Degrees of Freedom - UNSOLVED PRACTICE SET

Q7. Define degrees of freedom. How many degrees of freedom does a particle moving in space have?

Q8. Explain why a monoatomic gas has only 3 degrees of freedom while a diatomic gas has 5 degrees of freedom at room temperature.

Q9. What are vibrational degrees of freedom? Why are they usually not counted at room temperature?

Q10. A linear triatomic molecule (like CO₂) has 3N – 5 vibrational modes, while a non-linear triatomic molecule (like H₂O) has 3N – 6 vibrational modes. Explain the difference.

Q11. Explain the concept of "freezing" of degrees of freedom at low temperatures from a quantum mechanical perspective

Q12. Calculate the total degrees of freedom for:

(i) O₂ (diatomic)

(ii) CO₂ (linear triatomic)

(iii) NH₃ (non-linear, 4 atoms)

(iv) CH₄ (non-linear, 5 atoms)

Q13. Explain the concept of degrees of freedom in detail. For a molecule with N atoms, derive the general formula for:

(i) Total degrees of freedom

(ii) Translational degrees of freedom

(iii) Rotational degrees of freedom (distinguish between linear and non-linear molecules)

(iv) Vibrational degrees of freedom

Apply these formulas to calculate degrees of freedom for: He, N₂, CO₂, H₂O, NH₃, and CH₄.

Q14. Discuss how degrees of freedom affect the specific heats of gases. Using the equipartition theorem, derive expressions for:

(i) C_v for a gas with f degrees of freedom

(ii) C_p for the same gas

(iii) The ratio γ = C_p/C_v

Calculate these values for monoatomic (f = 3), diatomic (f = 5), and polyatomic (f = 6) gases. Explain any discrepancies with experimental values.

Q15. The degrees of freedom of a gas can change with temperature. Explain this phenomenon with reference to:

(i) Hydrogen gas at very low temperatures (~50 K)

(ii) Hydrogen gas at room temperature (~300 K)

(iii) Hydrogen gas at very high temperatures (~2000 K)

For each case, state the active degrees of freedom, the value of C_v, and explain why the change occurs. Draw a graph showing C_v vs T for hydrogen and explain its features.

Q16. Calculate the degrees of freedom and predict C_v, C_p, and γ for the following gases:

(i) Neon (Ne) — monoatomic

(ii) Oxygen (O₂) — diatomic at 300 K

(iii) Chlorine (Cl₂) — diatomic at 1000 K

(iv) Carbon dioxide (CO₂) — linear triatomic

(v) Ammonia (NH₃) — non-linear

(Given: R = 8.31 J mol⁻¹ K⁻¹)

Q17. A gas has the following properties: at 300 K, C_v = (5/2)R, and at 1000 K, C_v = (7/2)R.

(i) How many degrees of freedom are active at 300 K?

(ii) How many degrees of freedom are active at 1000 K?

(iii) What type of gas is this likely to be? Explain.

(iv) Calculate the energy required to heat 2 moles of this gas from 300 K to 1000 K at constant volume.

(v) Calculate γ at both temperatures.

Q18. A certain gas has 6 degrees of freedom per molecule.

(i) Is the molecule linear or non-linear? How many atoms does it contain?

(ii) Calculate C_v, C_p, and γ for this gas.

(iii) If 3 moles of this gas are heated from 250 K to 350 K at constant pressure, calculate the heat supplied and the work done.

(iv) At what temperature would vibrational modes start contributing significantly? Explain.

(Given: R = 8.31 J mol⁻¹ K⁻¹)