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For a monoatomic gas
1. Molar heat capacity at constant volume, \(C_v = \dfrac{3}{2} R\)
2. Molar heat capacity at constant pressure, \(C_P=\dfrac{3}{2} R\)
3. the ratio of Cp and Cv is \(\dfrac{3}{2}\)
4. the difference between Cp and Cv is 2R

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Correct Answer - Option 1 : Molar heat capacity at constant volume, \(C_v = \dfrac{3}{2} R\)

CONCEPT:

  • The molar specific heat capacity of a gas at constant volume is defined as the amount of heat required to raise the temperature of 1 mol of the gas by 1 °C at the constant volume.

\({C_v} = {\left( {\frac{\Delta Q}{{n\Delta T}}} \right)_{constant\;volume}}\)

  • The molar specific heat of a gas at constant pressure is defined as the amount of heat required to raise the temperature of 1 mol of the gas by 1 °C at the constant pressure.

\({C_p} = {\left( {\frac{{\Delta Q}}{{n\Delta T}}} \right)_{constant\;pressure}}\)

Monatomic Gas:

  • A monatomic gas is one in which atoms are not bound to each other.
  • Monatomic gas consists of single atoms.
  • Example: argonkrypton, and xenon.

EXPLANATION:

  • For a monoatomic gas, the molar heat capacity at constant volume is given as,

\(⇒ C_v = \dfrac{3}{2} R\)

  • Hence, option 1 is correct.

  • For a monoatomic gas, the molar heat capacity at constant pressure is given as,

\(⇒ C_P = \dfrac{5}{2} R\)

  • For a monoatomic gas, the ratio of Cp and Cv is,

\(⇒ \frac{C_{p}}{C_{v}}=\frac{5}{2}\times\frac{2}{3}\)

\(⇒ \frac{C_{p}}{C_{v}}=\frac{5}{3}\)

\(⇒ \frac{C_{p}}{C_{v}}=1.67\)

The difference between Cp and Cv is given by Mayer's formula,

⇒ Cp - Cv = R

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