Question

${\mathrm{C}}_{\mathrm{p}}-{\mathrm{C}}_{\mathrm{v}}$ for an ideal gas is $\mathrm{R}$. State whether the statement is true or false.

• A. True
• B. False

Answer 1

The correct option is A True

Derivation for the relation between ${\mathrm{C}}_{\mathrm{p}}$ and ${\mathrm{C}}_{\mathrm{v}}$

At constant pressure and constant volume, the heat capacities are denoted by ${\mathrm{C}}_{\mathrm{p}}$ and ${\mathrm{C}}_{\mathrm{v}}$ respectively.

• Therefore, the heat at constant pressure can be defined as:
  ${\mathrm{q}}_{\mathrm{p}}={\mathrm{C}}_{\mathrm{p}}\cdot \mathrm{\Delta T}$ ; where $\mathrm{T}$ is absolute temperature.
• We know that in a chemical change, the released heat at constant pressure is defined as change in enthalpy.
• Thus, $\mathrm{\Delta H}={\mathrm{q}}_{\mathrm{p}}={\mathrm{C}}_{\mathrm{p}}\cdot \mathrm{\Delta T}$
• The heat at constant volume can be defined as:
  ${\mathrm{q}}_{\mathrm{v}}={\mathrm{C}}_{\mathrm{v}}\cdot \mathrm{\Delta T}$
• We know that in a chemical change, the released heat at constant volume is defined as change in internal energy. Thus,
  $\mathrm{\Delta U}={\mathrm{q}}_{\mathrm{v}}={\mathrm{C}}_{\mathrm{v}}\cdot \mathrm{\Delta T}$
• Now, for a mole of ideal gas, enthalpy is equal to the sum of internal energy and the product of pressure and volume:
  $\mathrm{H}=\mathrm{U}+\mathrm{PV}$ ; where $\mathrm{P}$ is pressure and $\mathrm{V}$ is volume.
• The expression for change in these quantities will be:
  $\mathrm{\Delta H}=\mathrm{\Delta U}+\mathrm{\Delta }\left(\mathrm{PV}\right)$
• From ideal gas equation, substituting $PV=\mathrm{RT}$ in this expression:
  $\mathrm{\Delta H}=\mathrm{\Delta U}+\mathrm{\Delta }\left(\mathrm{RT}\right)$
  $\mathrm{R}$ is universal gas constant
• Thus: $\mathrm{\Delta H}=\mathrm{\Delta U}+\mathrm{R\Delta T}$
• Substituting the values of $\mathrm{\Delta H}$ and $\mathrm{\Delta U}$ in the above expression, we get:
  ${\mathrm{C}}_{\mathrm{p}}\mathrm{\Delta T}={\mathrm{C}}_{\mathrm{v}}\mathrm{\Delta T}+\mathrm{R\Delta T}$
• Dividing this equation by $\mathrm{\Delta T}$ , we get:
  ${\mathrm{C}}_{\mathrm{p}}={\mathrm{C}}_{\mathrm{v}}+\mathrm{R}$
• Rearranging the above expression:
  $C_p-C_v=R$
  Hence, the given statement is true.