Question

Which equation shows the correct form of Berthelot equation?
A. $\left( {{{P}} + \dfrac{{{a}}}{{{{T}}{{\left( {{{V}} + {{C}}} \right)}^2}}}} \right)\left( {{{V}} - {{b}}} \right) = {{RT}}$
B. $\left( {{{P}} + \dfrac{{{a}}}{{{{T}}{{\left( {{{V - C}}} \right)}^2}}}} \right)\left( {{{V}} - {{b}}} \right) = {{RT}}$
C. $\left( {{{P}} + \dfrac{{{a}}}{{{{T}}{{{V}}^2}}}} \right)\left( {{{V}} - {{b}}} \right) = {{RT}}$
D. $\left( {{{P}} + \dfrac{{{a}}}{{{{T}}{{{V}}^2}}}} \right)\left( {{{V + b}}} \right) = {{RT}}$

Answer 1

Hint:Berthelot equation is also called Berthelot equation of state. Equation of state explains about the relationship among the volume, pressure and the temperature of any substance. This equation is almost similar to van der Waals equation of state. There are four types of equations of state.

Complete answer:
Equation of state is an equation which is related to thermodynamics. The variables like volume, temperature and pressure are called state variables. We can determine the characteristics of fluids and solids using these equations of state. These equations of state are for real gases. There occurs inelastic collisions between the gas particles. Thus pressure will be less than that of ideal gases.
The types of equations of state are van der Waals, Bealltie-Bridgeman, Benedict-Webb-Rubin and the Virial equation of state.
The equation in option A is called Clausius’s equation. The equation is given below:
$\left( {{{P}} + \dfrac{{{a}}}{{{{T}}{{\left( {{{V}} + {{C}}} \right)}^2}}}} \right)\left( {{{V}} - {{b}}} \right) = {{RT}}$
The Berthelot equation of state is slightly different from van der Waals equation of state. Berthelot equation of state is given below:
$\left( {{{P}} + \dfrac{{{a}}}{{{{T}}{{{V}}^2}}}} \right)\left( {{{V}} - {{b}}} \right) = {{RT}}$
${{P}}$ is the pressure of the gas
${{a}}$ is the correction for attractive forces between molecules
${{T}}$ is the temperature
${{V}}$ is the volume
${{b}}$ is the volume correction of molecules
${{R}}$ is the ideal gas constant
${{a}}$ and ${{b}}$ are also known as the van der Waals coefficients.
Berthelot introduced a pressure correction along with the volume correction. He placed $\dfrac{{{a}}}{{{{T}}{{{V}}^2}}}$ instead of $\dfrac{{{a}}}{{{{{V}}^2}}}$ in the van der Waals equation of state.

Hence, option C is correct.

Note:
Van der Waals equation of state is given below:
$\left( {{{P}} + \dfrac{{{a}}}{{{{{V}}^2}}}} \right)\left( {{{V}} - {{b}}} \right) = {{RT}}$
This equation corrects volume and force of attraction between the gas molecules from the ideal gas equation.