President UniversityErwin SitompulThermal Physics 6/1 Lecture 6 Thermal Physics Dr.-Ing. Erwin Sitompul President University

President UniversityErwin SitompulThermal Physics 6/2 The Distribution of Molecular Speeds The root-mean-square speed v rms gives us a general idea of molecular speeds in a gas at a given temperature. But, we often want to know more about how the possible values of speed are distributed among the molecules. The speed distribution for oxygen molecules at room temperature (T = 300 K) is shown below. Chapter 19Kinetic Theory

President UniversityErwin SitompulThermal Physics 6/3 The Distribution of Molecular Speeds Chapter 19Kinetic Theory In 1852, Scottish physicist James Clerk Maxwell first solved the problem if finding the speed distribution of gas molecules. The resulting Maxwell’s speed distribution law can be written as: M : molar mass of the gas R: gas constant T: gas temperature v : molecular speed The quantity P(v) is a probability distribution function. For any speed v, the product P(v)dv (an area, dimensionless) is the fraction of molecules with speed in the interval dv centered on speed v.

President UniversityErwin SitompulThermal Physics 6/4 The Distribution of Molecular Speeds Chapter 19Kinetic Theory As shown in the figure, the fraction of molecules with speeds in the interval dv is equal to the area of a strip with height P(v) and width dv. The total area under the distribution curve corresponds to the fraction of the molecules whose speeds lie between zero and infinity. Since all molecules fall into this category, the value of this total area is unity: The fraction (frac) of molecules with speeds in an interval of v 1 and v 2 is then:

President UniversityErwin SitompulThermal Physics 6/5 The Distribution of Molecular Speeds Chapter 19Kinetic Theory As the speed is a function of temperature, the speed distribution also varies with temperature. The distribution at T = 300 K is compared with the one at T = 80 K in the following figure.

President UniversityErwin SitompulThermal Physics 6/6 Average, RMS, and Most Probable Speeds Chapter 19Kinetic Theory In principle, we can find the average speed v avg of the molecules in a gas by evaluating: After substituting P(v) and performing the integral, Average Speed Similarly, we can find the average of the square of the speeds v 2 avg with After substituting P(v) and performing the integral,

President UniversityErwin SitompulThermal Physics 6/7 Average, RMS, and Most Probable Speeds Chapter 19Kinetic Theory Thus, RMS Speed The most probable speed v P is the speed at which P(v) is maximum. To calculate v P, we set dP/dv = 0 and then solve for v. Doing so, we find: Most Probable Speed A molecule is more likely to have speed v P than any other speed, but some molecules will have speeds that are many times v P. These molecules lie in the high-speed tail of a distribution curve.

President UniversityErwin SitompulThermal Physics 6/8 Problem A container is filled with oxygen gas maintained at room temperature (300 K). What fraction of the molecules have speeds in the interval 599 to 601 m/s? The molar mass M of oxygen is kg/mol. Chapter 19Kinetic Theory The interval Δv = 2 m/s is very small compared to the center speed v = 600 m/s. Thus the integration can be approximated through:

President UniversityErwin SitompulThermal Physics 6/9 The Molar Specific Heats of an Ideal Gas Chapter 19Kinetic Theory In this section, we want to derive an expression for the internal energy E int of an ideal gas from molecular consideration. In other words, we want an expression for the energy associated with the random motions of the atoms or molecules in the gas. We shall then use that expression to derive the molar specific heats of an ideal gas.

President UniversityErwin SitompulThermal Physics 6/10 The Molar Specific Heats of an Ideal Gas Chapter 19Kinetic Theory Internal Energy E int Let us firs assume that the ideal gas is a monatomic gas such as helium, neon, or argon. The internal energy E int of the ideal gas is simply the sum of the translational kinetic energies of its atoms. The average translational kinetic energy of a single atom depends only on the gas temperature and is given as K avg =3/2·kT. A sample of n moles of such a gas contains nN A atoms. The internal energy of the sample is then: Since k = R/N A, we can rewrite this as: Monatomic Ideal Gas The internal energy E int of an ideal gas is a function of the gas temperature only.

President UniversityErwin SitompulThermal Physics 6/11 The Molar Specific Heats of an Ideal Gas Chapter 19Kinetic Theory Molar Specific Heat at Constant Volume The figure on the right shows n moles of an ideal gas at pressure p and temperature T, confined to a cylinder of fixed volume V. This initial state is denoted as i. Suppose a small amount of energy to the gas as heat Q is added to the gas. The gas temperature rises a small amount T+ΔT, and its pressure rises to p+Δp. This final state is denoted as f. We would find that the heat Q is related to the temperature change ΔT by: Constant Volume C V is a constant called the molar specific heat at constant volume [J/mol·K].

President UniversityErwin SitompulThermal Physics 6/12 The Molar Specific Heats of an Ideal Gas Chapter 19Kinetic Theory The first law of thermodynamics can now be written as: Monatomic Gas With the volume held constant, the gas cannot do any work, W = 0. This yields: Thus, We can now generalize the equation for the internal energy as: Any Ideal Gas

President UniversityErwin SitompulThermal Physics 6/13 The Molar Specific Heats of an Ideal Gas Chapter 19Kinetic Theory When an ideal gas that is confined to a container undergoes a temperature change ΔT, then we can write the resulting change in its internal energy as: Any Ideal Gas, Any Process The change in the internal energy E int of a confined ideal gas depends on the change in the gas temperature only. It does not depend on what type of process produces the change in the temperature.

President UniversityErwin SitompulThermal Physics 6/14 The Molar Specific Heats of an Ideal Gas Chapter 19Kinetic Theory Molar Specific Heat at Constant Pressure We now assume that the temperature of our ideal gas is increased by the same small amount ΔT as previously but now the necessary energy (heat Q) is added with the gas under constant pressure. From such experiments we find that the heat Q is related to the temperature change ΔT by: Constant Pressure C P is a constant called the molar specific heat at constant pressure [J/mol·K]. C P > C V, because energy must now be supplied not only to raise the temperature of the gas but also for the gas to do work.

President UniversityErwin SitompulThermal Physics 6/15 The Molar Specific Heats of an Ideal Gas Chapter 19Kinetic Theory To relate the molar specific heats C P and C V, we start with the first law of thermodynamics: We will find that:

President UniversityErwin SitompulThermal Physics 6/16 The Molar Specific Heats of an Ideal Gas Chapter 19Kinetic Theory

President UniversityErwin SitompulThermal Physics 6/17 Checkpoint The figure here shows five paths traversed by a gas on a p-V diagram. Rank the paths according to the change in internal energy of the gas, greatest first. 5, then tie of 1, 2, 3, and 4. Chapter 19Kinetic Theory

President UniversityErwin SitompulThermal Physics 6/18 Problem A bubble of 5.00 mol of helium is submerged at a certain depth in liquid water when the water (and thus the helium) undergoes a temperature increase ΔT of 20.0°C at constant pressure. As a result, the bubble expands. The helium is monatomic an ideal. (a)How much energy is added to the helium as heat during the increase and expansion? Chapter 19Kinetic Theory (b)What is the change ΔE int in the internal energy of the helium during the temperature increase? (c)How much work W is done by the helium as it expands against the pressure of the surrounding water during the temperature increase?

President UniversityErwin SitompulThermal Physics 6/19 Class Group Assignments 1.The rms velocity of oxygen molecules at 27°C is 318 m/s. Thus, the rms velocity of hydrogen molecules at 127 °C is: (a) 1470 m/s(b) 1603 m/s (c) 1869 m/s(d) 2240 m/s(e) 3211 m/s 2.An ideal gas of N monoatomic molecules is in thermal equilibrium with an ideal gas of the same number of diatomic molecules and equilibrium is maintained as the temperature is increased. The ratio of the changes in the internal energies ΔE dia /ΔE mon is: (a) 1/2 (b) 3/5(c) 1/1(d) 5/3(e) 2/1 3.An ideal gas has molar specific heat C P at constant pressure. When the temperature of n moles is increased by ΔT the increase in the internal energy is: (a) nC P ΔT (d) n(C P +R)ΔT (b) n(C P –R)ΔT (e) n(2C P +R)ΔT (c) n(2C P –R)ΔT Chapter 19Kinetic Theory

President UniversityErwin SitompulThermal Physics 6/20 No Homework This Week Chapter 19Kinetic Theory Prepare well for the midterm exam.