Question

What is the average kinetic energy of protons at the center of a star where the temperature is $1.00\cdot {10}^7\mathrm{K}?$ What is the average velocity of those protons?

Short answer

Answer: The average kinetic energy of protons is $2.07\cdot {10}^{-16}\mathrm{J}$, and their average velocity is $4.98\cdot {10}^5\mathrm{m}/\mathrm{s}$.

Step-by-step solution

Calculate the average kinetic energy of protons

Using the Equipartition theorem, the average kinetic energy per particle is given by: $KE=\frac{3}{2}kT$ Given temperature $T=1.00\cdot {10}^7\mathrm{K}$ and Boltzmann constant $k=1.38\cdot {10}^{-23}\mathrm{J}/\mathrm{K}$, we can find the average kinetic energy as: $KE=\frac{3}{2}\times 1.38\cdot {10}^{-23}\mathrm{J}/\mathrm{K}\times 1.00\cdot {10}^7\mathrm{K}=2.07\cdot {10}^{-16}\mathrm{J}$

Calculate the average velocity of protons

Now that we have the average kinetic energy, we can use the kinetic energy formula $\frac{1}{2}m{v}^2$ to find the average velocity of protons: $2.07\cdot {10}^{-16}\mathrm{J}=\frac{1}{2}(1.67\cdot {10}^{-27}\mathrm{kg})(v^2)$ Solving for $v$, we get: $v^2=\frac{2(2.07\cdot {10}^{-16}\mathrm{J})}{1.67\cdot {10}^{-27}\mathrm{kg}}$ $v^2=2.48\cdot {10}^{11}{\mathrm{m}}^2/{\mathrm{s}}^2$ $v=\sqrt{2.48\cdot {10}^{11}{\mathrm{m}}^2/{\mathrm{s}}^2}$ $v=4.98\cdot {10}^5\mathrm{m}/\mathrm{s}$ So, the average velocity of protons at the center of the star is $4.98\cdot {10}^5\mathrm{m}/\mathrm{s}$.

Key concepts

Equipartition Theorem

The Equipartition Theorem is a foundation in statistical mechanics. It tells us how energy is distributed among various degrees of freedom in a system. Simply put, it states that every degree of freedom contributes equally to the total energy. For gases, this translates into kinetic energy being shared among different ways a particle can move, like translational motion.
For any particle, such as a proton, the average kinetic energy due to translational motion can be calculated by the formula:

• $KE=\frac{3}{2}kT$
where:
• $k$ represents the Boltzmann constant, a crucial factor in various thermodynamic calculations.
• $T$ represents the temperature of the system.
The theorem simplifies understanding how temperature affects the kinetic energy of particles, especially in extreme environments like the center of a star.

Protons

Protons are subatomic particles with a significant role in the atomic nucleus. They carry a positive charge and, together with neutrons, form atomic nuclei. Protons are incredibly tiny yet are fundamental to the identity of elements.
In the context of stars, protons are critically important due to their abundance and role in nuclear reactions. The high temperature at a star's center means these protons have significant kinetic energy, allowing for vital interactions and reactions, such as nuclear fusion. Understanding the behavior of protons in these intense conditions helps us comprehend the energy production processes in stars.

Average Velocity

The average velocity of particles, like protons, is closely related to their kinetic energy. From physics, we know that the kinetic energy of a particle

• $KE=\frac{1}{2}m{v}^2$

where:

• $m$ is the mass of the particle, and
• $v$ is its velocity.
This formula helps us compute the average velocity of protons once we have their kinetic energy. After finding the kinetic energy, solving for $v$ gives insight into the speed at which these protons are moving.
In the star's high-temperature environment, protons achieve incredibly high velocities, emphasizing the energetic nature of stellar centers.

Boltzmann Constant

The Boltzmann constant $k$ is a fundamental parameter in statistical mechanics, key to connecting temperature with energy. It facilitates the transition from microscopic properties to macroscopic phenomena, playing a crucial role in thermodynamics.
When calculating kinetic energy, the Boltzmann constant allows us to relate the temperature of an object to the kinetic energy of its constituent particles. Its value is $1.38\times {10}^{-23}\mathrm{J}/\mathrm{K}$. This means for each kelvin of temperature, there's this tiny amount of energy per degree of freedom per particle.
Understanding the Boltzmann constant is fundamental because it builds a bridge between the physical behavior of atoms/molecules and observable thermodynamic behavior.

Star Temperature

Star temperature is a critical factor in determining the kinetic energy of particles within a star. At the core, stars have immense temperatures, often millions of Kelvin, leading to energetic and dynamic environments.
The temperature of a star, especially at its core, drives nuclear fusion, where protons and other particles collide with considerable energy leading to fusion reactions. These reactions are essential as they generate the energy that stars emit as light and heat.
The temperature also affects motion within the star. Higher temperatures mean greater kinetic energy for particles, causing them to move faster. This concept is vital in astrophysics as it helps understand phenomena like energy production, star evolution, and even elemental synthesis. Thus, knowing a star's core temperature allows us to comprehend not just its current state but also its lifecycle and influence in the universe.