Question

For a gas having $X$ degrees of freedom, $\gamma$ is

• A. $\frac{1+X}{2}$
• B. $1+\frac{X}{2}$
• C. $1+\frac{2}{X}$
• D. $1+\frac{1}{X}$

Hint:

Logic Tip: This formula helps calculate the adiabatic constant for monoatomic, diatomic, and polyatomic gases based on their rotational and vibrational freedom.

Answer 1

The Correct Option is C

Concept:
For an ideal gas, the ratio of specific heats is: \[ \gamma=\frac{C_P}{C_V} \] Using kinetic theory, if the gas molecule has $f$ degrees of freedom, then: \[ C_V=\frac{f}{2}R \] and \[ C_P=C_V+R=\frac{f}{2}R+R=\frac{f+2}{2}R \] Therefore: \[ \gamma=\frac{C_P}{C_V} \] \[ \gamma=\frac{\frac{f+2}{2}R}{\frac{f}{2}R} \] \[ \gamma=\frac{f+2}{f} \]
Step 1: Given degrees of freedom
\[ f=X \] Substitute into formula: \[ \gamma=\frac{X+2}{X} \]
Step 2: Simplify the expression
\[ \gamma=\frac{X}{X}+\frac{2}{X} \] \[ \gamma=1+\frac{2}{X} \]
Step 3: Final Answer
The ratio of specific heats is: \[ \boxed{\gamma=1+\frac{2}{X \] Quick Tip:
For any ideal gas, once degrees of freedom are known, use directly: \[ \gamma=\frac{f+2}{f} \]