2. Show that 1 1/2 (ɛ — eq)²4(r) = [p² — 2e(A · p) + e²A² + M²c²]y(r) · c²
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![The Klein-Gordon equation for a particle in external electromagnetic field is
2
ih d
Let
C at
e
—
C
p)
y = [(p - eA)² + M²c²]y
(r, t) = (r)e-iet/h](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3a11ce3d-21c0-472b-9212-b4fe5760ed67%2F272a4cf1-9dec-41fc-9e9f-3ee228cc8ff8%2F711jgry_processed.png&w=3840&q=75)
![2. Show that
1
(ɛ — eq)²4(r) = [p² – 2e(A · p) + e²A² + M²c²]y(r)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3a11ce3d-21c0-472b-9212-b4fe5760ed67%2F272a4cf1-9dec-41fc-9e9f-3ee228cc8ff8%2F6embvs9_processed.png&w=3840&q=75)

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- Find the ratio f(vp)/f(vrms) for hydrogen gas (M = 2.02 g/mol) at a temperature of 77.0 K?The density of gray and white tin are 5.77×103 kg/m3 and 7.37×103 kg/m3. The latent heat for the transition between gray and white tin is 2.20×103 J/mol. The atomic weight of tin is 118.7. Calculate the slope of the tangent of the coexistence curve of gray and white tin at T=376 K. Select one: a.dP/dT=−14.2 atm/K b.dP/dT=−12.9 atm/K c.dP/dT=−14.6 Pa/K d.dP/dT=−12.3 Pa/KProblem 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) What is the average kinetic energy in joules of a hydrogen atom on the 5500 °C surface of the Sun? The Boltzmann's constant is k =1.38×10¯2³ J/K KE av . 3.99 x10-21 J (b) What is the average kinetic energy of a helium atom in a region of the solar corona where the temperature is 6 x10³°K? KE av . 3.27 x10-18 ! No, that's not the correct answer.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.The volume of an automobile tire is 2.5 × 10−2m3 . the pressure of the air in the tire is 3 atm and the temperature is 37C ◦ . what is is the mass of air in grams ? The mean molecular mass of air is 29g. 1 atm = 1.01 × 105 Pa ; Calculate to 2 decimals.
- 2. For T = 300 K, calculate the pressure (in bars) at which the mean free path of a hydrogen molecule will be each of the lengths given here. For H₂, o = 2.30 x 10-1⁹ m². (a) 100 μm (b) 1.00 mm (c) 1.00 m LLthe 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.2 Thermo Physics The ground state of Cyanogen gas (Energy = 0 eV) and 1st excited state is (E = 4.7x10^-4eV) can be separately identified by their spectra. An astronomer determines that on interstellar gas has a ratio of 10:1. 10 molecules of cyanogen gas to every 1 excited state. Use this to determine the temperature of the gas?
- One mole of hydrogen has a mass of 2 g. Calculate the mean speed of the hydrogen molecules at T = 300 K using the Maxwell- Boltzmann distribution. Select one: O a. 1579 m/s O b. 1782 m/s O c. O d. 459 m/s 1934 m/sPlease asapWhat is the average thermal velocity of a hydrogen molecule H at 27 \deg C? (Assume: Boltzmann constant = 1.38 x 10-23 J/K and the mass of H is m = 3.32 x 10-27 kg)