# For a monoatomic gas

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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