Question

Calculate $\gamma$ (ratio of $C_p$ and $C_v$ ) for triatomic linear gas at high temperature. Assume that the contribution of the vibrational degree of freedom is 75%?

• A. 1.222
• B. 1.121
• C. 1.18
• D. 1.33

Answer 1

Answer: (C)

1.18

For Triatomic molecule $N=3$

Degree of freedom $=3N=9$

3$⟶$Translation degree of freedom

For linear, 2$⟶$Rotational degree of freedom

$9-(3+2)=4⟶$Vibrational degree of freedom

$U=3\times \frac{1}{2}kT+2\times \frac{1}{2}kT+4\times \frac{75}{100}kT$

as vibrational contributes only 75%

$U=\frac{3}{2}kT+kT+3kT=\frac{11}{2}kT$

$C_V=\frac{dU}{dT}=\frac{11}{2}k$ or for 1 mole $\frac{11}{2}R$

As for ideal gas,

$C_P-C_V=R{C}_P=C_V+R=\frac{11}{2}R+R=\frac{13}{2}Rg=\frac{C_P}{C_V}=\frac{13}{2}R\times \frac{2}{11R}=1.18$

Thus, the ratio of $C_p$ and $C_v$ is 1.18.