Physics (I) for Robotics and Mechatronics Engineering Students - Helwan University PDF

Physics (I) For Robotics and Mechatronics Engineering Students By D\ Afaf Mahmoud Abd-Rabou Associated Professor Physics Department-Faculty of Science – Helwan University Evaluation Semester work 20 Practical Exam 30 Final E...

Physics (I) For Robotics and Mechatronics Engineering Students By D\ Afaf Mahmoud Abd-Rabou Associated Professor Physics Department-Faculty of Science – Helwan University Evaluation Semester work 20 Practical Exam 30 Final Exam 50 Content Chapter 1 Units and dimension Chapter 2 Viscosity Chapter 3 Elasticity Chapter 4 Heat and Heat transferer Chapter 5 Thermodynamics Chapter 6 Application of first and Second law of thermodynamics  Chapter one Units and dimension 1.1 Introduction. What are the main objective of physics? Find laws that govern natural phenomena. Use laws to develop theories What is the basis of physics? Experimental observation Quantities measurements What are the physical laws expressed by? How can it be described? What happens when a discrepancy be between theories And experiments? 1.2 Standards of length, mass, and time How can the physical law be described? Are these physical quantities base or derived? Physical quantities: Standard Physical quantities Derived Physical quantities (Length Mass Time) (Velocity, Acceleration, Density,....) How can the results of physical quantities be reported? What if you know that: Mass of an object is 75 Fankosh A building is 3 glitch tall Three Base Units (SI) Three Base Units (SI)  ✓ Length yard In A.D. 1120 (7th century) king of England decide that the standard of length in his country. Foot length of the royal foot of King Louis XIV. 1799, legal standard of length in France became Meter one ten-millionth the distance from the equator to the North Pole along one particular longitudinal line that passes through Paris. 1960 The distance between two lines on a specific platinum–iridium bar stored under controlled conditions in France. 1970 1 650 763.73 wavelengths of orange-red light emitted from a krypton- 86 lamp. 1983 The distance travelled by light in vacuum during a time of 1/2999792458 second Mass Kg -1887 mass of a specific platinum–iridium alloy cylinder kept at the International Bureau of Weights & National Institute of Standards and Technology (NIST) in Gaithersburg, Maryland Time Before 1960 In terms of the mean solar day for the year 1900 In 1967 the second was redefined using the high precision attainable in an atomic clock, the characteristic frequency of the cesium-133 atom as the “reference clock.” The second (s) is now defined as 9 192 631 770 times the period of vibration of radiation from the cesium atom.  Example 2 Astronomical distances are sometimes described in terms of light- years (ly). A light-year is the distance that light will travel in one year (yr). How far in meters does light travel in one year. Solution: 1 year = 365.25 days, 1 day = 24 hours, 1 hour = 60 minutes, 1 minute = 60 seconds 1 year =365.25 day *24 hours* 60 min * 60 s =31,557,600 s The distance that light travels in a one year is 299,792,458 m 31,557,600 s 𝑙𝑦 =. (1 yr) = 9.461× 1015 m. 𝑠 𝑦𝑟 Systems of units. Units System Length Mass Time CGS Centimeter Gram Second MKS (I.S) Meter Kilogram Second System international (SI) FPS (British) Foot Pound Second 1 Kg = 2.20462 Pound 1 m = 3.28084 Foot 1.2 Matter and Model Building Imagine model that is related to the phenomenon. Make predictions about the behavior of the system. What is the base of the model? based on the interactions among the components of the system and/or the interaction between the system and the environment outside the system. 1.3 Density and Atomic Mass ✓ Density (ρ) 𝑴 𝝆= 𝑽 Are Equal volumes of different elements, have the same mass? ✓ Atomic Mass Mass of single atom is measured in atomic mass units (u) 1 u = 1.660 538 7 x 10-27 kg. Check Your Understanding In a machine shop, two cams are produced, one of aluminum and one of iron. Both cams have the same mass. Which cam is larger? (a) the aluminum cam (b) the iron cam (c) Both cams have the same size. Example A solid cube of aluminum (density 2.70 g/cm3) has a volume of 0.200 cm3. It is known that 27.0 g of aluminum contains 6.02 x 1023 atoms. How many aluminum atoms are contained in the cube? Solution: m (cube)= ρ V=2.7x0.2=0.54 g 27 g 6.02 x 1023 atoms 0.54 g ? No. of atoms in the cube = 6.02 x 1023 x 0.54/27= 1.2 x 1022 atoms 1.4-Dimensional Analysis What is meant by “Dimension” in physics? How it different from unit? Physical Length Mass Time Quantity Dimension [L] [M] [T] What are the usage of the dimension? What are the rules ? (1)Check the validity of physical expression. Example: 𝟏 2 𝒙 = 𝒂t 𝟐 Where x is the distance, a is the acceleration and t is the time Solution: 𝑳𝑯𝑺= 𝑳 𝑳 2 R𝑯𝑺 = T L.H.S = R.H.S T2 ∴ 𝒕𝒉𝒊𝒔 𝒆𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝒊𝒔 𝒄𝒐𝒓𝒓𝒆𝒄𝒕. (2) Deduce a physical equations Example 1: Use dimensional analysis to set up an expression of the form 𝒙 ∝ 𝑎𝑛 𝑡 𝑚 Where n and m are exponents that must be determined and α indicates a proportionality. This relationship is correct only if the dimensions of both sides are the same. Solution: x ∝ 𝑎𝑛 𝑡 𝑚 𝑛 𝐿 𝑚 RHS = 2 𝑇 𝑛 𝑚−2𝑛 RHS = 𝐿 𝑇 𝑇 1 0 𝐿𝐻𝑆 = 𝐿 𝑇 𝑛=1 m−2=0 m=2 x = const. 𝑎 𝑡 2 No numerical values can be expected Example 2: Suppose we are told that the acceleration a of a particle moving with uniform speed v in a circle of radius r is proportional to some power of r,say rn , and some power of v, say vm. Determine the values of n and m and write the simplest form of an equation for the acceleration. Solution: a ∝ 𝑟𝑛𝑣𝑚 a = k 𝑟𝑛𝑣𝑚 2 𝑚 a = k 𝑣 /𝑟 𝐿 RHS = 𝐿 𝑛 𝐿 𝑇 LHS = 2 𝑇 𝑛+𝑚 −𝑚 RHS = 𝐿 𝑇 LHS = 𝐿 1 𝑇 −2 𝑛+𝑚 =1 −𝑚 = −2 𝑚 = 2 & 𝑛 = −1 1.5 Conversion of Units 1 mile = 1 ft = = 1 609 m =1600 m =0.3048 m = 1.609 km =1.6 km = 30.48 cm =30 cm 1 in. = 0.025 4 m =2.54 cm =2.5 cm Example 1 Convert 15 in to cm Solution 1 in. =2.5 cm 15 in x 2.5=37.5 cm Quick Quiz 1.3 The distance between two cities is 100 mi. The number of kilometers between the two cities is (a) smaller than 100 (b) larger than 100 1 mile =1.600 km (c) equal to 100. Example 2 Is He Speeding? On an interstate highway, a car is traveling at a speed of 38.0 m/s. Is this car exceeding the speed limit of 75.0 mi/h? Solution: We first convert meters to miles: 𝟑𝟖 𝟑𝟖 𝒎/𝒔𝒆𝒄 = = 0.02375 mi/sec 1600 Now we convert seconds to hours: 𝒎𝒊 𝟎. 𝟎𝟐𝟑𝟕𝟓 ∗ 𝟔𝟎 𝐦𝐢𝐧 ∗ 𝟔𝟎 𝐡 = 𝟖𝟓. 𝟓 𝐦𝐢/𝐡 𝐬𝐞𝐜 Yes this car exceeding the speed limit 1.6 Order-of-Magnitude Calculations Example Estimate the number of breaths taken during an average life span. Solution We start by guessing that - Life span is about 70 years - 10 breaths / minute The number of minutes in a year is = 1*365*24*60= 525,600 min.= 𝟓. 𝟐𝐱𝟏𝟎5min The number of minutes in one years is = 𝟒𝟎𝟎*25*60=𝟔𝐱𝟏𝟎5 min Comment? The number of minutes in 70 years is = 𝟓. 𝟐𝐱𝟏𝟎5min*70=36,792,000=𝟑. 𝟔𝐱𝟏𝟎7min The number of minutes in 70 years is = 6𝐱𝟏𝟎5min*70=4. 𝟐𝐱𝟏𝟎7min The number of breaths/minute in 70 years is =𝟑. 𝟔𝐱𝟏𝟎7min*10=𝟑. 𝟔𝐱𝟏𝟎8 breaths Or =4. 𝟐𝐱𝟏𝟎7min*10=4. 𝟐𝐱𝟏𝟎8 breaths What If? life span = 80 years? The number of breaths/minute in 80 years is =𝟒. 𝟏𝐱𝟏𝟎8 breaths This is still on the order of 𝟏𝟎8 breaths, so The estimated order of magnitude would be unchanged Thank You for Your attention  Chapter II Heat and Matter Heat and Matter 1. Introduction. 2. Change of state(phase). 3. Phase diagram. 4. Phase equilibria and phase rule. 5. Effect of pressure on phase transformation. 6. Thermal expansion.  2.1 Introduction. What is Heat? Heat and temp are the same? Dependent variables Cause & (effect) Concept and Nature of Heat. Sensation of Heat Heat Hot Cold Is the size of the matter a factor? When does the flow stop? Concept of Heat Flow of energy (energy in transit) Energy transfers between two systems only due to temp. diff Concept of Temperature. Number related to K.E-units How to represent temperature qualitatively ? Is it macroscopic or microscopic? Is each rise in temperature associated with heat transferer? What is the difference between heat temperature? What Happen when change in temperature take place? (A) Change in state. (B) Change in phase. (C) Change in size. (D) Change in properties What is the difference between state and phase? 2.2 Change of State. 2.2.1 Heating and Cooling Curves. What about the behavior of a substance during heating and cooling? 100 evaporation 80 & condensation 60 40 20 0 Melting -20 & solidification  Add Q Add Q Remove Q Remove Q =thermal energy 2.2.3 Phase Diagram. What is the importance of studing phase transformation and phase diagram? What is the phase diagram? Shows the phases that exist in equilibrium for a material of given chemical composition. Unary, binary, ternary diagrams What are the types of Phase diagram? (a) H2O (b) CO2 (c) Cu Water Phase diagram P B C S L 1 atom G O A T 1 atom =1.01325 x 105 pascal =76 cm 0 oC 100 oC ? What does different areas, lines, line intersection represent? What is the condition of boiling point? Is it constant? Simple phase diagram of one component is divided into two classes: 1st Class: Expand on solidification (water – bismuth) -ve OB slope 2nd Class: Contract on solidification (Co2 – O2) +ve OB slope  Co2 Phase diagram B C B C O O A A Reason of expansion of water during solidification Liquid water molecules Solid water molecules Molecular structure Solid Liquid Gas 2.2.2 Phase Equilibrium and Phase Rule. Gibbs phase rule. ✓ Degree of Freedom(F) Number of independently variable (ways) affecting the range of states in which a system may exist at eq.. ✓Phase(P) Physically distinct portion of a system, separated from other portions by bounding surface. ✓Number of components (C) Minimum number of independent species necessary to define the composition of all phases of the system. Three phase system Ice, water, & vapor Three phase system CaCo3 , CaO & Co2 CaO = CaCo3 - Co2 Phase Rule (Gibbs) 𝐹 = 𝐶 − 𝑃 +2 How can the degree of freedom increase? Example(1) 𝑳 & 𝒗𝒂𝒑𝒐𝒖𝒓 𝒂𝒕 𝒆𝒒𝒖𝒍𝒊𝒃𝒓𝒊𝒖𝒎 𝑪=𝟏 𝑷=𝟐 𝑭=𝟏 Example(2) 𝑳 , 𝒗𝒂𝒑𝒐𝒖𝒓 & 𝒔𝒐𝒍𝒊𝒅 𝒂𝒕 𝒆𝒒𝒖𝒍𝒊𝒃𝒓𝒊𝒖𝒎 𝑪=𝟏 𝑷=𝟑 𝑭=𝟎 What is the meaning that, the degree of freedom of a System is equal to one? What is the meaning that, the degree of freedom of a System is equal to zero?  Number Degrees System of of Comments phases freedom gas, liquid 1 F=1-1+2=2 Bivariant system: lies or solid anywhere within the area marked G, L or S. gas-liquid, 2 F=1-2+2=1 Univariant system: lies liquid-solid anywhere along a line or between two phases gas-solid regions (AO, BO or CO). gas-liquid- 3 F=1-3+2=0 Invarient system: can solid only lie at the triple point (O). Thank you for your attention  Chapter II Heat and Matter ✓ Effect of pressure on melting or freezing point Contracting Expanding Pressure Pressure constrains the process helps the process ✓ What about contracting material? ✓ What is the effect of pressure on triple point? Triple point pressure and temperature examples: Substance Temperature (K) Pressure (mmHg) Ammonia 195.40 45.57 Carbondioxide 216.55 3880.00 Deuterium 18.63 128.00 Hellum (He4) 2.17 37.80 Hydrogen 13.96 54.10 Nitrogen 63.18 94.00 Oxygen 54.36 1.14 Sulfurdioxide 197.68 1.26 Water 273.16 4.58 2.3 Change in Size: Fundamentals: No Phase transition What are the importance of change in size? What is the cause of thermal expansions? Change in the average atomic separation Dependence of amplitude on T Lattice spacing is mean figure  Mechanical model of crystalline solid What the atomic spacing at room temperature? How can the degree of expansion or contraction be calculated? Thermal expansion due to asymmetric nature of potential energy 2.3.2 coefficient of thermal expansion. Coefficient: Number measure for property What is the thermal expansion? Is the thermal expansion coefficient +ve or –ve? What are the types of thermal expansion? (A) Linear thermal expansion. What is the difference between thermal expansion and due to mechanical one? ∆𝐿 ⋉ 𝐿𝑜 ∆𝑇 1 𝑑𝐿 𝛼= 𝐿 𝑑𝑇 ∆𝐿 = 𝛼 𝐿𝑜 ∆𝑇 units 𝐿 − 𝐿𝑜 = 𝛼 𝐿𝑜 ∆𝑇 𝐿 = 𝐿𝑜 (1 + 𝛼 ∆𝑇) (B) Thermal expansion of area. 𝑑𝐴 =𝝎  L 𝛽= /𝑑𝑇 𝐴 𝑑𝐴 𝛽= /𝐴 𝝎 𝑑𝑇 𝝎o What is the relation between 𝛽and α ? (L)2 1 𝑑𝐴 𝛽= 𝐴 𝑑𝑇 2 1 𝑑𝐿 𝑑𝐿 𝛽= 2. L= Lo+ L 𝐿 𝑑𝐿 𝑑𝑇  1 𝑑𝐿 𝛽 = 2 2𝐿. 𝐿 𝑑𝑇 2 𝑑𝐿 𝛽= = 2α 𝐿 𝑑𝑇  = 𝝎  L 𝐿 = 𝐿𝑜 (1 + 𝛼 ∆𝑇) 𝐿𝑜 𝜔 = 𝜔𝑜 (1 + 𝛼 ∆𝑇) 𝝎 𝜔𝑜 2 2 A= 𝐿𝑜 𝜔𝑜 (1 + 2𝛼Δ𝑇 + 𝛼 Δ𝑇 )  Substan  (10-5 𝐴 = 𝐴𝑜 (1 + 2𝛼Δ𝑇) ce K-1) 𝐴 = 𝐴𝑜 (1 + 𝛽Δ𝑇) Quartz 0.12 Glass 1.2-2.7 Copper 4.2 Substance  (10-5 K-1) Substance  (10-5 K-1) Brass 6.0 Glycerin 49.0 Aluminu Ethylalco 7.2 75.0 m hol Steel l3.0 Carbon 115.0 Mercury 18.0 disulfide Thank you for your attention  Chapter II Heat and Matter (B) Thermal expansion of volume. 𝑑𝑉 𝛾= /𝑑𝑇 𝑉 3 𝑑𝑉 1 𝑑𝐿 𝑑𝐿 𝛾= /𝑉 𝛾= 3. 𝑑𝑇 𝐿 𝑑𝐿 𝑑𝑇  3 1 𝑑𝐿 𝑑𝐿 𝛾= 3. 𝐿 𝑑𝐿 𝑑𝑇 1 2 𝑑𝐿 𝛾 = 3 3𝐿. 𝐿 𝑑𝑇 3 𝑑𝐿 𝛾= = 3α 𝐿 𝑑𝑇 𝑉 = 𝑉𝑜 (1 + 3𝛼Δ𝑇) 𝑉 = 𝑉𝑜 (1 + 𝛾Δ𝑇) 2.3.3 Negative thermal expansion. Anomalous behavior Range of temperature What is thermal expansion compensator? What is the origin negative expansion? (1)Phase transitions, (2)Transverse vibrational modes and (3)Rigid Unit modes. (1)Phase transitions. Liquid water molecules Solid water molecules Molecular structure  (2)Transverse vibrational modes. For Simple diatomic molecule, what about interatomic distance by heating? What about more complex molecules? At low Temperature Longitudinal Vibration Transverse Vibration  (3)Rigid Unit modes. At high temperature minerals framework 573°C 870°C 1470° 1705° α- β- β- β- Quartz Quartz Tridymit Cristo Trigona Hexago exago balite l nal nal Cubic 2.65 2.53 2.25 2.20 3 3 3 3 Thermal expansion of quartz. The α  β phase  Tetrahedral molecule geometry SiO4 coupled rotation of relatively rigid corner sharing SiO4tetrahedra 2.3.4 Change in volume in water. (A) Anomalous behavior. P constant VT What about the volume of water? What about its density from 0 °C to 4 °C? What about its density from 4 °C to 100°C? What about its density & volume at 4 °C? ✓ Importance of water anomalous behavior. What is the decreasing ratio of density from 4 to 0 Co ? Water freezes from top to down. 90% of iceberg sink. ✓ How do sea animal and plants live? What is the reason of –ve thermal expansion of water? (B)Structural consideration of water expansion. What is the difference between the molecular structure between Water and ice? Water crystallizes into an open hexagonal form. What The structure of water? Net dipole on the molecule In its liquid state, the molecules are free to roam around wherever they wish. (B)Dry ice. What is the chemical composition of dry ice? Sublimation Point 2.2.5 Application of Thermal Expansion. (A) Positive Thermal Expansion. ✓ What is the principal? ✓ Automatic control.  Large Small Expansion Expansion Heating is off Coil for compactness (B) Negative Thermal Expansion. ✓ What are the problems due to thermal expansion? ✓ Dental fillings ✓ Optical mirrors ✓ R.I and dimension of fiber optics ✓ Electronic industry -Circuit board ✓ Overcome problems of thermal expansion. ✓ Get a desired thermal expansion. Zirconium Tungstate (ZrW2O8) & tooth enamel 2.2.6 Thermal stresses. ✓ What is the reason of thermal stresses? ✓ Geometry-external constrains- T gradient Stresses due to expanding constraints.  ∆𝐿𝜏 = 𝛼 𝐿𝑜 ∆𝑇 ✓ Below the elastic limit: 𝑠𝑡𝑟𝑒𝑠𝑠 𝜎 Y= = 𝑠𝑡𝑟𝑎𝑖𝑛 𝜀  𝜎 Y= Δ𝐿𝑚 /𝐿𝑜 𝜎 Δ𝐿𝑚 = 𝑌/𝐿0 1 Δ𝐿𝑚 = 𝜎 𝐿𝑜 𝑌 Δ𝐿𝑚 + ∆𝐿𝜏 = 0 1 𝜎 𝐿𝑜 + 𝛼 𝐿𝑜 ∆𝑇 = 0 𝑌 𝜎 = −𝛼 ∆𝑇 𝜎 = −𝛼 𝑌∆𝑇 𝑌  𝐹 = −𝛼𝐴𝑌∆𝑇 Why F is –ve? When does it be +ve?  No electronic components needed 2.4 Smart Materials. (A) Smart Resistor. does T depend on the ambient temp.? (B) Thermochromatic Materials. What are the type of materials used? How can the transition temperature change? Uses? Thank You For your Attention Thermodynamics Therme-dynamis  Evaluation Assignment 10 M Semester work 10 M Mid-term 20 M Final 60  Course Content Ch.1 The Kinetic Gas Theory. Ch.2 Heat and the First Low of Thermodynamics. Ch.3 Entropy and Heat Engine.  Chapter 1. Content: (1) Introduction. (2)The universe and its parts. (3)Molecular model for the pressure of an ideal gas. (4)Van Der Waal’s equation of state. (5) Estimate of mean free path. (1) Introduction. Why thermodynamics field? Is this study microscopic? What is the most important feature Of that system? What are the properties of the system that affect the thermodynamic investigation? Composition Melting burn Expansion Structure What if a gas is not free to expand? Conditions of the system  Examples of a thermodynamic system (1) a fluid (a gas or a liquid) confined to a beaker of a certain volume, subjected to a certain pressure at a certain temperature. (2) a solid subjected to external stresses, at a given temperature. Example(2) A macroscopic system undergoes a series of transformations its initial state. Absorb Q of heat releases a net amount W. What is the efficiency of the transformation, ratio W/Q?  (2)The universe and its parts. What are system, boundary and surroundings? ✓ System Collection of matter within prescribed identifiable boundaries. Any macroscopic system, for which thermal eff. are important parameters, is an example of a thermodynamic system. What are the types of systems? ✓ Boundary. Thermodynamic system is always confined within some boundary. Surface enveloping the system and separating it from the surroundings. What is the importance of the boundary? ✓ Surrounding Outside this boundary. universe How to express an amount of gas (m) in moles? Mole Specific number(NA)of elementary entities of a substance. NA=6.022 x 1023 Are one mole of mercury atoms and another of water have the same mass? Volume? NA is the no carbon atoms in 12 gm M is molecular weight g/mole 𝑚 𝑛= (3) Macroscopic Description of an ideal gas. What are the features of gas to be studied? Most gases at room temperature and atmospheric pressure is ideal How the P, T and V related? What is the equation of the state? Is it simple? Description of the system No leakage No moles=const. (i) Const. T T = const. P 1/V Boyle’s low What if P is constant? (ii) Const. P P = const. V T Charle’s low V V Gay-Lussac's law (iii) Const. V  V1 V2 V3 V = const. P T What if P & T are constant? (iv) Const. T and P ✓At P=constant T=constant: V  amount of gas  R = 8.314 J mol–1 K–1 𝑃𝑉 = 𝑛𝑅𝑇 R = 0.0821 Litre atm K–1 mol–1 R = 2 cal K–1 mol–1 1 joule = 0.0099 L atm. 1 joule = 0.2390057361 cal 1 cal = 4.184 kJ = 4184 J. 1 atm = 1.01293 x10^5 Pascals = 76 cm Hg x13.534 g/cm^3x 980 cm/s2 1 Pascal = ???? N/m2  𝑃𝑉 = 𝑛𝑅𝑇 R Universal gas constant P0 PV/nT const. for all gases Volume of one mole STP P=Pa T= 0 C =273 K R=8.31 J/mol.K V =22.4 L 1 psi= 6894.76 pascal How to express ideal gas in total no molecules? 𝑁 𝑁 = 𝑛𝑁𝐴 𝑃𝑉 = 𝑅𝑇 𝑁𝐴 𝑅 =𝐾 𝑃𝑉 = 𝑁𝐾𝑇 𝑁𝐴 𝐾 = 1.38 x −23 10 𝐽/𝐾 What is the specific gas constant? Unit?  𝑃𝑉 = 𝑁𝐾𝑇 At fixed volume 𝑃𝛼𝑁𝑇 An ideal gas obey the equation of state 𝑃𝑉 = 𝑛𝑅𝑇 Is an ideal gas exist? (3)Molecular model for the pressure of an ideal gas. Kinetic theory or kinetic molecular theory The ideal gas is considered under the assumption: (1)Pure gas -Large number –Large distances (2)Molecules occupy negligible volume compared with container. (3) Molecules obey Newton’s low of motion. one molecule move randomly in all direction (vav= constant) (4) Molecules undergo elastic collision. ( energy & momentum are conserved) (5) Forces(interaction) between molecules negligible except at collision. (6) The gas in thermal equilibrium with walls. Walls reject as absorb. Kinetic gas Theory. Aim: x x x y d What is the pressure of all particles on the wall? Molecule of mass (m)collide with wall. ∆𝑝𝑥 = −𝑚𝑣𝑥 −(𝑚𝑣𝑥 ) = −2𝑚𝑣𝑥 (1) Distance between 2 successive collision: 2𝑑 d Time between 2 successive collision: 2𝑑 ∆𝑡 = (2) 𝑣𝑥  Pressure(F/A) of all particles on the wall: σ𝐹 P= (3) 𝐴 Average force exerted by a molecule on the wall in t: ∆p 2m𝑣𝑥 2m𝑣𝑥 m𝑣𝑥 2 m 𝐹= = = = = 𝑣𝑥 2 (4) ∆𝑡 ∆𝑡 2𝑑/𝑣𝑥 𝑑 𝑑 𝑚 P= 𝑣𝑥 2 (5) 𝑑3 𝑚 P= 𝑣𝑥12 + 𝑣𝑥22 + 𝑣𝑥32 + ⋯ + 𝑣𝑥𝑁 2 (6) 𝑑3  𝑚 2 2 2 2 P= 𝑣𝑥1 + 𝑣𝑥2 + 𝑣𝑥3 + ⋯ + 𝑣𝑥𝑁 (6) 𝑑3 𝑣𝑥12 +𝑣𝑥22 +𝑣𝑥32 +⋯+𝑣𝑥𝑁2 ∵ 𝑣𝑥 2 = (7) 𝑁 𝑚 ∴𝑃= 3 𝑁 × 𝑣𝑥 2 (8) 𝑑 𝑁𝑚 2 𝑃= 𝑣𝑥 (9) 𝑉  2 2 𝑣 𝑣𝑥 = (10) 3 1 𝑁𝑚 2 P= 𝑣 (11) 3 𝑉 N𝑚 = 𝑛𝑀 1 𝑛𝑀 2 (12) P= 𝑣 3 𝑉 Thank you for your attention  Ch. 3 Thermodynamics Molecular Pressure of an ideal gas (Continued) What about the collision between molecules? What about the shape of the container? How to express P as K.E? 1𝑁 2 𝑃= 𝑚𝑣 (13) 3𝑉 2𝑁 1 2 𝑃= 𝑚𝑣 (14) 3𝑉 2 There is a key link between the microscopic world of the gas molecules and the macroscopic one The total translational av. U of N molecules: 1𝑁 2 1 𝑃= 𝑚 𝑣 𝐸𝑡 = 𝑁 𝑚𝑣 2 3𝑉 2 2𝑁 1 2 𝑃= 𝑚𝑣 3𝑉 2 2 1 𝑃= 𝑁 𝑚𝑣 2 3𝑉 2 2 3 3 3 𝑃𝑉 = 𝑈𝑡 𝑈𝑡 = 𝑃𝑉= N𝑘𝑇 = n𝑅𝑇 3 2 2 2 Molecular interpretation of Temperature In x-direction and in space. 2𝑁 1 𝑃= 𝑚𝑣 2 𝑃𝑉 = 𝑁𝑘𝑇 3𝑉 2 2 1 2 𝑁𝑘𝑇 = 𝑁 𝑚𝑣 3 2 1 2 3 𝑚𝑣 = 𝐾𝑇 (15) 2 2  2 2 (16) 𝑣 = 3𝑣𝑥 3 2 3 𝑚 𝑣𝑥 = 𝑘𝑇 (17) 2 2 1 1 𝑚 𝑣𝑥 2 = 𝑘𝑇 (18) 2 2 1 ഥ = 𝑘𝑇 𝑈 2  1 2 1 𝑚 𝑣𝑦 = 𝑘𝑇 (19) 2 2 1 1 2 𝑚 𝑣𝑧 = 𝑘𝑇 (20) 2 2 Each translation degree of freedom contributes an equal amount of energy to the gas From kinetic gas theory to law of gases. 2 1 2 𝑃= 𝑁 𝑚𝑣 3𝑉 2 1 2 3 𝑚𝑣 = 𝐾𝑇 2 2 2 3 𝑃= 𝑁 𝐾𝑇 3𝑉 2 𝑁 𝑃𝑉 = 𝑁 𝐾𝑇 𝑃 = 𝐾𝑇 𝑉 P  T at constant V 𝑃𝑉 = 𝑁 𝐾𝑇 𝑁 = 𝑛 𝑁𝐴 𝑃𝑉 = 𝑛𝑅 𝑇 The root mean square of the molecules speed 3 1 𝑘𝑇 = 𝑚 𝑣 2 2 2 3𝐾𝑇 3𝑅𝑇 𝑣𝑟𝑚𝑠 = = 𝑚 𝑀 Which type of molecule moves faster? Relation between  and rms speed 3𝐾𝑇 𝑣𝑟𝑚𝑠 = 𝑃𝑉 = 𝑁𝐾𝑇 𝑚 3𝑃𝑉 3𝑃 𝑣𝑟𝑚𝑠 = 𝑣𝑟𝑚𝑠 = 𝑁𝑚 𝑁𝑚/𝑉 3𝑃 𝑣𝑟𝑚𝑠 = 𝜌 ✓ Real gas versus ideal gas What are the conditions under which the real gas behave as an ideal gas? 2𝑁 1 2 𝑃= 𝑚𝑣 3𝑉 2 Assumptions of kinetic theory of gas: ✓ Neglect the volume of the gas ✓ Neglect the attraction force between molecules  For Ideal gas: 𝑇1 < 𝑇2 < 𝑇3 P v  For real gas: T3 T2 𝑇1 < 𝑇2 < 𝑇3 T1 What is the reason of this behavior? (4)Van Der Waal’s equation of state. (1)The volume occupied by the gas(b): What if V is comparable with b? Fintermolecular can’t be neglected What is b? (2)The inter-molecular force Pressure of real gas versus real one? What happen when the molecules are close together? P. E versus K.E liquidity gas 1 1 Finward  Finward  𝑉 𝑉 1 Finward  density Finward  𝑉 1 𝑎 Pα 2 P= 2 𝑉 𝑉  𝑎 P+ 2 𝑉 − 𝑏 = 𝑛𝑅𝑇 𝑉 a and b are empirical constants for a practical gas Low and high temperature against the behavior Of the molecules! ✓ No simple equation can describe real gas. ✓ Low pressure and heigh temperature is best to approximate the situation of ideal gas. (5)Mean Free Path What is the mean free path? Is the molecule a point mass? What is the effect of gravity on the molecules? What are the factors affect mfp ? ✓ Density of the gas. ✓ Diameter of molecule. ✓ How to estimate mean free path? What is the effective cross-section for collision ? 2 𝜋(𝑟𝐴 + 𝑟𝐵 ) What is the average distance between molecules required for collision ? In time interval t, Speed 𝑣ҧ it travel a distance 𝑙 = 𝑣tҧ 𝑛𝑣 = 𝑚𝑜𝑙𝑒𝑐𝑢𝑙𝑒 𝑝𝑒𝑟 𝑢𝑛𝑖𝑡 𝑣𝑜𝑙𝑢𝑚𝑒 - 𝑙 = 𝑣tҧ How many particles in the molecule path? V 𝜋𝑑 2 𝑣𝑡ҧ nv no particle /volume Number of collision = number of particles 2 𝑣𝑡ҧ 𝜋𝑑 𝑣𝑡ҧ 𝑛𝑣 ℓ= 𝜋𝑑 2 𝑣𝑡ҧ 𝑛𝑣 1 ℓ= 𝜋𝑑 2 𝑛𝑣  1 ℓ= 2 𝜋𝑑 𝑛𝑣 no. collision 𝑓= = 𝜋𝑑 2 𝑣ҧ 𝑛𝑣 time 𝑓= 2 𝜋𝑑 𝑣𝑛 ҧ 𝑣 𝑓ℓ = 𝑣ҧ 1/f mean free time Due to the motion of particles, there is correction as: 1 ℓ= 2𝜋𝑑 2 𝑛𝑣 2 𝑓 = 2𝜋𝑑 𝑣𝑛 ҧ 𝑣  Ch. 4 Heat and the First Law of Thermodynamic  Outline 2.1 Introduction 2.2 Units of heat and unit conversion 2.3 The first low of thermodynamics. 2.4 The equipartition of energy. 2.5 Heat capacity. 2.6 Enthalpy. 2.7 Isothermal Process. 2.8 Adiabatic Process. 2.1 Introduction. What is the difference between these concepts: Heat Heat is thermal energy on move Heat flow: W or cal/sec The amount of heat transfer per unit time that takes place due to temperature difference. Internal energy: ✓ is the energy that a system has because its T The internal energy of ideal gas is related to T 1 1 𝑚 𝑣𝑥 2 = 𝑘𝑇 2 2 The higher the temperature of the gas, the greater its internal energy Work The work done by or on the system is a measure of energy transfer between the system and its Surroundings. 2.2 Units of heat Calorie The amount of heat necessary to rise 1gm of water one degree(14.5 to 15.5). Joule The amount of energy exerted when a force of one Newton is applied over a displacement of one meter. Unit conversion. 1 Calorie= 4.186 J= 3.968e-3 Btu 1 J=0.2389 Cal =9.478 x10-4 Btu 1 Btu = 1055 J = 252 Cal 1 Btu = 0.293 W joules = watts × seconds Thank you for your attention  Ch. 5 Heat and the First Law of Thermodynamic  Outline 5.1 Introduction 5.2 Units of heat and unit conversion 5.3 The first low of thermodynamics. 5.4 The equipartition of energy. 5.5 Heat capacity. 5.6 Enthalpy. 5.7 Isothermal Process. 5.8 Adiabatic Process. 5.1 Introduction. What is the difference between these concepts: Heat Heat is thermal energy on move Heat flow: W or cal/sec The amount of heat transfer per unit time that takes place due to temperature difference. Internal energy: ✓ is the energy that a system has because its T The internal energy of ideal gas is related to T 1 1 𝑚 𝑣𝑥 2 = 𝑘𝑇 2 2 The higher the temperature of the gas, the greater its internal energy Work The work done by or on the system is a measure of energy transfer between the system and its Surroundings. 5.2 Units of heat Calorie The amount of heat necessary to rise 1gm of water one degree(14.5 to 15.5). Joule The amount of energy exerted when a force of one Newton is applied over a displacement of one meter. Unit conversion. 1 Calorie= 4.186 J= 3.968e-3 Btu 1 J=0.2389 Cal =9.478 x10-4 Btu 1 Btu = 1055 J = 252 Cal 1 Btu = 0.293 W joules = watts × seconds Work and Heat in Thermodynamic process. Gas in equilibrium Expand quasi-statically  dy 𝑑𝑊 = 𝐹𝑑𝑦 𝑑𝑊 = 𝑃𝐴𝑑𝑦 𝑉𝑓 𝑑𝑊 = 𝑃𝑑𝑉 𝑊 = න 𝑃𝑑𝑉 𝑉 dy As conclusion we have converted an amount of heat Q into W or W to Q As Q=W 𝑑𝑊 = 𝑃𝑑𝑉 𝑉𝑓 𝑊 = න 𝑃𝑑𝑉 𝑉𝑖 P is a function of V  𝑉𝑓 𝑊 = න 𝑃𝑑𝑉 𝑉𝑖  𝑉𝑓 𝑊 = න 𝑃𝑑𝑉 𝑉𝑖  𝑉𝑓 𝑊 = න 𝑃𝑑𝑉 𝑉𝑖 (Pi,Vi) (Pf,Vf) (Pf,Vf) (Pi,Vi) What about work at constant V? Is the work affected by path from initial to final state only? Is the work done by a system depends on the process by which (Pi,Vi) (Pf,Vf)? Insulating wall Case(1) Sufficient heat to maintain Ti At P> Pa due to absorbed heat Gas expand slowly Piston rise Reach Pf and Vf In isothermal process heat transfer between the system and the surrounding. Case(2) In case of broken membrane As expand rapidly Reach Pf and Vf Gas does no work Adiabatic wall In adiabatic process no heat transfer between the system and its surrounding. How Heat and work transfer to and from system? (Pi,Vi) (Pf,Vf) What about the difference between Q and W? What is the molar heat capacity? 𝑄 = 𝑚𝑡 𝐶∆𝑇 𝑄 = nM𝐶∆𝑇 M𝐶? ? 𝑄 = 𝑛𝐶𝑃 ∆𝑇 𝑄 = 𝑛𝐶𝑉 ∆𝑇 What the conditions under which Q+? Negative work Positive work Done on the gas Done by the gas 5.3 The First Low of Thermodynamics. What is the first low of thermodynamic? (Pi,Vi) (Pf,Vf) ∆𝑈 = 𝑄 − 𝑊 ∆𝑈 = 𝑄 − 𝑊 = 𝑈𝑓 − 𝑈𝑖 d𝑈 = 𝑑𝑄 − 𝑑𝑊 Is U depends on path and process? What about the units of all quantities? Some application of thermodynamics. Thermodynamics process. (1) Isochoric Process. (i)What about general law & P-V diagram 𝑃𝑖 𝑉 = 𝑁𝐾𝑇𝑖 𝑓 𝑃𝑓 𝑉 = 𝑁𝐾𝑇𝑓 𝑃𝑖 𝑇𝑖 𝑖 = 𝑃𝑓 𝑇𝑓 𝑉𝑖 = 𝑉𝑓 (ii)What about work in the isochoric process? 𝑉 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑑𝑉 = 0 𝑉𝑓 𝑊 = න 𝑃𝑑𝑉 𝑊=0 𝑉𝑖 What does it mean? Heat Capacity of an Ideal Gas at constant V. What is the total Ek for monoatomic gas? (iii)What about 1st law ? ∆𝑈 = ∆𝑄 − ∆𝑊 ∆𝑈 = ∆𝑄 3 3 𝑈 = 𝑁𝐾𝑇 ∆𝑈 = 𝑁𝐾∆𝑇 2 2 (vi)What about molar heat capacity at constant V ? 3 𝑄 = 𝑛𝐶𝑉 ∆𝑇 ∆𝑈 = 𝑁𝐾∆𝑇 2 ∆𝑈 = 𝑄 3 𝑛𝐶𝑉 ∆𝑇 = 𝑁𝐾∆𝑇 2 3 𝑛𝐶𝑉 ∆𝑇 = 𝑛𝑅∆𝑇 2 3 ∴ 𝐶𝑉 = 𝑅 2 (2) Isobaric Process. 𝑃𝑉𝑖 = 𝑁𝐾𝑇𝑖 𝑃𝑉𝑓 = 𝑁𝐾𝑇𝑓 𝑉𝑖 𝑇𝑖 = 𝑉𝑓 𝑇𝑓 What about work? 𝑃 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑉𝑓 i f 𝑊 = න 𝑃𝑑𝑉 𝑉𝑖 i f 𝑉𝑓 𝑊 = 𝑃 𝑉ቚ = 𝑃 𝑉𝑓 − 𝑉𝑖 𝑉𝑖 𝑊 = 𝑛𝑅 𝑇𝑓 − 𝑇𝑖 What about the kinetic energy? ∆𝑈 = 𝑄 − 𝑊 𝑊 = 𝑛𝑅 𝑇𝑓 − 𝑇𝑖 ∆𝑄 = 𝑛𝐶𝑃 ∆𝑇 ∆𝑈 = 𝑛𝐶𝑃 ∆𝑇 − 𝑛𝑅∆𝑇 3 ∆𝑈 = 𝑛𝑅∆𝑇 2 What about molar heat capacity at constant P ? 𝑑𝑄 = 𝑑𝑈 + dW 3 𝑛𝐶𝑃 ∆𝑇 = 𝑛𝑅∆𝑇 + 𝑛𝑅∆𝑇 2 3 5 𝐶𝑃 = 𝑅 + 𝑅 𝐶𝑃 = 𝑅 2 2 Thanks for Your Attention