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NEET PHYSICSMedium

The value γ=CPCV\gamma = \frac{C_P}{C_V} for hydrogen, helium, and another ideal diatomic gas XX (whose molecules are not rigid but have an additional vibrational mode), are respectively equal to:

A

75,53,97\frac{7}{5}, \frac{5}{3}, \frac{9}{7}

B

53,75,97\frac{5}{3}, \frac{7}{5}, \frac{9}{7}

C

53,75,75\frac{5}{3}, \frac{7}{5}, \frac{7}{5}

D

75,53,75\frac{7}{5}, \frac{5}{3}, \frac{7}{5}

Step-by-Step Solution

  1. Formula for γ\gamma: The adiabatic index is given by γ=1+2f\gamma = 1 + \frac{2}{f}, where ff is the number of degrees of freedom.
  2. Hydrogen (H2H_2): It is a diatomic gas. At ordinary temperatures, it behaves as a rigid rotator with f=3f = 3 (translational) +2+ 2 (rotational) =5= 5. γH2=1+25=75\gamma_{H_2} = 1 + \frac{2}{5} = \frac{7}{5}.
  3. Helium (HeHe): It is a monoatomic gas. It has only translational degrees of freedom, so f=3f = 3. γHe=1+23=53\gamma_{He} = 1 + \frac{2}{3} = \frac{5}{3}.
  4. Gas XX: It is a diatomic gas with an additional vibrational mode. A vibrational mode contributes 2 degrees of freedom (potential + kinetic energy) per mode. Thus, f=3f = 3 (translational) +2+ 2 (rotational) +2+ 2 (vibrational) =7= 7. γX=1+27=97\gamma_{X} = 1 + \frac{2}{7} = \frac{9}{7}.
  5. Conclusion: The respective values are 75,53,97\frac{7}{5}, \frac{5}{3}, \frac{9}{7}.
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