# The specific heats , $C_{p}$ and  $C_{v}$ of a gas of diatomic molecules ,A, are given ( in units of $J\:mol^{-1}$ ) by 29 and 22 , respectively . Another gas of diatomic molecules , B , has the corresponding value 30 and 21 . If they are treated as ideal gases , then : Option 1) A is rigid but B has a vibrational mode. Option 2) a has a vibrational mode but B has none. Option 3) A has one vibrational mode and B has two. Option 4) Both A and B have a vibrational mode each.

$\frac{C_{p}}{C_{v}}=1+\frac{2}{f}$

For gas A ,

$\frac{29}{22}=1+\frac{2}{f}\Rightarrow\frac{2}{f}=\frac{7}{22}$

$\Rightarrow f=\frac{44}{7}\simeq 6$

$\frac{C_{p}}{C_{v}}=\frac{30}{21}=1+\frac{2}{f} \Rightarrow \frac{2}{f}=\frac{9}{21}$

$\Rightarrow f=\frac{42}{9} \simeq 5$

Option 1)

A is rigid but B has a vibrational mode.

Option 2)

a has a vibrational mode but B has none.

Option 3)

A has one vibrational mode and B has two.

Option 4)

Both A and B have a vibrational mode each.

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