Two cylinders A and B of equal capacity are connected to each other via a stopcock.

A contains a gas at standard temperature and pressure. B is completely evacuated.

The entire system is thermally insulated. The stopcock is suddenly opened. Answer the following:

(a) What is the final pressure of the gas in A and B?

(b) What is the change in internal energy of the gas?

(c) What is the change in the temperature of the gas?

(d) Do the intermediate states of the system (before settling to the final equilibrium state) lie on its P-V-T surface?

Now here two insulated cylinders are connected to each other via a stop cock A contains a gas at standard temperature and pressure, and cylinder B is empty.

as shown in figure

Now as the cock is suddenly opened the gas from A will rush towards B, i.e. free expansion of gas will take place and the process will be very fast and not Quasi-static

(a) For free expansion of a gas we have the relation

P₁V₁ = P₂V₂

Where P₁ and P₂ are initial and final pressures of the system respectively, V₁ and V₂ are initial and final volume of the system respectively

Now we know initial system is container A only because gas is in A only but final system is Both container A and B combined and gas in also filled in container B uniformly

As shown in Figure

Now since initially system was at Standard Temperature and Pressure initial pressure of the system would be

P₁ = 1 atm

Let volume of both the cylinders be V

Since initially system was only container A so initial volume of the system would be

V₁ = V

We have to find final pressure which would be same in both the cylinders, let it be P₂

Now finally gas is distributed in both the cylinders so final volume will be the total volume of both the cylinders i.e.

V₂ = 2V

So applying

P₁V₁ = P₂V₂

We have

1atm × V = P₂ × 2V

or P₂ = 1/2 atm = 0.5 atm

so the final pressure in both the cylinders would be half of initial pressure in cylinder A and will be 0.5 atm

(b) now in free expansion no work is done by gas and system is already insulated so no heat transfer has taken place to or from the system, for internal energy of a gas to change either work should be done by or on the gas or heat should be transferred to or from the gas , but here none has happened so internal energy would remain same or constant or change in internal energy of gas is 0, i.e.

ΔU = 0

(c) In case of free expansion of gas no heat is transferred to or from the system, no work is done i.e. no change of internal energy of gas and temperature of a gas changes when its internal energy changes so the temperature remain constant, i.e. no change in temperature or we can say the change in temperature is 0, i.e.

ΔT = 0

(d) Free expansion is a very fast process , and not Quasi-static so no stable intermediate equilibrium states exist and hence the intermediate states do not satisfy the gas equation, they do not lie on the P-V-T surface of the system.