2) Assuming an ideal gas EoS, what temperature is required for 2 protons to collide (neglecting quantum tunneling)? Use that r- 2 fm and a) for the velocity, use that nuclei have the rms value from the Maxwell-Boltzmann distribution. b) now, assume that nuclei have instead 10x the rms value. c) How does this value compare to the temperature inside the Sun? What is missing in your calculation?
2) Assuming an ideal gas EoS, what temperature is required for 2 protons to collide (neglecting quantum tunneling)? Use that r- 2 fm and a) for the velocity, use that nuclei have the rms value from the Maxwell-Boltzmann distribution. b) now, assume that nuclei have instead 10x the rms value. c) How does this value compare to the temperature inside the Sun? What is missing in your calculation?
Chemistry
10th Edition
ISBN:9781305957404
Author:Steven S. Zumdahl, Susan A. Zumdahl, Donald J. DeCoste
Publisher:Steven S. Zumdahl, Susan A. Zumdahl, Donald J. DeCoste
Chapter1: Chemical Foundations
Section: Chapter Questions
Problem 1RQ: Define and explain the differences between the following terms. a. law and theory b. theory and...
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attached is the problem and I have also written the formulas that will help you. Thank you!
Transcribed Image Text:**Problem 2:**
Assuming an ideal gas equation of state (EoS), what temperature is required for two protons to collide, neglecting quantum tunneling? Use the following conditions:
- \( r \approx 2 \, \text{fm} \)
**a)** For the velocity, assume that nuclei have the root mean square (rms) value from the Maxwell-Boltzmann distribution.
**b)** Now, assume that nuclei have instead 10 times the rms value.
**c)** How does this value compare to the temperature inside the Sun? What is missing in your calculation?
![**Kinetic Energy: Replacing the Ideal Gas Equation of State in the Massive/Non-Relativistic Pressure Equation**
The image outlines the process of incorporating the kinetic energy into the pressure equation for a massive, non-relativistic gas. Below are the steps and explanations of the formulas presented.
1. **Equation for Pressure**:
\[
n k T = \frac{1}{3} \int_0^\infty n m v^2 \, dv
\]
- This equation relates the pressure in terms of number density (\(n\)), Boltzmann constant (\(k\)), temperature (\(T\)), and velocity distribution of particles.
2. **Integral of Kinetic Energy**:
\[
\frac{1}{n} \int_0^\infty n v v^2 \, dv = \frac{3 k T}{m}
\]
- Integrating over the velocity yields the mean kinetic energy per particle.
3. **Mean Squared Velocity**:
- Derived from the Maxwell-Boltzmann distribution:
\[
\overline{v^2} = \frac{3 k T}{m}
\]
- Alternatively expressed in terms of kinetic energy:
\[
\frac{1}{2} m \overline{v^2} = \frac{3}{2} k T
\]
- Defined as \( v_{\text{rms}}^2 \):
- This represents the square of the root-mean-square velocity (\(v_{\text{rms}}\)) of particles in the gas.
This document provides insight into how kinetic energy can be used to modify the ideal gas equation of state, especially in cases involving non-ideal behavior in gas mechanics.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6214fda8-f992-4e88-b320-5195339361f5%2F45b0cbaf-ff73-4bfc-8038-f7597d96dd23%2Ffrss24x_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Kinetic Energy: Replacing the Ideal Gas Equation of State in the Massive/Non-Relativistic Pressure Equation**
The image outlines the process of incorporating the kinetic energy into the pressure equation for a massive, non-relativistic gas. Below are the steps and explanations of the formulas presented.
1. **Equation for Pressure**:
\[
n k T = \frac{1}{3} \int_0^\infty n m v^2 \, dv
\]
- This equation relates the pressure in terms of number density (\(n\)), Boltzmann constant (\(k\)), temperature (\(T\)), and velocity distribution of particles.
2. **Integral of Kinetic Energy**:
\[
\frac{1}{n} \int_0^\infty n v v^2 \, dv = \frac{3 k T}{m}
\]
- Integrating over the velocity yields the mean kinetic energy per particle.
3. **Mean Squared Velocity**:
- Derived from the Maxwell-Boltzmann distribution:
\[
\overline{v^2} = \frac{3 k T}{m}
\]
- Alternatively expressed in terms of kinetic energy:
\[
\frac{1}{2} m \overline{v^2} = \frac{3}{2} k T
\]
- Defined as \( v_{\text{rms}}^2 \):
- This represents the square of the root-mean-square velocity (\(v_{\text{rms}}\)) of particles in the gas.
This document provides insight into how kinetic energy can be used to modify the ideal gas equation of state, especially in cases involving non-ideal behavior in gas mechanics.
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