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Chapter 9: Problem 7

What is the molar specific heat at constsnt volume of a diatomic gas at room temperature \(T_{0}\) ? Use the fact that for practically all diatomic molecules the spacing between rotational energy levels is sroall compared to \(k T_{0}\), while that between their vibrational energy levela is large compared to \(k T_{0}\).

Short Answer

Expert verified
The molar specific heat at constant volume for a diatomic gas is \( \frac{5}{2} R \).

Step by step solution

01

Understanding Degrees of Freedom

For a diatomic gas, we consider translations, rotations, and vibrations. The gas has 3 translational degrees of freedom and 2 rotational degrees of freedom. Vibrations are not excited under room temperature because the spacing between vibrational energy levels is large compared to thermal energy, implying vibrations do not contribute to the specific heat at room temperature.
02

Calculate Total Degrees of Freedom

The diatomic gas molecule has a total of 5 active degrees of freedom at room temperature: 3 translational and 2 rotational.
03

Apply Equipartition Theorem

According to the equipartition theorem, each degree of freedom contributes \( \frac{1}{2} R \) to the molar specific heat capacity. For 5 degrees of freedom, the total contribution is \( \frac{5}{2} R \).
04

Identify Molar Specific Heat at Constant Volume

Thus, the molar specific heat at constant volume \( C_v \) for a diatomic gas is \( \frac{5}{2} R \), where \( R \) is the universal gas constant.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Diatomic Gas
Diatomic gases consist of molecules composed of two atoms. These atoms can be the same, such as in molecular nitrogen ( N_2 ), or different, like carbon monoxide ( CO ). Diatomic molecules have unique properties that influence their thermodynamic behavior.
  • At room temperature, these molecules display characteristics that alter their specific heat values, notably different from monoatomic gases.
  • Understanding the molecular structure is vital for predicting how these molecules behave under and contribute to different physical changes.
This is crucial when we consider specific heat calculations, as the arrangement and motion of the atoms dictate how energy is stored and transferred.
Degrees of Freedom
Degrees of freedom refer to the number of independent ways in which the molecules of a system can move and store energy. For diatomic gases, this concept is crucial in understanding their specific heat at room temperature.
  • Translation: Diatomic molecules can move in three spatial dimensions, delivering 3 translational degrees of freedom.
  • Rotation: These molecules can rotate about two axes perpendicular to the line joining their atoms, adding 2 rotational degrees of freedom.
  • Vibration typically does not contribute at room temperature because the gap between vibrational energy levels is too large, restricting this mode's activation.
This total of 5 active degrees of freedom significantly influences the energy distribution within the gas and subsequently determines the specific heat.
Equipartition Theorem
The Equipartition Theorem is a fundamental principle in thermal physics that sheds light on the distribution of energy in different degrees of freedom for a system in thermal equilibrium.
  • It posits that each degree of freedom contributes equally to the system's overall thermal energy, specifically allocating energy of \( \frac{1}{2} k_B T \) per degree of freedom, where \( k_B \) is the Boltzmann constant and \( T \) is the temperature.
  • For molar pots, where \( R \) (the ideal gas constant) is used instead for consistency with a mole-based framework, this contribution translates to \( \frac{1}{2} R \).
The theorem allows us to predict that a diatomic gas with 5 degrees of freedom (3 translational + 2 rotational) will have a molar specific heat of \( \frac{5}{2} R \) at constant volume.
Molecular Rotation
Molecular rotation is a key contributor to the degrees of freedom for diatomic gases. Unlike monoatomic gases, where no rotational motion is possible, diatomic molecules can rotate about two distinguishable axes that lie perpendicular to the bond axis.
  • This rotation gives rise to two additional degrees of freedom, augmenting the gas's ability to store thermal energy.
  • At room temperature, these rotational modes are sufficiently excited, making them significant to the analysis and calculation of the gas's specific heat.
Understanding these rotations is essential for explaining why diatomic gases exhibit higher specific heat capacities compared to their monoatomic counterparts.
Thermal Physics
Thermal physics is the overarching study of temperature, heat, and their effects on matter. In the context of our discussion on molar specific heat, it involves how energy is distributed among molecular motions and how these impact macroscopic thermodynamic properties.
  • This field combines principles from physics and chemistry to explain how specific molecular attributes, like degrees of freedom, influence a substance's response to thermal energy.
  • For diatomic gases, thermal physics explains why vibrations do not influence specific heat at room temperature 鈥� the vibrational energy levels are not excited enough to store thermal energy under these conditions.
Such insights underline the importance of thermodynamics and statistical mechanics in calculating and understanding heat capacities, particularly highlighting the significance of molecular composition and structure.

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Most popular questions from this chapter

(a) An ideal gas of \(N\) atoms of mass \(m\) is contained in a volume \(V\) at absolute temperature T. Calculate the ohemical potential \(\mu\) of this gas. You may use the olassical approximation for the parlition function, taking into aecount the indistinguishability of the particles. (b) A gas of \(N^{\prime \prime}\) such weakly interscting particles, adsorbed on s surface of area \(A\) on which they are free to move, can form a two- dimensional ideal gas on buch a surface. The energy of an adsorbed molecule is then \(\left(p^{2} / 2 m\right)-c_{0}\) where p denotes its (two-eomponent) roomenturn vector and \(\epsilon_{0}\) is the binding energy which holds a molecule on the surface, Calculate the chernical potential \(\mu^{\prime}\) of this adsorbed ideal gas. The partition funetion can again be evaluated in the classical spproximation. (c) At the tempersture \(T\), the equilibrium condition between rolecules adeorbed on the surface and molecules in the surrounding three-dimengional gas can be exprased in terms of the respective chemical potentisls. Uge this condition to find at temperatare \(T\) the mean number \(n^{\prime}\) of molecules adsorbed per unit area of the surface when the mean pressure of the surrounding gas is \(\tilde{p}\).

What is the total number 9 of molecules thet escspe per unit time from a unit areq of the surfsce of s llquid which is at tempersture \(T\), where its vapor preszure is p? Use detailed bulstce argumenk by considering a eituation where the Ilqund is in thermal equilibrium with its vupor et this temperature and presbure. Treat the vapor as an ideal gas, and nasume that molecules striking the gurfioe of the liqnid are not appreciably reflected. Calculate the number 94 of molecules escaping per unit time from unit area of the surface of water in a glass at \(25^{\circ} \mathrm{C}\). The vapor pressere of water at this temperature is \(23.8 \mathrm{~mm} \mathrm{Hg}\).
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