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Heat and Thermodynamics question

2024 · Q197
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Heat and Thermodynamics question

2024 · Q197

NEETPhysicsHeat and ThermodynamicsMCQ+4 / −1

According to the law of equipartition of energy, the number of vibrational modes of a polyatomic gas of constant γ=CpCv\gamma=\frac{C_p}{C_v}γ=Cv​Cp​​ is (CPC_PCP​ where CVC_VCV​ are the specific heat capacities of the gas at constant pressure and constant volume, respectively):

  1. A
    4+3γγ−1\frac{4+3 \gamma}{\gamma-1}γ−14+3γ​
  2. B
    3+4γγ−1\frac{3+4 \gamma}{\gamma-1}γ−13+4γ​
  3. C
    4−3γγ−1\frac{4-3 \gamma}{\gamma-1}γ−14−3γ​
  4. D
    3−4γγ−1\frac{3-4 \gamma}{\gamma-1}γ−13−4γ​
View written solutionFree

Correct answer: C

A polygamic gas has 3 translational, 3 rotational and $f$ vibration modes

$$\begin{aligned} & U=\frac{3}{2} k_B T+\frac{3}{2} k_B T+f k_B T \\ & U=(3+f) k_B T \\ & C_V=(3+f) R \\ & C_p=(4+f) R \\ & \frac{C_p}{C_v}=\frac{4+f}{3+f}=\gamma \\ & 4+f=3 \gamma+f \gamma \\ & 4-3 \gamma=f(\gamma-1) \Rightarrow f=\frac{4-3 \gamma}{\gamma-1} \end{aligned}$$

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