In a free expansion of a perfect gas also called Joule expan

In a free expansion of a perfect gas (also called Joule expansion), we know U does not change, and no work is done. However, the entropy must increase because the process is irreversible. Are
these statements compatible with the rst law dU = TdS pdV ?

Solution

U does not change means dU=0

so dU=TdS-pdV

0=TdS-pdV

TdS=pdV

dS= pdV / T

from pv=nRT

p/T =nR/v

from dS= pdV / T = nRdV / V = nR(lnV) from this equation we can say entropy will increase