Question

In this problem, we will investigate how the molar volume of a gas depends on the temperature in two real gas representations. Recall that the van der Waals equation for a real gas takes the form p= V m

​
−b
RT
​
+ V m
2
​
a
​
, where a and b are constants and V m
​
is the molar volume. The virial expansion (up to second virial coefficient) is p= V m
​
RT
​
[1+ V m
​
B
​
], where B is the second virial coefficient. Evaluate the derivative ( ∂T
∂V m
​
​
) p
​
for each gas and show that, for the van dew Waals equation you get ( ∂T
∂V m
​
​
) p
​
= T− RV m
3
​
2a(V M
​
−b) 2
​
V m
​
−b
​
and for the virial equation you get (assuming that B is temperature independent) ( ∂T
∂V m
​
​
) p
​
= T
V m
​
​
(V m
​
+2B)
(V m
​
+B)
​
. Also, verify that the derivatives reduce to the value of the derivative for the ideal gas law. NOTE: You cannot differentiate V m
​
(T,P) because the state functions are not linear in V m
​
. Instead, review the two rules from the first set of lecture notes (reciprocal rule and cyclic rule). These will be required.

Answer 1

To evaluate the derivatives (∂T/∂Vm)p for the van der Waals equation and the virial equation, we can apply the reciprocal rule and the cyclic rule, respectively. The derivatives can be obtained by differentiating the equations and they can be used to verify the derivative for the ideal gas law.

Step-by-step explanation:

To evaluate the derivative ( ∂T/∂Vm )p for the van der Waals equation, we can apply the reciprocal rule. Using this rule, we can write the equation as (∂Vm/∂T)p = (∂T/∂Vm)p-1.

Similarly, for the virial equation, we can apply the cyclic rule to evaluate the derivative (∂T/∂Vm)p = (∂Vm/∂T)p-1.

By differentiating the van der Waals equation and the virial equation using these rules, we can obtain the respective derivatives and verify that they reduce to the derivative for the ideal gas law.

Learn more about Derivatives of gas equations