JEE Main - RMS speed, degrees of freedom and equipartition Online Test (MCQ Quiz)

1. What is the RMS speed of gas molecules directly proportional to?

A) The square root of its absolute temperature B) Its absolute temperature C) The square of its absolute temperature D) The inverse of its absolute temperature

2. The RMS speed of gas molecules is given by the formula:

A) v_rms = sqrt(3RT/M) B) v_rms = sqrt(RT/M) C) v_rms = sqrt(3kT/m) D) v_rms = sqrt(kT/m)

3. In the formula for RMS speed, v_rms = sqrt(3RT/M), what does 'M' represent?

A) Mass of a single molecule B) Molar mass of the gas C) Boltzmann constant D) Universal gas constant

4. If the absolute temperature of a gas is doubled, how does its RMS speed change?

A) It increases by a factor of sqrt(2) B) It doubles C) It increases by a factor of 4 D) It remains the same

5. What is the unit of Boltzmann constant (k)?

A) Joule per Kelvin (J/K) B) Joule per mole Kelvin (J/mol·K) C) Pascal meter cubed per Kelvin (Pa·m³/K) D) Watt per Kelvin (W/K)

6. The kinetic energy of a gas is directly proportional to:

A) Absolute temperature B) Pressure C) Volume D) Number of moles

7. What is the average kinetic energy of one molecule of an ideal gas at absolute temperature T?

A) 3/2 kT B) kT C) 3/2 RT D) RT

8. The ratio of RMS speed of hydrogen molecules to that of oxygen molecules at the same temperature is:

A) 4:1 B) 1:4 C) 2:1 D) 1:2

9. Which of the following gases will have the highest RMS speed at a given temperature?

A) Oxygen (O2) B) Nitrogen (N2) C) Hydrogen (H2) D) Carbon Dioxide (CO2)

10. What is meant by the 'degrees of freedom' of a molecule?

A) The number of ways a molecule can move B) The number of atoms in a molecule C) The number of bonds in a molecule D) The number of collisions a molecule makes

11. For a monatomic gas molecule, what is the number of degrees of freedom?

A) 1 B) 2 C) 3 D) 5

12. A diatomic molecule typically has how many degrees of freedom at moderate temperatures?

A) 3 B) 4 C) 5 D) 6

13. At very high temperatures, a diatomic molecule can have how many degrees of freedom?

A) 5 B) 6 C) 7 D) 9

14. What does the equipartition theorem state?

A) Each degree of freedom of a molecule contributes 1/2 kT to its average energy B) Each degree of freedom contributes kT to its average energy C) Each degree of freedom contributes 3/2 kT to its average energy D) The total energy is equally distributed among all molecules

15. According to the equipartition theorem, the average energy associated with translational motion for a molecule is:

A) 3/2 kT B) kT C) 5/2 kT D) 7/2 kT

16. For a diatomic molecule in thermal equilibrium, the total average internal energy per molecule is:

A) 5/2 kT B) 3/2 kT C) 7/2 kT D) 9/2 kT

17. What is the specific heat at constant volume (Cv) for a monatomic ideal gas per mole?

A) 3/2 R B) 5/2 R C) 7/2 R D) R

18. What is the specific heat at constant volume (Cv) for a diatomic ideal gas per mole at moderate temperatures?

A) 5/2 R B) 3/2 R C) 7/2 R D) 9/2 R

19. The ratio of specific heats (Cp/Cv) for a monatomic ideal gas is known as the adiabatic index, γ. What is its value?

A) 1.67 B) 1.40 C) 1.33 D) 1.20

20. The ratio of specific heats (Cp/Cv) for a diatomic ideal gas at moderate temperatures is approximately:

A) 1.40 B) 1.67 C) 1.33 D) 1.20

21. Which of the following is NOT a degree of freedom for a molecule?

A) Translational motion B) Rotational motion C) Vibrational motion D) Intermolecular attractive force

22. The equipartition theorem is applicable under which condition?

A) Thermal equilibrium B) High pressure C) Low temperature D) Non-ideal conditions

23. For a diatomic molecule, vibrational degrees of freedom contribute how much energy per molecule according to equipartition theorem?

A) kT B) 1/2 kT C) 3/2 kT D) 2 kT

24. If a gas has f degrees of freedom, its molar specific heat at constant volume is given by:

A) f/2 R B) f R C) (f+1)/2 R D) (f+2)/2 R

25. At higher temperatures, the contribution of vibrational modes to the specific heat of a diatomic gas becomes significant. This leads to:

A) An increase in Cv B) A decrease in Cv C) No change in Cv D) An increase in Cp

26. The RMS speed of gas molecules is independent of:

A) Temperature B) Molar mass C) Number of molecules D) Universal gas constant

27. Which type of gas molecule has the maximum number of degrees of freedom at room temperature among monatomic, diatomic, and triatomic (linear)?

A) Monatomic B) Diatomic C) Triatomic (linear) D) All have the same

28. The average translational kinetic energy of gas molecules is directly proportional to:

A) Absolute temperature B) Pressure C) Volume D) Density

29. If the temperature of a gas is increased by 2 K, the increase in its average kinetic energy per molecule is:

A) 2k B) 3/2 k C) 3k D) 3/2 R

30. The average energy of a photon in black body radiation is given by kT. This is an example of:

A) Equipartition theorem B) Maxwell-Boltzmann distribution C) Ideal gas law D) Dalton's law

31. For a rigid diatomic molecule, rotational degrees of freedom contribute:

A) kT B) 1/2 kT C) 3/2 kT D) 2 kT

32. What is the value of the universal gas constant (R)?

A) 8.314 J/mol·K B) 1.38 x 10^-23 J/K C) 6.022 x 10^23 mol^-1 D) 9.8 m/s²

33. If the pressure of an ideal gas is doubled while keeping the temperature constant, the RMS speed of its molecules will:

A) Remain unchanged B) Double C) Halve D) Increase by a factor of sqrt(2)

34. The mean free path of gas molecules depends on:

A) Temperature and pressure B) Molecular diameter and number density C) Volume and number of moles D) RMS speed and temperature

35. A triatomic linear molecule (like CO2) has how many degrees of freedom related to vibration?

A) 1 B) 2 C) 3 D) 4

36. The adiabatic process is described by PV^γ = constant. What does γ represent?

A) Ratio of specific heats (Cp/Cv) B) Ratio of molar masses C) Ratio of absolute temperatures D) Ratio of pressures

37. The specific heat capacity of a gas depends on:

A) Degrees of freedom B) Molecular weight C) Temperature D) Pressure

38. For a monatomic gas, the internal energy per mole is:

A) 3/2 RT B) 5/2 RT C) 7/2 RT D) RT

39. If the volume of a gas is kept constant, and its temperature is increased, the RMS speed of the molecules:

A) Increases B) Decreases C) Remains constant D) Becomes zero

40. The equipartition theorem is a consequence of:

A) Classical mechanics B) Quantum mechanics C) Relativistic mechanics D) Thermodynamics

41. Which statement is incorrect regarding the RMS speed of gas molecules?

A) It increases with temperature. B) It decreases with increasing molar mass. C) It is directly proportional to the pressure. D) It is proportional to the square root of temperature.

42. A non-linear triatomic molecule has how many degrees of freedom at moderate temperatures?

A) 6 B) 5 C) 7 D) 9

43. The internal energy of an ideal gas depends only on:

A) Temperature B) Pressure C) Volume D) Number of moles

44. In the equation v_rms = sqrt(3kT/m), what does 'm' represent?

A) Mass of a single molecule B) Molar mass C) Boltzmann constant D) Universal gas constant

45. At absolute zero temperature, the RMS speed of gas molecules is:

A) Zero B) Minimum C) Maximum D) Undefined

46. The average kinetic energy of a diatomic molecule at temperature T is approximately:

A) 5/2 kT B) 3/2 kT C) kT D) 7/2 kT

47. Which gas will have the lowest RMS speed at 27°C?

A) Helium (He) B) Neon (Ne) C) Argon (Ar) D) Krypton (Kr)

48. The equipartition theorem applies to systems in:

A) Thermal equilibrium B) Mechanical equilibrium C) Chemical equilibrium D) Phase equilibrium

49. For a monatomic gas, Cp - Cv = R. For a diatomic gas, Cp - Cv is also R. This is an application of:

A) Mayer's relation B) Ideal gas law C) Dalton's law D) Avogadro's law

50. The number of degrees of freedom for translational motion is always:

A) 3 B) 2 C) 1 D) 5

RMS speed, degrees of freedom and equipartition - Online Test

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