The first law of thermodynamics relates internal energy

to heat added

and work added

:

. (1)

For an adiabatic system, heat transfer is zero:

. (2)

By the first law, applied to both tanks together, no work is done across the system boundary (the walls of the tanks):

. (3)

, (4)

. (5)

, (6)

where

represents the number of moles,

is the constant-volume heat capacity,

is absolute temperature,

is a reference temperature and

is the ideal gas constant. Subscript

represents initial conditions, and subscripts

and

represent the final conditions in tanks A and B.

Equation (6) simplifies to

. (7)

Equation (7) means the average temperature of the system (

) does not change, so the final pressure (

) is determined from the ideal gas law:

,

where the initial volume of tank A is 22.4 L.

The total moles in the system does not change:

. (8)

The expansion of the gas that remains in tank A is modeled as a reversible adiabatic expansion. This gas does work in pushing the rest of the gas out of tank A and through the valve. For an ideal gas, the final temperature in tank A is related to its initial temperature

and pressure

and its final pressure

by:

, (9)

where

, the heat capacity ratio.