A rigid container is divided into two compartments by a very thin frictionless piston that is held in place by a latch. The left side contains $1$ mole gas $A$ at $10$ bar and $750K.$ The right side contains $2$ mole gas B at $1$ bar and $400K.$ You may treat both gases as ideal gas with the same $C_v.$ After removal of the latch, the piston moves until both the temperature and pressure of the two compartments are equal. If the container is well$-$insulated,

$(1)$ What is the final temperature? $(15$ pts$)$

$(2)$ What is the final pressure? $(10$ pts$)$