Three equal-mass molecules have speeds 200, 400 and 600 m s⁻¹. Calculate ⟨c⟩, ⟨c²⟩ and c_rms. A student uses (1/2)m⟨c⟩² for mean translational kinetic energy. Explain the error and state the correct expression.

Question

Accepted Answer

⟨c⟩ = 400 m s⁻¹; ⟨c²⟩ = 1.87 × 10⁵ m² s⁻²; c_rms = 432 m s⁻¹; use (1/2)m⟨c²⟩. ⟨c⟩ = 1200/3 = 400 m s⁻¹. ⟨c²⟩ = (40000 + 160000 + 360000)/3 = 186667 m² s⁻². c_rms = √186667 = 432 m s⁻¹. The student's square of the mean, 160000 m² s⁻², is not the mean of the squares. Average each molecule's kinetic energy instead: ⟨E_trans⟩ = (1/2)m⟨c²⟩.

Apply the statistical definition

Three equal-mass molecules have speeds 200, 400 and 600 m s⁻¹. Calculate ⟨c⟩, ⟨c²⟩ and c_rms. A student uses (1/2)m⟨c⟩² for mean translational kinetic energy. Explain the error and state the correct expression.

What a complete response should do

Your self-assessment

Three equal-mass molecules have speeds 200, 400 and 600 m s⁻¹. Calculate ⟨c⟩, ⟨c²⟩ and crms. A student uses (1/2)m⟨c⟩² for mean translational kinetic energy. Explain the error and state the correct expression.

What a complete response should do

Your self-assessment

Three equal-mass molecules have speeds 200, 400 and 600 m s⁻¹. Calculate ⟨c⟩, ⟨c²⟩ and c_rms. A student uses (1/2)m⟨c⟩² for mean translational kinetic energy. Explain the error and state the correct expression.