mRT/PV - divide by pressure and volume. We can also see density in that last equation, m/V (grams/liter). The same equation, but with density(d) in place of mass per volume (m/V), is: molecular mass = dRT/P To solve just for density, the equation would become: density = (molecular mass x pressure)/(constant x temperature) So far, we have been skirting the concept of an ideal gas. What exactly is an ideal gas? An ideal gas is one that exactly conforms to the kinetic theory. The kinetic theory, as stated by Rudolf Clausius in 1857, has five key points. These are: 1. Gases are made of molecules in constant, random movement. Gases like Argon have 1-atom molecules. 2. The large portion of the volume of a gas is empty space. The volume of all gas molecules, in comparison, is negligible. 3. The molecules show no forces of attraction or repulsion. 4. No energy is lost in collision of molecules; the impacts are completely elastic. 5. The temperature of a gas is the average kinetic energy of all of the molecules. Non-Ideal Behavior The Kinetic Theory makes several assumptions about an ideal gas. These cause problems because real gases are not ideal. The main causes of error are related to pressure and temperature. Pressure At high pressures, the behavior of real gases changes dramatically from that predicted by the Ideal Gas Law. Under 10 atmospheres of pressure or less, Ideal Gas Law predictions are very close to real amounts and do not generate serious error. 3 Temperature When the temperature of a gas is close to its liquefaction point, the behavior is very different from Ideal Gas Law predictions. With increasing temperatures, the Ideal Gas Law predictions become close to real values. Why? The answer is simple: ideal gases have molecular volume and show no attraction between molecules at any distance; real gas molecules have volume and show attraction at short distances. Let us first consider what pressure does. Pressure at high degrees will bring the molecules very close together. This causes more collisions and also allows the weak attractive forces to come into play. With low temperatures, the molecules do not have enough energy to continue on their path to avoid that attraction. 4