7) Why is the heat capacity of a diatomic gas different than a monatomic gas at room temperature ?

Elements Of Electromagnetics
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Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
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Here are some advanced questions related to Thermodynamics:

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### Questions

7) Why is the heat capacity of a diatomic gas different than a monatomic gas at room temperature?

8) Why is it that the probability of a molecule occupying a higher energy state is lower than the probability that the molecule occupies a high energy state?

9) Can a diatomic gas have a heat capacity at constant volume that is greater than 5/2 Nk? Explain.

10) What is the ratio of the number of molecular states with a speed of 50.0 m/s to the number of states with a speed of 150.0 m/s? Show work.

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### Solutions

#### Question 7:
The heat capacity of a diatomic gas is different from a monatomic gas at room temperature due to the different degrees of freedom available to the molecules. A monatomic gas molecule has only translational degrees of freedom, while a diatomic gas molecule has translational, rotational, and possibly vibrational degrees of freedom. These additional degrees of freedom allow for more ways to store energy, resulting in a higher heat capacity for diatomic gases compared to monatomic gases at room temperature.

#### Question 8:
The probability of a molecule occupying a higher energy state is lower due to the Boltzmann distribution. According to the Boltzmann factor \(e^{-E/kT}\), where \(E\) is the energy, \(k\) is the Boltzmann constant, and \(T\) is the temperature, the exponential term decreases as \(E\) increases. This results in a lower probability for higher energy states compared to lower energy states.

#### Question 9:
A diatomic gas can have a heat capacity at constant volume greater than \(\frac{5}{2} Nk\) if it involves additional internal degrees of freedom, such as vibrational modes being excited. At higher temperatures, these vibrational modes can be excited, contributing additional capacity to store heat, thereby increasing the heat capacity beyond \(\frac{5}{2} Nk\).

#### Question 10:
To find the ratio of the number of molecular states with speeds of 50.0 m/s and 150.0 m/s, we use the Maxwell-Boltzmann distribution:

\[ f(v) = 4\pi \left( \frac{m}{2kT} \right)^{3/2} v^2 e^{-\frac{mv^2}{2
Transcribed Image Text:Here are some advanced questions related to Thermodynamics: --- ### Questions 7) Why is the heat capacity of a diatomic gas different than a monatomic gas at room temperature? 8) Why is it that the probability of a molecule occupying a higher energy state is lower than the probability that the molecule occupies a high energy state? 9) Can a diatomic gas have a heat capacity at constant volume that is greater than 5/2 Nk? Explain. 10) What is the ratio of the number of molecular states with a speed of 50.0 m/s to the number of states with a speed of 150.0 m/s? Show work. --- ### Solutions #### Question 7: The heat capacity of a diatomic gas is different from a monatomic gas at room temperature due to the different degrees of freedom available to the molecules. A monatomic gas molecule has only translational degrees of freedom, while a diatomic gas molecule has translational, rotational, and possibly vibrational degrees of freedom. These additional degrees of freedom allow for more ways to store energy, resulting in a higher heat capacity for diatomic gases compared to monatomic gases at room temperature. #### Question 8: The probability of a molecule occupying a higher energy state is lower due to the Boltzmann distribution. According to the Boltzmann factor \(e^{-E/kT}\), where \(E\) is the energy, \(k\) is the Boltzmann constant, and \(T\) is the temperature, the exponential term decreases as \(E\) increases. This results in a lower probability for higher energy states compared to lower energy states. #### Question 9: A diatomic gas can have a heat capacity at constant volume greater than \(\frac{5}{2} Nk\) if it involves additional internal degrees of freedom, such as vibrational modes being excited. At higher temperatures, these vibrational modes can be excited, contributing additional capacity to store heat, thereby increasing the heat capacity beyond \(\frac{5}{2} Nk\). #### Question 10: To find the ratio of the number of molecular states with speeds of 50.0 m/s and 150.0 m/s, we use the Maxwell-Boltzmann distribution: \[ f(v) = 4\pi \left( \frac{m}{2kT} \right)^{3/2} v^2 e^{-\frac{mv^2}{2
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