A monoatomic ideal gas is adiabatically compressed to 1 8 of its initial volume. If the initial root-mean-square (rms) speed of the gas molecules is v₀ , the final rms speed of the molecules after compression will be :

Step-by-step solution

For an adiabatic process, the temperature and volume are related by: T₁ V₁^ -1 = T₂ V₂^ -1 For a monoatomic gas, the ratio of specific heats is = 5 3 . Given V₂ = V₁ 8 , we can find the ratio of the final temperature to the initial temperature: T₂ T₁ = ( V₁ V₂ )^ -1 = (8)^ 5 3 -1 = (8)^ 2 3 = (2^3 )^ 2 3 = 4 The root-mean-square (rms) speed of gas molecules is given by v_ rms = 3RT M , which implies v_ rms T . Therefore, the ratio of the final to initial rms speed is: v_ final v₀ = T₂ T₁ = 4 = 2 v_ final = 2v₀ Answ