At what temperature would the rms speed of hydrogen atoms equal the following speeds? (Note: The mass of a hydrogen atom is 1.66 x 10-27 kg.) (a) the escape speed from Earth, 1.12 x 104 m/s K (b) the escape speed from the Moon, 2.37 x 10³ m/s K
At what temperature would the rms speed of hydrogen atoms equal the following speeds? (Note: The mass of a hydrogen atom is 1.66 x 10-27 kg.) (a) the escape speed from Earth, 1.12 x 104 m/s K (b) the escape speed from the Moon, 2.37 x 10³ m/s K
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Question
![**Problem Description**
At what temperature would the rms (root mean square) speed of hydrogen atoms equal the following speeds? (Note: The mass of a hydrogen atom is \(1.66 \times 10^{-27}\) kg.)
**Questions**
(a) The escape speed from Earth, \(1.12 \times 10^4\) m/s
\[ \boxed{\phantom{K}} \] K
(b) The escape speed from the Moon, \(2.37 \times 10^3\) m/s
\[ \boxed{\phantom{K}} \] K
---
The above content presents a physics problem that seeks to find the temperature at which the root mean square (rms) speed of hydrogen atoms would equal the given escape speeds from Earth and the Moon. The mass of a hydrogen atom is provided, which is \(1.66 \times 10^{-27}\) kg. Students are expected to calculate the temperatures for the hydrogen atoms' rms speeds to match specified velocities.
**Scientific Background**
This exercise involves concepts from thermodynamics and kinetic theory of gases, where the rms speed of gas particles can be linked to temperature using the formula:
\[ v_{rms} = \sqrt{\frac{3k_BT}{m}} \]
Where:
- \( v_{rms} \) is the root mean square speed of the gas particles.
- \( k_B \) is Boltzmann's constant (\(1.38 \times 10^{-23}\) J/K).
- \( T \) is the temperature in Kelvins.
- \( m \) is the mass of a single gas particle.
By rearranging this formula to solve for temperature, \( T \), we get:
\[ T = \frac{m v_{rms}^2}{3 k_B} \]
Students will use this formula to find the required temperatures for the helium atoms' rms speeds to match the escape speeds from Earth and Moon.
**Instructions for Students**
1. Use the provided formula to solve for temperature:
\[ T = \frac{m v_{rms}^2}{3 k_B} \]
2. Substitute the given values for \( m \), \( v_{rms} \), and \( k_B \) to find the temperatures.
3. Make sure to convert the escape speeds provided in m/s correctly within your calculations.
4. Insert your final](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa5d0ed69-4925-4403-a796-f6681910a505%2Fe3f637f1-7f66-4651-8154-56da172f9d91%2F41ingo8_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Description**
At what temperature would the rms (root mean square) speed of hydrogen atoms equal the following speeds? (Note: The mass of a hydrogen atom is \(1.66 \times 10^{-27}\) kg.)
**Questions**
(a) The escape speed from Earth, \(1.12 \times 10^4\) m/s
\[ \boxed{\phantom{K}} \] K
(b) The escape speed from the Moon, \(2.37 \times 10^3\) m/s
\[ \boxed{\phantom{K}} \] K
---
The above content presents a physics problem that seeks to find the temperature at which the root mean square (rms) speed of hydrogen atoms would equal the given escape speeds from Earth and the Moon. The mass of a hydrogen atom is provided, which is \(1.66 \times 10^{-27}\) kg. Students are expected to calculate the temperatures for the hydrogen atoms' rms speeds to match specified velocities.
**Scientific Background**
This exercise involves concepts from thermodynamics and kinetic theory of gases, where the rms speed of gas particles can be linked to temperature using the formula:
\[ v_{rms} = \sqrt{\frac{3k_BT}{m}} \]
Where:
- \( v_{rms} \) is the root mean square speed of the gas particles.
- \( k_B \) is Boltzmann's constant (\(1.38 \times 10^{-23}\) J/K).
- \( T \) is the temperature in Kelvins.
- \( m \) is the mass of a single gas particle.
By rearranging this formula to solve for temperature, \( T \), we get:
\[ T = \frac{m v_{rms}^2}{3 k_B} \]
Students will use this formula to find the required temperatures for the helium atoms' rms speeds to match the escape speeds from Earth and Moon.
**Instructions for Students**
1. Use the provided formula to solve for temperature:
\[ T = \frac{m v_{rms}^2}{3 k_B} \]
2. Substitute the given values for \( m \), \( v_{rms} \), and \( k_B \) to find the temperatures.
3. Make sure to convert the escape speeds provided in m/s correctly within your calculations.
4. Insert your final
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