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Chapter 13: Problem 71

If the upper atmosphere of Jupiter has a temperature of \(160 \mathrm{K}\) and the escape speed is \(60 \mathrm{km} / \mathrm{s}\), would an astronaut expect to find much hydrogen there?

Short Answer

Expert verified
Based on the calculations, the root mean square speed of hydrogen molecules in Jupiter's upper atmosphere is 1.77 x 10^3 m/s, which is significantly lower than the escape speed of 60,000 m/s. Therefore, hydrogen can be found in the upper atmosphere of Jupiter.

Step by step solution

01

Identify the given information

We are given the following information: - Temperature of Jupiter's upper atmosphere: \(T = 160 \mathrm{K}\) - Escape speed: \(v_\text{escape} = 60 \mathrm{km} / \mathrm{s} = 60,000 \mathrm{m} / \mathrm{s}\) - We need to determine if the hydrogen (H\(_2\)) can be found in the upper atmosphere.
02

Convert temperature to energy

First, we need to find the average kinetic energy of hydrogen molecules in the atmosphere. We will use the following equation: $$\frac{3}{2} k_\mathrm{B} T = \frac{1}{2} m_\mathrm{H_2} v_\mathrm{rms}^{2},$$ where \(k_\mathrm{B}\) is the Boltzmann constant (\(1.38 \times 10^{-23} \mathrm{J} / \mathrm{K}\)), \(T\) is the temperature of the atmosphere, \(m_\mathrm{H_2}\) is the mass of a hydrogen molecule, and \(v_\text{rms}\) is the root mean square speed of the hydrogen molecules.
03

Calculate the mass of a hydrogen molecule

For H\(_2\), we need to find the molar mass using the atomic mass of hydrogen (approximately \(1 \mathrm{amu}\)): $$m_\mathrm{H_2} = 2 \times \frac{1 \mathrm{g}}{\text{mol}} \times \frac{1 \text{kg}}{1000 \mathrm{g}} \times \frac{1 \text{mol}}{6.022 \times 10^{23} \text{particles}},$$ which simplifies to: $$m_\mathrm{H_2} = 3.32 \times 10^{-27} \text{kg}$$.
04

Calculate the root mean square speed

Now, we use the temperature-energy equation to solve for \(v_\text{rms}\): $$v_\mathrm{rms} = \sqrt{\frac{3 k_\mathrm{B} T}{m_\mathrm{H_2}}}.$$ Plug in the values: $$v_\mathrm{rms} = \sqrt{\frac{3 \times 1.38 \times 10^{-23} \mathrm{J / K} \times 160 \mathrm{K}}{3.32 \times 10^{-27} \text{kg}}} = 1.77\times 10^3 \mathrm{m} / \mathrm{s}.$$
05

Compare root mean square speed with escape speed

Now we need to compare the calculated root mean square speed with the escape speed of Jupiter. If the root mean square speed is significantly lower than the escape speed, hydrogen will not be able to escape Jupiter's atmosphere: $$v_\mathrm{rms} = 1.77 \times 10^3 \mathrm{m} / \mathrm{s} \quad \text{ vs.} \quad v_\text{escape} = 60,000 \mathrm{m} / \mathrm{s}.$$ Given that the root mean square speed of hydrogen is far lower than the escape speed, we can expect to find hydrogen molecules in the upper atmosphere of Jupiter.

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Most popular questions from this chapter

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