The solar system was born about 4.6 billion years ago out of an interstellar gas cloud, which is mos — Analytical Chemistry Chemistry Question
Atmosphere of the planets
The solar system was born about 4.6 billion years ago out of an interstellar gas cloud, which is mostly hydrogen and helium with small amounts of other gases and dust.
The age of the solar system can be estimated by determining the mass ratio between Pb–206 and U–238 in lunar rocks. Write the overall nuclear reaction for the decay of U–238 into Pb–206.
Model Answer
238 92U → 206 82Pb + 8 4 2He + 6 0 -1e
The half–life for the overall reaction is governed by the first alpha–decay of U–238 ( 238 92U → 234 90 Th + 4 2 He), which is the slowest of all reactions involved. The half–life for this reaction is 4.51×109 years. Estimate the mass ratio of Pb–206 and U–238 in lunar rocks that led to the estimation of the age of the solar system.
Model Answer
After almost one half–life, the molar ratio between 206Pb and 238U is 1.
Mass ratio: m(206Pb) / m(238U) = 206 / 238 = 0.87
Elemental hydrogen and helium are rare on Earth, because they escaped from the early Earth. Escape velocity is the minimum velocity of a particle or object (e.g., a gas molecule or a rocket) needed to become free from the gravitational attraction of a planet. Escape velocity of an object with mass m from the Earth can be determined by equating the gravitational potential energy, –G M m / R, to the kinetic energy, (1/2) m v2, of the object. Note that the m’s on both sides cancel and, therefore, the escape velocity is independent of the mass of the object. However, it still depends on the mass of the planet.
The universal constant of gravitation G = 6.67×10–11 N m2 kg–2
The Earth’s mass M = 5.98×1024 kg
The Earth's radius R = 6.37×106 m
Calculate the escape velocity for the Earth.
Model Answer
(1/2) m ve^2 = G M m / R
ve^2 = 2 G M / R = (2 × 6.67×10–11 N m2 kg–2 × 5.98×1024 kg) / (6.37×106 m)
ve = 1.12×104 m s–1
Calculate the average speed, (8RT / πM)1/2, of a hydrogen atom and a nitrogen molecule at ambient temperature. Compare these with the escape velocity for the Earth. Note that the temperature of the upper atmosphere where gases can escape into space will be somewhat different. Also note that photolysis of water vapor by ultraviolet radiation can yield hydrogen atoms. Explain why hydrogen atoms escape more readily than nitrogen molecules even though the escape velocity is independent of the mass of the escaping object.
Model Answer
Hydrogen atom: v = ((8 × 8.3145 kg m2 s–2 mol–1 K–1 × 298 K) / (3.14 × 1.008×10–3 kg mol–1))^1/2 = 2500 m s–1 (22 % of the escape velocity)
Nitrogen molecule: 2500 m s–1 × (1/28)^1/2 = 470 m s–1 (4 % of the escape velocity)
The fraction with speed exceeding the escape velocity is much greater for hydrogen atoms than for nitrogen molecules.
The chemical composition of the atmosphere of a planet depends on the temperature of the planet’s atmosphere (which in turn depends on the distance from the sun, internal temperature, etc.), tectonic activity, and the existence of life. As the sun generated heat, light, and solar wind through nuclear fusion of hydrogen to helium, the primitive inner planets (Mercury, Venus, Earth, and Mars) lost most of their gaseous matter (hydrogen, helium, methane, nitrogen, water, carbon monoxide, etc.). As the heavy elements such as iron and nickel were concentrated at the core through gravity and radioactive decay produced heat, internal temperature of the planets increased. Trapped gases, such as carbon dioxide and water, then migrated to the surface. The subsequent escape of gases from the planet with a given escape velocity into space depends on the speed distribution. The greater the proportion of gas molecules with speed exceeding the escape velocity, the more likely the gas is to escape over time.
Circle the planet name where the atmospheric pressure and composition are consistent with the given data. Explain.
The average surface temperatures and the radii of the planets are as follows:
Venus: 730 K; 6 052 km Earth: 288 K; 6 378 km Mars: 218 K; 3 393 km
Jupiter: 165 K; 71,400 km Pluto: 42 K; 1,160 km
pressure (in atm) composition (%) planet
a. > 100 H2(82); He(17) (Venus, Earth, Mars, Jupiter, Pluto)
b. 90 CO2(96.4); N2(3.4) (Venus, Earth, Mars, Jupiter, Pluto)
c. 0.007 CO2(95.7); N2(2.7) (Venus, Earth, Mars, Jupiter, Pluto)
d. 1 N2(78); O2(21) (Venus, Earth, Mars, Jupiter, Pluto)
e. 1. 10–5 CH4(100) (Venus, Earth, Mars, Jupiter, Pluto)
Model Answer
a. Jupiter: large mass, low temperature, H/He retained at high pressure
b. Venus: lost light elements, rich in carbon dioxide, high pressure
c. Mars: small mass, rich in carbon dioxide, low pressure
d. Earth: lost light elements, carbon dioxide converted to oxygen through photo–synthesis
e. Pluto: very small mass, lost light elements, very low atmospheric pressure
Write the Lewis structures for H2, He, CO2, N2, O2, and CH4. Depict all valence electrons.
All of the above atmospheric components of the planets are atoms and molecules with low boiling point. Boiling point is primarily determined by the overall polarity of the molecule, which is determined by bond polarity and molecular geometry. Non–polar molecules interact with dispersion force only and, therefore, have low boiling points. Yet there are differences in boiling points among nonpolar molecules.
Arrange H2, He, N2, O2, and CH4 in the order of increasing boiling point. Explain the order.
Model Answer
He (4 K) < H2 (20 K) < N2 (77 K) < O2 (90 K) < CH4 (112 K)
Dispersion force is greater for larger molecules.
Nitrogen with the triple bond has a smaller bond length than oxygen.
Nitrogen also has less lone pair electrons to be involved in dispersion.