Solution The rms speed of hydrogen gas is 3RT√23×8.315×300 12=193×10°(m/s) 2×10 The rms speed of hydrogen gas is less than the escape speed from the Earth(11.2 km/s). Since hydrogen is the least massive gas, hydrogen molecules have the highest rms speeds at a given temperature. Hydrogen have highest escape rate in the atmosphere of the Earth, so there is very few hydrogen gas in the atmosphere of the Earth 5. The average kinetic energy of a particle in a gas at temperature T is given by Equation 14.13 KE=二kT.or kT. The special theory of relativity states that there is an upper limit on the speed of an particle: the speed of light c=3.00 x 10 m/s. For a gas of hydrogen atoms, the immediately preceding equation implies an upper limit on the temperature. The absolute temperature such that(v") for a gas of hydrogen atoms is equal to the square of the speed of light is 4.35x10 K In fact, there is no upper limit on the temperature, so the classical expression for the kinetic energy cannot be valid for speeds approaching the speed of light Solution According to the problem (v*)=c,so T 2×10-3×(3×103)2 3×1.38×10-23=4.35×1036(K) I. Give the solutions of the following problems Consider helium gas at temperature 300 K near the surface of the earth (a) Calculate the average kinetic energy of one of the helium atoms (b) Calculate the gravitational potential energy of a single helium atom near the surface of the Earth Choose the zero of gravitational potential energy to be infinitely far away from the earth (c)What is the absolute value of the ratio of the gravitational potential energy of the helium atom to its average kinetic energy? Is it justifiable to neglect the gravitational potential energy in theory? Why or why not (a) The average kinetic energy of one of the helium atoms KE 38×10-2×300=6.21×1021(J) (b) The gravitational potential energy of a single helium atom near the surface of the Earth is 981×6374×10°=-416×10-(J) 6.02×10Solution: The rms speed of hydrogen gas is ) 1.93 10 (m/s) 2 10 3 8.315 300 ) ( 3 ( 1/ 2 3 3 1/ 2 = × × × × = = − M RT vrms The rms speed of hydrogen gas is less than the escape speed from the Earth (11.2 km/s). Since hydrogen is the least massive gas, hydrogen molecules have the highest rms speeds at a given temperature. Hydrogen have highest escape rate in the atmosphere of the Earth, so there is very few hydrogen gas in the atmosphere of the Earth. 5. The average kinetic energy of a particle in a gas at temperature T is given by Equation 14.13: KEave kT 2 3 = , or m v kT 2 3 2 1 2 = . The special theory of relativity states that there is an upper limit on the speed of an particle: the speed of light c = 3.00 × 108 m/s. For a gas of hydrogen atoms, the immediately preceding equation implies an upper limit on the temperature. The absolute temperature such that 〈v 2 〉 for a gas of hydrogen atoms is equal to the square of the speed of light is 4.35×1036 K . In fact, there is no upper limit on the temperature; so the classical expression for the kinetic energy cannot be valid for speeds approaching the speed of light. Solution: According to the problem 〈v 2 〉 = c 2 , so 4.35 10 (K) 3 1.38 10 2 10 (3 10 ) 3 3 36 23 2 3 8 2 2 = × × × × × × = = = − − k mc k m v T III. Give the Solutions of the Following Problems 1. Consider helium gas at temperature 300 K near the surface of the Earth. (a) Calculate the average kinetic energy of one of the helium atoms. (b) Calculate the gravitational potential energy of a single helium atom near the surface of the Earth. Choose the zero of gravitational potential energy to be infinitely far away from the Earth. (c) What is the absolute value of the ratio of the gravitational potential energy of the helium atom to its average kinetic energy? Is it justifiable to neglect the gravitational potential energy in theory? Why or why not? Solution: (a) The average kinetic energy of one of the helium atoms is 1.38 10 300 6.21 10 (J) 2 3 2 3 −23 −21 KEave = kT = × × × = × (b) The gravitational potential energy of a single helium atom near the surface of the Earth is 9.81 6.374 10 4.16 10 (J) 6.02 10 4 10 6 19 23 3 − − × × × = − × × × PEgrav = −mgREarth = −