Problem 76 - University Physics with Modern Physics

Question

Hydrogen on the Sun. The surface of the sun has a temperature of about 5800 \(\mathrm{K}\) and consists largely of hydrogen atoms. (a) Find the rms speed of a hydrogen atom at this temperature. (The mass of a single hydrogen atom is \(1.67 \times 10^{-27}\) kg. (b) The escape speed for a particle to leave the gravitational influence of the sun is given by \((2 G M / R)^{1 / 2},\) where \(M\) is the sun's mass, \(R\) its radius, and \(G\) the gravitational constant \((\) see Example 13.5 of Section 13.3\() .\) Use the data in Appendix \(F\) to calculate this escape speed. (c) Can appreciable quantities of hydrogen escape from the sun? Can any hydrogen escape? Explain.

Step-by-Step Solution

Answer

The RMS speed of a hydrogen atom is about 11800 m/s.

Step 1: Calculate RMS Speed of Hydrogen

To find the root mean square speed, we use the formula \( v_{rms} = \sqrt{\frac{3kT}{m}} \), where \( k \) is the Boltzmann constant \(1.38 \times 10^{-23} \ \text{J/K}\), \( T \) is the temperature in Kelvin, and \( m \) is the mass of a hydrogen atom. Substituting the values:\[ v_{rms} = \sqrt{\frac{3 \times 1.38 \times 10^{-23} \times 5800}{1.67 \times 10^{-27}}} \]Calculating this gives:\[ v_{rms} \approx \sqrt{\frac{2.39724 \times 10^{-19}}{1.67 \times 10^{-27}}} \approx \sqrt{1.435385 \times 10^8} \approx 11800 \ \text{m/s} \].

Key Concepts

Kinetic Theory of Gases

The kinetic theory of gases offers an insightful perspective on understanding gas molecules' behavior. This theory suggests that gases are composed of many small particles, such as atoms or molecules, constantly moving in random directions.
Imagine these particles like tiny, fast-moving dots bouncing around in a jar. Their speed and movement are influenced by temperature. As the temperature rises, these particles move faster.
In our sun's example, the surface temperature is about 5800 K. Using the formula for root mean square (RMS) speed, \( v_{rms} = \sqrt{\frac{3kT}{m}} \), we calculate how quickly a hydrogen atom moves at this temperature.

• \( k \) stands for the Boltzmann constant, a fundamental figure in physics that helps relate energy at the molecular scale to temperature.
• \( T \) is the temperature, in this context, 5800 K, representing the sun's surface temperature.
• \( m \) is the mass of a hydrogen atom, approximately \( 1.67 \times 10^{-27} \) kg.

Plugging these into the formula allows us to determine the hydrogen atom's RMS speed, resulting in an impressive \( 11800 \ m/s \).
This rapidly moving atom isn’t just part of the solar surface’s relentless dynamism—each molecule's movement is a tiny piece of the broader energetic puzzle of our sun.

Solar Physics

Solar physics delves into the study of the sun, our closest stellar neighbor and a crucial component of our solar system. The sun operates not only as a source of light and heat but also affects our entire solar system's environment.
Its surface temperature of about 5800 K means that even ordinary hydrogen, the simplest and most prevalent element in the universe, becomes extremely energetic.

• The sun's energy output, fundamentally powered by nuclear fusion reactions in its core, radiates outward, influencing the solar atmosphere where these hydrogen atoms exist.
• The extreme conditions on the sun are why many familiar earth-based concepts, like atmospheric escape, require adjusted understandings.

In these searing conditions, hydrogen atoms don't merely abide by earth-like behaviors. They become part of processes far beyond typical earthly physics, like high-energy solar winds and other complex solar phenomena.
Understanding solar physics enables scientists to unravel mysteries about solar activity like solar flares, sunspots, and the sun's magnetic field—all essential for predicting and understanding space weather.

Gravitational Escape Velocity

Gravitational escape velocity is a critical concept when considering whether particles, like hydrogen atoms on the sun, can escape the gravitational pull. This velocity refers to the minimum speed an object must reach to break free from the gravitational attraction of a celestial body without further propulsion.
For the sun, the formula to find this escape speed is \((2 G M / R)^{1 / 2},\) where

• \( G \) is the gravitational constant, offering a measure of the strength of gravity.
• \( M \) represents the mass of the sun, an immense figure given the sun's vastness.
• \( R \) is the sun's radius.

Calculating this shows that hydrogen atoms on the sun's surface, traveling at their RMS speed of \( 11800 \ m/s \), do not reach the solar escape velocity, which is significantly higher.
Consequently, single hydrogen atoms don’t generally escape the sun simply due to their speed. However, solar phenomena like solar winds consist of particles accelerated by other forces, giving them an extra push off the sun’s surface into the solar system.
This distinction is crucial in explaining why, despite not having the escape speed on their own, hydrogen atoms can still move away from the sun influenced by external forces.