What is the final temperature of an isentropic expansion with a given function?

[avocadogirl]

Homework Statement

I have a system u =Av^-2exp(s/R)

N moles of this substance initially at T₀ and P₀ are expanded isentropically until the pressure is halved. What is the final temperature?

Homework Equations

du = $$\partial$$u$$\partial$$s ds + $$\partial$$u$$\partial$$v dv
du = Tds - Pdv

The Attempt at a Solution

If this function is changing isentropically, ds would have to be zero, correct? The derivative of v would have to be such as to multiply P times 1/2, correct? But, if ds --> zero, how do I find my final temperature as a multiple of my initial temperature? Do I have to start plugging values which would satisfy these conditions back into the original equation?

 

[Mapes]

How about expressing T as $(\partial U/\partial S)_V$ and P as $-(\partial U/\partial V)_S$ and looking at how the volume changes during the process?