Molecular degrees of freedom and heat capacity: For an ideal monatomic gas, the constant-volume specific heat Cv equals which multiple of the gas constant R?

Introduction / Context:
Heat capacity of gases is tied to the number of accessible degrees of freedom. For monatomic ideal gases (e.g., He, Ne, Ar), only three translational degrees of freedom contribute at ordinary temperatures. Equipartition results lead directly to simple ratios between Cv, Cp, and the gas constant R.

Given Data / Assumptions:

• Monatomic ideal gas; translational degrees of freedom = 3.
• Equipartition theorem: each quadratic degree contributes (1/2)R per mole to internal energy.
• Cp − Cv = R for ideal gases.

Concept / Approach:
Total molar internal energy u for a monatomic ideal gas is (3/2)R T. Therefore, Cv = (∂u/∂T)_v = (3/2)R. From Cp − Cv = R, Cp = (5/2)R and γ = Cp/Cv = 5/3 ≈ 1.667. These benchmarks are widely used in compressible-flow and thermodynamics calculations.

Step-by-Step Solution:

Verification / Alternative check:
Experimental Cp and Cv values for He, Ne, Ar at room temperature match the (5/2)R and (3/2)R predictions within small deviations, validating the ideal monatomic model.

Why Other Options Are Wrong:

• R: too low; would imply only two degrees or non-ideal assumptions.
• 2R or 3R: correspond to additional active degrees (rotations/vibrations) not present for monatomic gases at ordinary T.
• 0.5 R: physically impossible for monatomic ideal gases.

Common Pitfalls:
Applying diatomic/polyatomic heat capacities to monatomic gases; overlooking Cp − Cv = R.

Final Answer:
1.5 R