Specific-heat ratio – monoatomic ideal gas Select the correct value of the ratio of molar specific heats gamma = cp/cv for a monoatomic ideal gas:

Introduction / Context:The specific-heat ratio gamma influences wave speeds, nozzle flows, and thermodynamic relations. Its value depends on molecular degrees of freedom.

Given Data / Assumptions:

• Monoatomic ideal gas (e.g., He, Ne, Ar).
• Equipartition theorem applies at ordinary temperatures.
• Molar specific heats are constant over the range considered.

Concept / Approach:For a monoatomic ideal gas, only three translational degrees of freedom contribute at room temperature. Therefore cv = (3/2) * R and cp = cv + R = (5/2) * R, giving gamma = cp/cv = (5/2)R / [(3/2)R] = 5/3 ≈ 1.67.

Step-by-Step Solution:Write cv = 3R/2.Compute cp = cv + R = 5R/2.Form the ratio gamma = cp/cv = (5/2)/(3/2) = 5/3.Convert to decimal: 5/3 ≈ 1.67.

Verification / Alternative check:Measured gamma for noble gases at room temperature matches ~1.66–1.67, validating the ideal-gas, constant-heat-capacity assumption.

Why Other Options Are Wrong:

• 1.4 is typical for diatomic gases (e.g., air) at moderate temperatures.
• 1 or 1.3 contradict monoatomic degrees of freedom.
• 1.87 exceeds typical values even for restricted rotational modes.

Common Pitfalls:Using air's gamma for all gases; gamma depends strongly on molecular structure.

Final Answer:1.67