Question

The gas equation for a real gas is: P(V-b) = RT. Here the parameter ‘b’ is Van der Waals constant. The graph of pressure against temperature (isochore) will give a straight line of slope:
[A] Zero
[B] $\dfrac{R}{\left( V-b \right)}$
[C] $\dfrac{R}{P}$
[D] Negative

Answer 1

HINT: Gas equation gives us a relation between pressure temperature and volume for real gases. Real gases do not follow the gas equation of ideal gases. To solve this compare the given equation with the equation of a straight line.


COMPLETE STEP BY STEP SOLUTION: The ideal gas law equation is an equation of state variables of a hypothetical ideal gas. We know that the ideal gas equation is – PV = nRT where, P is the pressure, V is the volume of the gas, n is the number of moles, R is the universal gas constant which has a fixed value and T is the temperature.
However, for a real gas we make some changes to this ideal gas equation. We add a pressure term and subtract a volume term.
However, excluding the extra pressure term it is given as- P (V-b) = RT.
We know that the equation of a straight line is- y = mx + c
Where, m is the slope and c is the intercept and x and y depicts the two axes.
Rearranging equation P (V-b) = RT we can write that –
\[P=\dfrac{R}{(V-b)}T\]
Therefore, we have the relation between pressure and temperature for an isochoric curve.
Now if we compare this equation with the straight line equation we will see that the slope m is $\dfrac{R}{(V-b)}$ .
Hence, we can see from the above discussion that the slope of pressure against the temperature graph will be $\dfrac{R}{(V-b)}$.

Therefore, the correct answer is option [B] $\dfrac{R}{\left( V-b \right)}$.


NOTE: The ideal gas law equation is an equation of state variables of a hypothetical ideal gas. It has many limitations but still used for approximation of behaviour of a gas under certain conditions. It is a combination of Boyle’s law, Avogadro’s law, Charles’s law and Gay-Lussac’s law. Real gases do not obey the ideal gas law and therefore a gas equation for added pressure and subtracted volume is given which holds true for real gases. The equation is -
\[\left( P+\dfrac{a{{n}^{2}}}{{{V}^{2}}} \right)\left( V-nb \right)=RT\]