Problem 76 - University Physics with Modern Physics

Question

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 \(\times 10^{-27} \mathrm{kg} . )\) (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 12.5 of Section \(12.3 ) .\) Use the data in Appendix \(\mathrm{F}\) 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

RMS speed is much lower than escape speed; hydrogen cannot escape the sun significantly.

Step 1: Understanding the RMS Speed Formula

The root mean square (rms) speed of particles in a gas can be found using 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 the particle. In this problem, \( m = 1.67 \times 10^{-27} \text{ kg} \) and \( T = 5800 \text{ K} \).

Step 2: Calculate the RMS Speed of a Hydrogen Atom

Substitute the given values into the rms speed formula:\[ v_{rms} = \sqrt{\frac{3 \times 1.38 \times 10^{-23} \times 5800}{1.67 \times 10^{-27}}} \]Calculate \( v_{rms} \) to find the speed of hydrogen atoms on the sun's surface.

Step 3: Evaluating the Expression for the Escape Speed

The escape speed \( v_{escape} \) is given by \( v_{escape} = \sqrt{\frac{2GM}{R}} \). Using Appendix F, we find the following constants: \( G = 6.674 \times 10^{-11} \text{ m}^3\text{kg}^{-1}\text{s}^{-2} \), \( M = 1.989 \times 10^{30} \text{ kg} \), and \( R = 6.96 \times 10^{8} \text{ m} \).

Step 4: Calculate the Escape Speed from the Sun

Substitute the known values into the escape speed equation:\[ v_{escape} = \sqrt{\frac{2 \times 6.674 \times 10^{-11} \times 1.989 \times 10^{30}}{6.96 \times 10^{8}}} \]Calculate \( v_{escape} \) to determine the speed needed to escape the Sun's gravitational pull.

Step 5: Comparing RMS and Escape Speeds

Compare the calculated rms speed to the escape speed. Determine whether the rms speed is sufficient for hydrogen atoms to escape the sun's gravitational influence, and if not, whether some could still escape due to high-speed tail effects.

Key Concepts

RMS Speed

The concept of root mean square (RMS) speed is vital in understanding how molecules move in a gas. RMS speed gives us an average measure of the speed of particles in a gas at a given temperature. It's derived from the kinetic theory of gases, implying that each particle doesn't travel at the same speed but has a distribution of speeds.
To find RMS 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
• \( m \) is the mass of a single particle

In this context, for hydrogen atoms on the surface of the sun, we substitute the known values. The mass of a hydrogen atom is \( 1.67 \times 10^{-27} \text{ kg} \) and the temperature \( T \) is \( 5800 \text{ K} \). By plugging these into the formula above, we can calculate the RMS speed specific to a hydrogen atom at these conditions.

Escape Velocity

Understanding escape velocity is crucial when discussing how objects or particles can overcome a celestial body's gravitational pull. Escape velocity is the minimum speed needed for an object to "break free" from the gravitational attraction without further propulsion.
It is calculated using the formula:

• \[ v_{escape} = \sqrt{\frac{2GM}{R}} \]

Here:

• \( G \) is the gravitational constant, \( 6.674 \times 10^{-11} \text{ m}^3\text{kg}^{-1}\text{s}^{-2} \)
• \( M \) is the mass of the celestial body
• \( R \) is the radius of the celestial body

The exercise explores the escape velocity from the sun, where:

• \( M = 1.989 \times 10^{30} \text{ kg} \)
• \( R = 6.96 \times 10^{8} \text{ m} \)

By substituting these values into our escape velocity formula, we can compute the speed necessary for particles to leave the Sun's gravitational field. This calculation is essential to understand whether hydrogen atoms could naturally leave the sun.

Hydrodynamics

Hydrodynamics is the study of fluids in motion, fundamentally tied to both physics principles and real-world applications. In thermal physics, particularly when discussing stellar objects like the sun, hydrodynamics helps explain how particles such as gases move and behave.
Thermal and gravitational influences cause gases on the sun to exhibit behaviors examined using hydrodynamic principles. This links closely with the RMS speed and escape velocity concepts. While RMS speed helps us understand the motion of particles at a micro level, hydrodynamics considers the fluid as a whole. It looks at large-scale behaviors impacted by forces, energy transfer, and dynamics.

• Fluid motion is often turbulent and influenced by temperature gradients.
• Pressure variations within the fluid are significant, especially in celestial environments.
• Hydrodynamics tackles how these factors might cause some particles to reach speeds higher than average, potentially following phenomena like solar winds.

By integrating our understanding of RMS speeds and escape velocities with hydrodynamics, we provide a clearer picture of why, despite the high-energy environment of the sun, not all hydrogen atoms can escape its grasp. It’s the exceptional, high-speed hydrogen particles, possibly influenced by such dynamic processes, that may achieve the necessary escape velocity.