Question

Which of the following is correct?
(Internal pressure of gas in Van Der Waals equation)
A. \[{{C}_{v}}={{\left( \dfrac{\partial v}{\partial T} \right)}_{p}}\]
B. \[{{C}_{p}}={{\left( \dfrac{\partial H}{\partial T} \right)}_{v}}\]
C.\[{{C}_{p}}-{{C}_{v}}=R\]
D. \[{{\left( \dfrac{\partial v}{\partial T} \right)}_{^{t}}}=\dfrac{-a}{{{v}^{2}}}\]

Answer 1

Hint: Van Der Waals equation of gases is given below:
\[\left( P+\dfrac{{{n}^{2}}a}{{{V}^{2}}} \right)\left( V-nb \right)=nRT\]
Where P = Pressure of gas
n = number of moles
V = Volume occupied by gas
T = Temperature
R = Gas constant
a = attraction constant Vander Waals constant
b = Van Der Waals correction constant

Complete step by step solution:
From Maxwell’s relative we know that,
\[{{C}_{v}}={{\left( \dfrac{\partial V}{\partial T} \right)}_{v}}\]
\[{{C}_{v}}\] is heat capacity at constant volume.
And \[{{C}_{p}}\] is heat capacity at constant pressure so,
\[{{C}_{p}}={{\left( \dfrac{\partial H}{\partial T} \right)}_{p}}\]
Also, \[{{C}_{p}}\] and \[{{C}_{v}}\]are specific heat capacities and constant pressure and constant volume respectively.

It is observed that in this case:
\[{{C}_{p}}-{{C}_{v}}=R\] where R = universal gas constant
For one mole of an ideal gas,
\[\Delta Q=\Delta U+P\Delta V\]
At constant volume
\[\Delta V=0\therefore {{C}_{v}}={{\left( \dfrac{\partial Q}{\partial T} \right)}_{v}}={{\left( \dfrac{\partial U}{\partial T} \right)}_{v}}\text{ }.............\text{ 1}\]
At constant pressure,
\[{{C}_{p}}={{\left( \dfrac{\partial Q}{\partial T} \right)}_{p}}=P{{\left( \dfrac{\partial U}{\partial T} \right)}_{p}}\text{ }.............\text{ 2}\]
For mole of ideal gas \[\text{PV}=\text{RT}\]
\[\Rightarrow \dfrac{\text{PV}}{\text{T}}=R\]

Performing equation 2 - equation 1 we get,
\[{{C}_{p}}-{{C}_{v}}=\left( \dfrac{\partial V}{\partial T} \right)+P{{\left( \dfrac{\partial V}{\partial T} \right)}_{P}}=R\]
Hence, we get \[{{C}_{p}}-{{C}_{v}}=R\]


Note:
\[{{C}_{p}}-{{C}_{v}}=R\]
The above mentioned relation is known as the mayor's relation. JuluisVan Mayor derived this relation between with specific heat at constant pressure and the specific heat at constant volume.Although this is strictly true for an ideal gas it is a good approximation for real gases. Using this we can calculate heat capacity under constant pressure if we know it’s value at constant volume and vice-versa.