Negative or Positive Partial Derivative

[Saptarshi Sarkar]

Homework Statement

A non-ideal gas is described by the VdW law

##(P+an^2)(1-nb) = nkT##

where a,b and k are positive constants, n is the density of particles, P the pressure, and T the temperature.
How will n change (increase or decrease) if we

1) Hold P and increase T
2) Hold T and increase P

Relevant Equations

##(P+an^2)(1-nb) = nkT##

My attempt

I calculated the partial derivatives of n wrt P and T. They are given below.

##\frac {\partial n}{\partial P} = \frac{nb -1}{\left(2an-Pb-3abn^2-kT\right )}##
##\frac {\partial n}{\partial T}= \frac {nk}{\left(2an-Pb-3abn^2-kT \right ) }##

I know that if the partial derivative is positive, n should increase with the increase in the independent variable and if the partial derivative is negative, n should decrease with the increase in the independent variable. But, I am not sure how to determine if the above partial derivatives are positive or negative.

 

[vela]

Both ##(P+an^2)## and ##nkT## are positive, so you know that ##(1-nb)## is positive. Try expressing your partial derivatives in terms of those combinations.

 

[Saptarshi Sarkar]

Both ##(P+an^2)## and ##nkT## are positive, so you know that ##(1-nb)## is positive. Try expressing your partial derivatives in terms of those combinations.

Thanks a lot!