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Heat and Thermodynamics question

2021 · 17 Mar · Shift 1 · Q44
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Heat and Thermodynamics question

2021 · 17 Mar · Shift 1 · Q44

JEE MainPhysicsHeat and ThermodynamicsMCQ+4 / −1
A polyatomic ideal gas has 24 vibrational modes. What is the value of γ\gammaγ?
  1. A
    1.37
  2. B
    1.30
  3. C
    1.03
  4. D
    10.3
View written solutionFree

Correct answer: C

  1. Use the relation between γ\gammaγ and degrees of freedom

For an ideal gas,

γ=CPCV=f+2f\gamma = \frac{C_P}{C_V} = \frac{f+2}{f}γ=CV​CP​​=ff+2​

where fff is the total number of active degrees of freedom.

  1. Count the degrees of freedom

For a polyatomic molecule:

  • Translational degrees of freedom =3=3=3
  • Rotational degrees of freedom =3=3=3 (for a general non-linear polyatomic molecule)
  • Each vibrational mode contributes 2 degrees of freedom to energy

Given vibrational modes =24=24=24, so vibrational contribution to degrees of freedom is

2×24=482 \times 24 = 482×24=48

Hence total degrees of freedom:

f=3+3+48=54f = 3 + 3 + 48 = 54f=3+3+48=54
  1. Compute γ\gammaγ
γ=f+2f=54+254=5654=2827\gamma = \frac{f+2}{f} = \frac{54+2}{54} = \frac{56}{54} = \frac{28}{27}γ=ff+2​=5454+2​=5456​=2728​

Now,

2827≈1.037\frac{28}{27} \approx 1.0372728​≈1.037
  1. Match with options

The closest option is:

  • C: 1.031.031.03

Therefore, the correct answer is C.

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