Question

Which of the following equation establishes the correct relationship between molar specific heat at constant volume C_V, adiabatic constant γ and universal gas constant R?
1. \(C_V = R\gamma ^2\)
2. \(C_V = (R+1)\gamma\)
3. \(C_V=\frac{R}{\gamma - 1}\)
4. \(C_V = (\gamma +2)R^2\)

Answer 1

Correct Answer - Option 3 : \(C_V=\frac{R}{\gamma - 1}\)

The correct answer is option 3) i.e. \(C_V=\frac{R}{\gamma - 1}\)

CONCEPT:

• Molar specific heat of gas is defined as the amount of heat required to raise the temperature of one mole of any gas by 1^∘C.
  • The molar specific heat measured at constant volume is called molar specific heat at constant volume (C_V).
  • The molar specific heat measured at constant pressure is called molar specific heat at constant pressure (CP).
  • The relation between C_V, C_P, and universal gas constant R is: C_P - C_V = R

Adiabatic constant, \(\gamma = \frac{C_P}{C_V}\)

EXPLANATION:

Using, C_P - C_V = R

\(\Rightarrow C_V(\frac{C_P}{C_V} -1) = R\)

Substituting  \(\gamma = \frac{C_P}{C_V}\),

\(\Rightarrow C_V(\gamma-1) = R\)

\(\Rightarrow C_V=\frac{R}{\gamma - 1}\)