Physics-I Ideal Gas Laws PDF

Physics-I Ideal Gas Laws Bakranova Dina Igorevna PhD, Assistant professor Lecture 9 Equation of State for an Ideal Gas The pressure–volume law The volume–temperature law The pressure–temperature law Maxwell–Boltzmann speed distribution function Dalton’s Law ...

Physics-I Ideal Gas Laws Bakranova Dina Igorevna PhD, Assistant professor Lecture 9 Equation of State for an Ideal Gas The pressure–volume law The volume–temperature law The pressure–temperature law Maxwell–Boltzmann speed distribution function Dalton’s Law  Main assumptions for Ideal Gas Model 1.The number of molecules in the gas is large, and the average separation between them is large compared with their dimensions. This means that the molecules themselves occupy a negligibly small volume in the container. The molecules are considered to be point-like. 2.The molecules obey Newton’s laws of motion, but as a whole they move randomly. That is any molecule can move in any direction with any speed. At any given moment, one of the molecules move at high speeds, and others move at low speeds. 3.The molecules interact only by short-range forces during elastic collisions, no long-range forces on among molecules. 4.The molecules make elastic collisions with the walls. 5.All molecules of the gas are identical.  Molecular interpretation of temperature So we have Experimentally found the equation of state for an ideal gas: So we can define temperature as: It tells us that temperature is a direct measure of average molecular kinetic energy.  Theorem of equipartition of energy We can transform the last expression into Generalization of this result is known as the theorem of equipartition of energy: 1 2 k BT Each degree of freedom contributes to the energy of a system, where possible degrees of freedom in addition to translation arise from rotation and vibration of molecules. Equation of State for an Ideal Gas Found experimentally: m  or pV = M RT n is the number of moles of the gas R is the universal gas constant:  The pressure–volume law The equipment can be used to investigate the relationship between pressure and volume of a fixed quantity of gas that is kept at constant temperature. The pump forces oil to move higher, decreasing the volume of the air trapped in the tube above the oil. A pressure Apparatus for investigating the pressure– gauge reads the pressure of the trapped volume law. The pump forces oil to move up the tube, decreasing the volume of air. air and so the relationship between pressure and volume may investigated. The changes in pressure and volume must take place slowly so that the temperature stays the same.  The pressure–volume law  The results of a typical experiment are shown in graph. We have plotted pressure against the inverse of the volume and obtained a straight line. Graph of pressure against inverse volume at constant temperature.  The pressure–volume law  At constant temperature and with a fixed quantity of gas, pressure  is inversely proportional to volume, that is:  This relationship is known as the Boyle’s law. The relationship between pressure and volume at constant temperature for a fixed quantity of a gas. The product pV is the same for all points on the curve. ISOTHERMAL PROCESS T=CONST Boyle's laws For a gas of a given mass, the product of the gas pressure and its volume is constant if the gas temperature does not change. pV = const p↑ , V↓ inverse proportionality ISOTHERMAL PROCESS T=CONST  The volume–temperature law The dependence of volume on temperature of a fixed quantity of gas kept at constant pressure can be investigated with the special apparatus. Air is trapped in a thin capillary tube that is immersed in heated water. The air is trapped by a thin thread of very Apparatus for verifying the volume– concentrated sulfuric acid. temperature law. The thread is exposed to the atmosphere and so the pressure of the trapped air is constant.  The volume–temperature law  It is found that the volume increases uniformly with temperature. The  striking fact is that when the straight line is extended backwards it always  crosses the temperature axis at −273 °C, as in Graph. This suggests  that there exists a minimum possible temperature, namely −273 °C. With a real gas the experiment cannot be conducted at very low temperatures since the gas would liquefy – hence the dotted line. With an ideal gas there would be no such restriction. When the graph of volume versus temperature is extended backwards the line intersects the temperature axis at −273 °C.  The volume–temperature law  If this same experiment is repeated with a different quantity of gas, or a  gas at a different constant pressure, the result is the same. In each case, the  straight-line graph of volume versus temperature crosses the temperature  axis at −273 °C (a). And (b) the same graphs are drawn using the Kelvin temperature scale Graph of V versus T is extended backwards, all the lines intersect the temperature axis at the same point (a) and in kelvin, the lines start at zero temperature (b). ISOBARIC PROCESS P=CONST Charles's law For a gas of a given mass, the ratio of the volume of gas to temperature is constant, if the gas pressure does not change. V↑ , T↑ direct proportionality ISOBARIC PROCESS P=CONST  The pressure–temperature law What remains now is to investigate the dependence of pressure on temperature of a fixed quantity of gas in a fixed volume. This can be done with the apparatus shown in Figure. The gas container is surrounded by water whose temperature can be changed. Investigating the pressure– temperature law A pressure gauge measures the pressure of the gas.  The pressure–temperature law We find that pressure increases uniformly with increasing temperature, as shown by the graph The graph of P versus T is a straight line that, when extended backwards, again intersects the temperature axis at −273 °C.  The pressure–temperature law  For quantities of gases containing different numbers of moles at different volumes the results are the same, as shown in Figure. When the temperature is expressed in kelvin, the straight lines all pass through the origin When extended backwards, the graphs of pressure versus temperature for three different quantities of gas all intersect the temperature axis at the same point (a) and in kelvin, the lines start at zero temperature (b). ISOCHORIC PROCESS V=CONST Gay-lussac's law or Amontons’ law. For a gas of a given mass, the ratio of gas pressure to temperature is constant if the volume does not change. p↑ , T↑ direct proportionality ISOBARIC PROCESS P=CONST The Boltzmann Distribution Law We found average kinetic energy of a molecule. But all molecules move chaotically, collide with each other. So arises question what is distribution of molecules depending on their energy? The answer is given by statistical mechanics: PROBLEM-SOLVING STRATEGY for the Ideal Gas Law  Step 1. Examine the situation to determine that an ideal gas is involved.  Step 2. Make a list of what quantities are given or can be inferred from the problem as stated.  Step 3. Identify exactly what needs to be determined in the problem.  Step 4. Determine whether the number of molecules or the number of moles is known or asked for to decide whether to use the ideal gas law as where N is the number of molecules, or where n is the number of moles.  Step 5. Convert known values into proper SI units (K for temperature, Pa for pressure, for volume, molecules for N, and moles for n). If the units of the knowns are consistent with one of the non-SI values of R, you can leave them in those units. Be sure to use absolute temperature and absolute pressure.  Step 6. Solve the ideal gas law for the quantity to be determined. You may need to take a ratio of final states to initial states to eliminate the unknown quantities that are kept fixed.  Step 7. Substitute the known quantities, along with their units, into the appropriate equation and obtain numerical solutions complete with units.  Step 8. Check the answer to see if it is reasonable: Does it make sense? The Boltzmann Distribution Law Where n0 is defined such that n0dE is the number of molecules per unit volume having energy between E = 0 and E = dE. The Boltzmann law states that the probability of finding the molecules in a particular energy state varies exponentially as the negative of the energy divided by kBT: n is the number density Maxwell–Boltzmann speed distribution function If N is the total number of molecules, then the number of molecules with speeds between V and V+dV is dN = NV dV. Maxwell–Boltzmann speed distribution function dN = NV dV.  Gas molecules velocities Root mean square Mean Most probable  Calculating the Ratio of Numbers of Molecules Near Given Speeds  Problem. In a sample of nitrogen (N2, with a molar mass of 28 g/mol) at a temperature of 27 , find the ratio of the number of molecules with a speed very close to 300 m/s to the number with a speed very close to 100 m/s.  Strategy. Since we are looking at a small range, we can approximate the number of molecules near 100 m/s as  Then the ratio we want is  All we have to do is take the ratio of the two f values.  Solution.  1. Identify the knowns and convert to SI units if necessary. Calculating the Ratio of Numbers of Molecules Near Given Speeds  2. Substitute the values and solve. Calculating the Ratio of Numbers of Molecules Near Given Speeds  Figure shows that the curve is shifted to higher speeds at higher temperatures, with a broader range of speeds. you can view this simulation (https://openstax.org/l/21maxboltzdisim) to see the essential features that more massive molecules move slower and have a narrower distribution  Dalton’s Law Dalton’s Law of Partial Pressures states that the total pressure of a gas mixture is equal to the sum of the partial pressures of each individual gas within the mixture. (where P , P ,..., P represent the 1 2 n partial pressures of each component) Each gas in a mixture behaves independently, exerting pressure as if it were the only gas present.  Dalton’s Law Partial Pressure - the pressure exerted by a single gas in a mixture. where Xi is the mole fraction of the ith component in the total mixture of n components. Example Problem A container holds a mixture of oxygen (O₂), nitrogen (N₂), and carbon dioxide (CO₂) with the following information: Total pressure, Ptotal=1 atm Mole fractions: XO2=0.21, XN2=0.78, XCO2=0.01 Calculate the partial pressure of each gas in the mixture.  Dalton’s Law Solution Using Dalton’s Law, we find the partial pressure Pi of each gas by multiplying the mole fraction Xi of the gas by the total pressure Ptotal: Pi =Xi⋅Ptotal Step-by-Step Calculation: Partial Pressure of O₂: PO2=XO2⋅Ptotal=0.21*1 atm= 0.21 atm Partial Pressure of N₂: PN2=XN2⋅Ptotal=0.78*1 atm= 0.78 atm Partial Pressure of CO₂: PCO2= XCO2⋅Ptotal=0.01*1 atm=0.01 atm Each of these partial pressures adds up to the total pressure: Ptotal=PO2+PN2+PCO2=0.21+0.78+0.01=1 atm This calculation shows how the contributions from each gas relate to the total pressure in the mixture.  Applications of Dalton's Law Scuba Diving. Helps divers understand gas mixtures and avoid decompression sickness. Atmospheric Science. Calculating pressures of different gases in air. Medical Applications. Used in anesthesia and respiration therapies.  Dalton’s Law Key Notes Dalton's Law provides a simple way to calculate total pressure in gas mixtures. Important in various scientific and industrial applications. Knowing partial pressures helps predict gas behavior in different settings. Main terms: thermal energy ideal gas atomic mass Mole ideal gas law velocity distribution function Maxwell-Boltzmann distribution Gas Mixture Partial Pressure Mole Fraction  Control questions: 1. Derive the ideal gas law equation from the combination of the pressure-volume law, volume-temperature law, and pressure-temperature law. 2. For a gas at constant temperature, if the volume is halved, what happens to the pressure? 3. Describe what happens to the volume of a gas as its temperature increases at constant pressure. 4. Explain the relationship between pressure and temperature when the volume of a gas is held constant. What happens to the pressure if the temperature is doubled? 5. Define the Maxwell-Boltzmann distribution function and explain what it represents about gas molecules at a given temperature. 6. How is the average speed of gas molecules related to the temperature of the gas? Provide the formula and explain the role of mass and temperature. 7. State Dalton’s Law and provide an example of how to calculate the total pressure in a gas mixture with known partial pressures of its components. 8. Describe two conditions under which a real gas deviates from ideal gas behavior, and explain why the ideal gas law may not hold under these conditions.