Degrees of freedom, law of equipartition of energy and applications to specific heat capacities - One Line Questions

1. At very high temperatures, diatomic molecules can exhibit vibrational motion. How many additional degrees of freedom does this add? — 2
2. The ratio of specific heats (gamma = Cp/Cv) for a monoatomic gas is approximately: — 1.67
3. The ratio of specific heats (gamma = Cp/Cv) for a diatomic gas at moderate temperatures is approximately: — 1.4
4. The ratio of specific heats (gamma = Cp/Cv) for a triatomic gas (like water vapor) at high temperatures, assuming all modes are active, tends towards: — 1.33
5. A monoatomic gas molecule possesses how many degrees of freedom at any temperature? — 3
6. For a rigid diatomic molecule at moderate temperatures, what is the total number of degrees of freedom? — 5
7. The specific heat capacity of a solid is often explained using the equipartition theorem. How many degrees of freedom are typically associated with each atom in a solid? — 3 (vibrational in x, y, z directions)
8. For a triatomic linear molecule (like CO2) at moderate temperatures, how many degrees of freedom are typically considered? — 3 (translational) + 3 (rotational) = 6
9. What is the average energy per molecule for a rigid diatomic gas at temperature T? — 5kT/2
10. For a rigid diatomic gas at moderate temperatures, the internal energy per molecule is: — 5kT/2
11. If a gas has 6 degrees of freedom, its molar specific heat at constant volume (Cv) would be: — 3R
12. What is the molar specific heat at constant volume (Cv) for a monoatomic gas? — 3R/2
13. What is the molar specific heat at constant volume (Cv) for a rigid diatomic gas at moderate temperatures? — 5R/2
14. What is the molar specific heat at constant pressure (Cp) for a monoatomic gas? — 5R/2
15. What is the molar specific heat at constant pressure (Cp) for a rigid diatomic gas at moderate temperatures? — 7R/2
16. According to the equipartition theorem, the internal energy of one mole of a monoatomic gas is: — 3RT/2
17. What is the value of 'f' (degrees of freedom) for a rigid triatomic molecule like CO2 at room temperature? — 6
18. If vibrational degrees of freedom are also active for a diatomic molecule, its internal energy per molecule becomes: — 7kT/2
19. What is the molar specific heat at constant volume (Cv) for a diatomic gas when vibrational modes are active? — 7R/2
20. What is the molar specific heat at constant pressure (Cp) for a diatomic gas when vibrational modes are active? — 9R/2
21. At very low temperatures, quantum mechanical effects become significant. How does this affect the degrees of freedom considered? — Rotational and vibrational degrees of freedom freeze out
22. The concept of degrees of freedom is crucial for understanding the microscopic behavior of gases and their macroscopic properties like specific heat. Which law forms the basis for relating degrees of freedom to energy? — Law of Equipartition of Energy
23. Which of the following is NOT a direct application of the Law of Equipartition of Energy? — Explaining the photoelectric effect
24. The molar specific heat of an ideal gas is defined as the heat required to raise the temperature of one mole of the gas by one Kelvin at constant volume. This relates to: — Cv
25. The molar specific heat at constant pressure (Cp) is related to Cv by: — Cp = Cv + R
26. The molar specific heat at constant volume (Cv) for a gas is related to the internal energy U by: — Cv = dU/dT * 1/N_A
27. Which type of molecule has the fewest degrees of freedom? — Monoatomic molecule
28. Why is Cv for diatomic gases higher than for monatomic gases? — Diatomic molecules have rotational degrees of freedom.
29. Which statement best describes the contribution of rotational motion to the internal energy of a gas molecule? — Each rotational degree of freedom contributes kT/2.
30. For a system in thermal equilibrium, the average energy per molecule is distributed equally among all its active degrees of freedom. This is the essence of: — Law of Equipartition of Energy
31. Which of the following molecules has the highest number of degrees of freedom at room temperature, assuming vibrational modes are not active? — Methane (CH4)
32. The Law of Equipartition of Energy is a classical result. What is a major limitation of this law? — It does not account for quantum mechanical effects at low temperatures.
33. If the vibrational modes of a diatomic molecule are excited, how does it affect its specific heat capacity? — It increases.
34. According to the Law of Equipartition of Energy, how much energy does each translational degree of freedom contribute to the average energy of a molecule? — kT/2
35. The Law of Equipartition of Energy states that for a system in thermal equilibrium, the average energy associated with each degree of freedom is: — kT/2
36. According to the Law of Equipartition of Energy, the average translational kinetic energy per molecule in any gas is: — 3kT/2
37. For a monoatomic gas, the internal energy per molecule is equal to the average energy associated with its degrees of freedom. What is this value? — 3kT/2
38. Why does a triatomic molecule like ozone (O3) have more degrees of freedom than a diatomic molecule like oxygen (O2)? — Ozone has more atoms, leading to more possible vibrational modes.
39. The specific heat capacity of a polyatomic gas is generally higher than that of a diatomic gas because: — Polyatomic gases have more vibrational degrees of freedom.
40. The internal energy of an ideal gas depends only on: — Temperature and degrees of freedom
41. The Law of Equipartition of Energy is valid under which condition? — The system is in thermal equilibrium
42. What does 'T' represent in the context of the Law of Equipartition of Energy? — Absolute temperature
43. The specific heat capacity of a gas is a measure of: — The heat required to raise the temperature of a unit mass by one degree.
44. What does 'k' represent in the context of the Law of Equipartition of Energy? — Boltzmann constant
45. What is the definition of degrees of freedom for a molecule? — The number of independent ways in which a molecule can store energy.
46. What are the types of motion contributing to the degrees of freedom of a molecule? — Translational, rotational, and vibrational
47. For a non-rigid diatomic molecule, which degrees of freedom are active at moderate temperatures? — Translational and rotational only
48. Consider a system of N molecules, each with f degrees of freedom. The total internal energy U is given by: — U = N * f * kT/2
49. For a monatomic gas, Cv = 3R/2. This implies that its internal energy per mole is directly proportional to: — Temperature
50. The Dulong-Petit law, which approximates the molar specific heat of solids, is a consequence of: — The Law of Equipartition of Energy
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