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Thermodynamics I AP PHYSICS RECAP Thermal Physics Equations not on the equation sheet Q m cT c specific heat, units: J/(kg·K) Q mL L Latent Heat, units: J/kg The Mole Quantity in Physics mass – quantitative measure of object’s inertia mole – number of particles Mole & Mass Every substance has a unique relationship between its mass and number of moles Molar Mass (M) the ratio of the mass of a substance in grams to the number of moles of the substance How do you determine Molar Mass? the mass of 1 mole of a substance equals the atomic mass of the substance in units of grams rather than atomic mass units Ex. What is the molar mass of O2? Methane What is the molar mass (M) of CH4? What number of moles (n) are there in 40 g of methane gas? How many molecules (N) of CH4 does this include? What is the mass of 24.08 x 1023 molecules of ethanol (C2H5OH)? Note: n number of moles N number of particles Ideal Gases: Volume and Number The behaviors of ideal gases at low pressures are relatively easy to describe: The volume V is proportional to the number of moles n and thus to the number of molecules (this concept stems from Avogadro’s Law) Ideal Gases: Boyle’s Law Robert Boyle (1627 – 1691) Irish physicist and chemist who employed Robert Hooke as an assistant (you know the Hooke’s law guy and the “cell” guy) Boyle’s Law The volume V varies inversely with the pressure P when temperature (T) and amount of gas (n) are constant. Ideal Gases: Charles’ Law Jacques Charles (1746 – 1823) French inventor, physicist and hot air balloonist. Charles’ Law The pressure P is directly proportional to the absolute temperature T (temperature in Kelvin) when volume V and amount n are constant Ideal Gas Law Combining of Boyle’s Law and Charles’ Law Adding Avogadro’s Law yields: R is the ideal gas constant or R = 0.08206 L∙atm/(mol∙K) Gas at STP The condition called standard temperature and pressure (STP) for a gas is defined to be a temperature of 0oC and a pressure of 1 atm (1.013 x 105 Pa). If you want to keep 1 mole of an ideal gas in your room at STP, how big is the Tupperware that you need? [Answer in units of liters, 1 m3 = 1000 L] Kinetic Theory of an Ideal Gas Assumptions of the Kinetic-Molecular Model 1. 2. 3. 4. A container with volume V contains a very large number N of identical molecules, each with mass m. The container has perfectly rigid walls that do not move. The molecules behave as point particles; their size is small in comparison to the average distance between particles and to the dimensions of the container. The molecules are in constant random motion; they obey Newton’s laws. Each molecule occasionally makes a perfectly elastic collision with a wall of the container. During collisions, the molecules exert forces on the walls of the container; these forces create the pressure that the gas exerts. Kinetic Theory of an Ideal Gas For an ideal gas, the average kinetic energy Kavg per molecule is proportional to the absolute (Kelvin) temperature T. The ratio R/No occurs frequently in molecular theory and is known as the Boltzmann constant kB. What is the value of the Boltzmann constant including units?? Ludwig Boltzmann (1844 – 1906) was an Austrian physicist famous for his founding contributions in the fields of statistical mechanics and statistical thermodynamics. Molecular Speeds in an Ideal Gas If molecules have an average kinetic energy Kavg given by the equation: Kavg 32 kBT then what is their average speed??? Five Molecules Five ideal-gas molecules chosen at random are found to have speeds of 500, 600, 700, 800, and 900 m/s, respectively. Find the rms speed for this collection. Is it the same as the average speed of these molecules? The Boltzmann Constant Gets Around Starting with the typical ideal gas law (PV = nRT ) derive an expression for the gas law that includes both the Boltzmann Constant kB and the number of molecules of an ideal gas N. Kinetic Energy of A Molecule What is the average (translational) kinetic energy of a molecule of oxygen (O2) at a temperature of 27oC, assuming that oxygen can be treated as an ideal gas? b) What is the total (translational) kinetic energy of the molecules in 1 mole of oxygen at this temperature? c) Compare the root-mean-square speeds of oxygen and nitrogen molecules at this temperature (assuming that they can be treated as ideal gases). [mass of oxygen molecule mO2 = 5.31 x 10-26 kg, mass of nitrogen molecule mN2 = 4.65 x 10-26 kg] a) Internal Energy DWT Particles in a system (like an ideal gas) are in motion. Therefore: 1. the particles have some velocity 2. the particles have some kinetic energy 3. the system as a whole has some internal energy as a result of the individual particles’ kinetic energy This is true of material in any phase (solid, liguid, gas, plasma). Internal Energy & Temperature Kinetic Energy of the individual particles The system as a whole has some internal energy as a result of the individual particles’ kinetic energy Kinetic energy of all the particles First Law of Thermodynamics DWT Thermodynamics is the study of energy relationships that involve heat, mechanical work, and other aspects of energy and energy transfer. There are two ways to transfer energy to an object: 1. Heat the object 2. do Work on the object Both of these energy transfer methods add to the internal energy of the object. The First Law of Thermodynamics (1LT) Work Done during Volume Change Classic Thermodynamic System Gas in a cylinder confined by a piston. PV Diagrams Pressure Area under the curve is the work done by the gas. P Notice the arrow denoting direction of the process. Work Vo Vf Volume Thermodynamic Processes Four Processes 1. Isothermal 2. Isobaric 3. Isochoric (Isovolumetric) 4. Adiabatic Isothermal Pressure The curve represents pressure as a function of volume for an ideal gas at a single temperature. The curve is called an isotherm. Pa For the curve, PV is constant and is directly proportional to T (Boyle’s Law). Pb Va For Ideal Gases: Vb Volume Isobaric Pressure The curve is called an isobar. Ta The pressure of the system (system - constant amount of gas, n) changes as a result of heat being transferred either into or out of the system and/or work done on or by the system. Tb P Work Va Vb Volume isotherms Ta > T b Isochoric (Isometric or Isovolumetric) Pressure The curve is called an isochor. Pb There is no work done in this process. All of the energy added/subtracted as heat changes the internal energy. Pa isotherms V Volume Adiabatic Pressure The curve is called an adiabat. No heat is transferred into or out of the system. (An adiabatic curve Pb at any point is always steeper than the isotherm passing through the same point.) Pa Work Va isotherms Vb Volume Isobaric Example Thermodynamics of Boiling Water One gram of water (1 cm3) becomes 1671 cm3 of steam when boiled at a constant pressure of 1 atm (1.013 x 105 Pa). The latent heat of vaporization at this pressure is Lv = 2.256 x 106 J/kg. Compute: a) the work done by the water when it vaporizes. b) its increase in internal energy. Isochoric (Isovolumetric) Example Heating Water Water with a mass of 2.0 kg is held at constant volume in a container while 10,000 J of heat is slowly added by a flame. The container is not well insulated, and as a result 2,000 J of heat leaks out to the surroundings. a) What is the increase in internal energy? b) What is the increase in temperature? [the specific heat of water is 4186 J/kg∙oC] Molar Heat Capacities The amount of heat Q needed for a certain temperature change ΔT is proportional to the temperature change and to the number of moles n of the substance being heated; Q = nCΔT where C is a quantity, different for different materials, called the molar heat capacity of the material. Units of C: J/(mol∙K) Relation of specific heat (c) to molar heat capacity (C) is the molar mass C = Mc Because of 1LT the molar heat capacities are not the same for different thermodynamic processes… Molar Heat Capacities & 1LT The molar heat capacities are different for isochoric (constant volume) and isobaric (constant pressure) processes. 1LT ΔU = Q+W Q = ΔU – W Isochoric Isobaric W = -PΔV = 0 For pressure to remain constant the volume must change Q = ΔU – o W = -PΔV Q = ΔU All of the heat gained/lost results directly in a change in internal energy. Molar heat for a constant volume, Cv Q = ΔU + PΔV Some of the heat gained by the system is converted into work as the system expands. Molar heat for a constant pressure, Cp Relationship between Cv and Cp: Cp = Cv + R R universal gas constant