The value of gamma left( = frac{C_{p}}{C_{v}} | vedclass

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(A) The adiabatic index is given by $\gamma = 1 + \frac{2}{f}$,where $f$ is the number of degrees of freedom.$1$. Hydrogen $(H_2)$ is a diatomic gas a

Vedclass · Accepted Answer

$\frac{7}{5}, \frac{5}{3}, \frac{9}{7}$

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(A) The adiabatic index is given by $\gamma = 1 + \frac{2}{f}$,where $f$ is the number of degrees of freedom.$1$. Hydrogen $(H_2)$ is a diatomic gas at room temperature. Its degrees of freedom $f = 5$ ($3$ translational + $2$ rotational). Thus,$\gamma = 1 + \frac{2}{5} = \frac{7}{5}$.$2$. Helium $(He)$ is a monoatomic gas. Its degrees of freedom $f = 3$ ($3$ translational). Thus,$\gamma = 1 + \frac{2}{3} = \frac{5}{3}$.$3$. Gas $X$ is a diatomic gas with an additional vibrational mode. $A$ vibrational mode contributes $2$ degrees of freedom ($1$ kinetic + $1$ potential). So,$f = 5 + 2 = 7$. Thus,$\gamma = 1 + \frac{2}{7} = \frac{9}{7}$.Therefore,the values are $\frac{7}{5}, \frac{5}{3}, 	ext{and } \frac{9}{7}$.

The value of $\gamma \left( = \frac{C_{p}}{C_{v}} \right)$ for hydrogen,helium,and another ideal diatomic gas $X$ (whose molecules are not rigid but have an additional vibrational mode) are respectively equal to:

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