Kinetic Theory

ExplanationKey Concepts:

1. Degrees of Freedom:

   • A rigid diatomic molecule has 5 degrees of freedom (3 translational + 2 rotational).
   • A diatomic molecule with vibrational modes has additional vibrational degrees of freedom.

2. Specific Heats ($C_v$​ and $C_p$):

   • The internal energy is proportional to the degrees of freedom. For a rigid diatomic gas: $$C_v = \frac{5R}{2}$$​
   • When vibrational modes are included, the total degrees of freedom increase. Each vibrational mode adds $2R$, making: $$C_v = \frac{5R}{2} + 2R = \frac{9R}{2}$$​
   • $C_p$​ is related to $C_v$ by: $$C_p = C_v + R$$

3. Adiabatic Index ($\gamma$γ):

   • The adiabatic index $\gamma$γ is defined as: $$\gamma = \frac{C_p}{C_v} = 1 + \frac{R}{C_v}$$​
   • For the rigid diatomic gas: $$\gamma_1 = \frac{C_p}{C_v} = \frac{\frac{7R}{2}}{\frac{5R}{2}} = \frac{7}{5} = 1.4$$
   • For the diatomic gas with vibrational modes: $$\gamma_2 = \frac{C_p}{C_v} = \frac{\frac{11R}{2}}{\frac{9R}{2}} = \frac{11}{9} \approx 1.22$$

Step 2: Compare $\gamma_1$​ and $\gamma_2$​

Clearly, $\gamma_2 < \gamma_1$ because the inclusion of vibrational modes increases $C_v$​ more significantly than $C_p$, leading to a smaller $\gamma$.

Final Answer:

(A) $\gamma_2 < \gamma_1$​

Explanation

For a diatomic gas, the adiabatic index y is given by:

$\gamma =\frac{C_p}{C_v}=\frac{f+2}{f}$

without vibration: f = 5

${\gamma }_1=\frac{7}{5}=1.4$

When vibrational modes are included, $C_v$ increases, reducing ${\gamma }_2$.
Thus, ${\gamma }_2<{\gamma }_1$.

Explanation

The translational kinetic energy ($KE$) of an ideal gas is given by:

$$KE = \frac{3}{2} nRT$$

where:

• $n$ = number of moles of $CO_2$​
• $R$ = universal gas constant ($8.314$ J/mol·K)
• $T$T = absolute temperature in Kelvin ( $17^\circ C = 290K$)

Step 1: Calculate the number of moles of $CO_2$​

$$n = \frac{\text{Mass of gas}}{\text{Molar mass of } CO_2}$$

​ $$n = \frac{50}{44} = 1.136 \text{ moles}$$

Step 2: Compute Translational Kinetic Energy

$$KE = \frac{3}{2} \times 1.136 \times 8.314 \times 290$$

$$KE = \frac{3}{2} \times 1.136 \times 2411.06$$

$$KE = 4110 J$$

Thus, the correct answer is Option C (4110 J).