# deduce the relationship between critical constants and vander walls constants.

SAGAR SINGH - IIT DELHI
879 Points
12 years ago

Dear student,

.

Now we can take the derivatives in Equations 1 and 2 and set them (independently) equal to zero.

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In order to stress that from here on the problem is pure algebra, let's rewrite the simultaneous equations that must be solved for the two unknowns V and T (which solutions we will call VC and TC

),

There are several ways to solve simultaneous equations. One way is to multiply Equation (6) by,

to get

Now add equations (7) and (8). Note that in this addition the terms containing T will cancel out leaving,

Divide Equation (9) by 2an2 and multiply it by V 3 (and bring the negative term to the other side of the equal sign) to get,

which is easily solved to get

which "cleans up" to give,

or

The critical pressure is obtained by substituting VC and TC into the van der Waals equations of state as solved for p in Equation (3).

This simplifies to,