Concepts Of Physics MCQ Edition [Volume 2]PhysicsSpecific Heat Capacities of Gases
At 0^ C , the speed of sound in hydrogen is 1280 m s ⁻¹ . The density of hydrogen at STP is 0.089 kg m ⁻³ . Determine the molar heat capacities C_p and C_V of hydrogen.
Options
- AC_p = 26.3 J K ⁻¹ mol ⁻¹ , C_V = 18.0 J K ⁻¹ mol ⁻¹
- BC_p = 18.0 J K ⁻¹ mol ⁻¹ , C_V = 26.3 J K ⁻¹ mol ⁻¹
- CC_p = 29.1 J K ⁻¹ mol ⁻¹ , C_V = 20.8 J K ⁻¹ mol ⁻¹
- DC_p = 20.8 J K ⁻¹ mol ⁻¹ , C_V = 12.5 J K ⁻¹ mol ⁻¹
Correct answer
A. C_p = 26.3 J K ⁻¹ mol ⁻¹ , C_V = 18.0 J K ⁻¹ mol ⁻¹
Step-by-step solution
The speed of sound in a gas is given by the formula: v = P Rearranging for the adiabatic index : = v^2 P Given: v = 1280 m s ⁻¹ = 0.089 kg m ⁻³ At STP, taking standard pressure P = 10^5 Pa (1 bar): = (1280)^2 0.089 10^5 = 1638400 0.089 100000 1.458 We know the relations for molar heat capacities: C_p - C_V = R C_p C_V = From these, we get: C_V = R - 1 Using R = 8.3 J K ⁻¹ mol ⁻¹ : C_V = 8.3 1.458 - 1 = 8.3 0.458 18.1 J K ⁻¹ mol ⁻¹ Now, calculating C_p : C_p = C_V + R = 18.1 + 8.3 = 26.4 J K ⁻¹ mol ⁻¹ These values a