Question

The molar internal energy of a gas at temperature T is $ {{U}_{m}} $ [T]. The molar internal energy at T = 0 is $ {{U}_{m}} $ (0). The correct expression that relates these two with appropriate contributions is

• A. $ {{U}_{m}}(T)={{U}_{m}}(0)+3RT $ [Linear molecule, translation only]
• B. $ {{U}_{m}}(T)={{U}_{m}}(0)+\frac{5}{2}RT $ [Linear molecule translation and rotation only]
• C. $ {{U}_{m}}(T)={{U}_{m}}(0)+\frac{3}{2}RT $ [Non-linear molecule, translation and rotation only]
• D. $ {{U}_{m}}(T)={{U}_{m}}(0)+RT $ [Non-linear molecule, translation only]

Answer 1

Answer: (B)

$ \Delta {{U}_{m}}={{C}_{V}}\Delta T $ $ {{\Delta }_{m}}(T)-{{U}_{m}}(0)={{C}_{V}}(T-0) $ $ {{U}_{m}}(T)-{{U}_{m}}(0)=\frac{5}{2}RT $ [ $ Cv=\frac{5}{2}R, $ for linear molecule, by taking translational and rotational] $ {{U}_{m}}(T)={{U}_{m}}(0)+\frac{5}{2}RT $