1. Calculate both m(v2) and m(v)2 (that is, KE using the average of the square and the square of the average) for an ideal gas of particles with mass m at temperature T that follows Maxwell's speed distribution f(v)= =4T 3/2 (2TKT) ³/² +² exp[-2KT]
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- Problem 2: Helium is a very important element for both industrial and research applications. In its gas form it can be used for welding, and since it has a very low melting point (only 0.95 K under 2.5 MPa) it can be used in liquid form to cool superconducting magnets, such as those found in particle physics experiments. Say we have a cylinder of n = 145 moles of Helium gas at room temperature (T = 20° C). The cylinder has a radius of r = 17.5 cm and a height h = 1.45 m. Part (a) What pressure (in kPa) is the helium gas under? P = ______ Part (b) Helium is usually kept in the highest pressure gas cylinders, which can typically withstand at least 500 atm of pressure. Would the tank in part (a) be able to maintain its structural integrity? Part (c) In principle, these tanks could fail if the temperature of the Helium started rising - if they were stored in a hot environment, for instance. To determine how much of a danger this is, calculate the temperature (in C) the Helium gas…A container holds N molecules of nitrogen gas at T = 280 K. Find the number of molecules with kinetic energies between 0.0300 eV and 0.0312 eV.At what temperature would the rms speed of nitrogen atoms equal the following speeds? (Note: The mass of a nitrogen atom is 2.32 ✕ 10−26 kg.) (a) the escape speed from Earth, 1.12 ✕ 104 m/s K(b) the escape speed from the Moon, 2.37 ✕ 103 m/s K
- A hypothetical speed distribution of gas molecules is defined as follows: P(v) = 0 for 0≤v < vo P(v) = 0.21 for vo ≤ v << 2vo P(v) = 0 for 2v0 < v where P(v) is the probability distribution as a function of speed, v. a) Use the normalisation condition to find the value of v. b) What percentage of the gas molecules has its speed between vo and/vo? c) What percentage of the gas molecules has its speed between 0 and 2 ?Much of the gas near the Sun is atomic hydrogen. Its temperature would have to be 1.50 107 K for the average speed vrms to equal the escape speed of the Sun. What is that speed? The atomic mass of hydrogen is 1.008 10-3 kg/mol.During a physics experiment, helium gas is cooled to a temperature of 17.0 K at a pressure of 5.00×10−2 atm. What is the mean free path in the gas? What is the rms speed of the atoms? What is the average energy per atom?
- In 1995 a research group led by Eric Cornell and Carl Wiemann at the University of Colorado successfully cooled Rubidium atoms to the 20-200 nk temperature range. Assuming (incorrectly) that the Rubidium atoms behave like particles of a classical ideal gas, calculate the RMS speed of a Rubidium atom at a temperature of 36.0 nk. In the experiments one particular isotope of Rubidium was used, Rubidium-87. The molar mass of this isotope is 86.91 g/mol. Submit Answer Tries 0/12C A F D B E VThe most probable speed an atom/molecule in an ideal gas in thermodynamic equilibrium at temperatureT will have is vmp=(2kgT/M)/2. What is the most probable speed in a gas of hydrogen molecules (H2) at 77 °C? m,=1.67x10 27 kg, kg=1.38x10 23 J/K.
- Problem 2: Atmospheric composition A particle near earth's surface traveling faster than about 11km/s has enough kinetic energy to completely escape from the earth, despite earth's gravitational pull. Molecules in the upper atmosphere that are moving faster than this will therefore escape if they do not collide on the way out. a) The temperature of earth’s upper atmosphere is actually quite high, around 1000K. Calculate the probability of a nitrogen molecule at this temperature moving faster than 11km/s, and comment on the result. b) Repeat the calculation for a hydrogen molecule (H2) and for a helium atom, and discuss the implications. c) Escape speed from the moon's surface is only about 2.4km/s. Explain why the moon has no atmosphere.At what temperature would the rms speed of the helium atoms (mass=6.66x10^-27kg) equal its escape speed from the earth which is 1.12x 10^4 m/s? Answer is given as 60509 K but cannot figure out how that could be it.the partition function of an ideal gas of diatomic molecules in an external electric field & is [g(V, T, 8)]" Q(N, V, T, 8) N! where (2mmkT 312 (87 IkT -hv/2kT e q(V,T, 8)= V{ h2 (kT' (µ8 sinh kT) h2 (1 – e-hv/kT) Here I is the moment of inertia of the molecule; v is its fundamental vibrational frequency; and u is its dipole moment. Using this partition function along with the thermodynamic relation, dA = -S dT –p dV – M de where M=Nū, where u is the average dipole moment of a molecule in the direction of the external field &, show that kT] coth kT, Sketch this result versus & from & =0 to & =∞ and interpret it.