text: Entropy is a function of state, and therefore the entropy change can be computed directly from the knowledge of the final and initial equilibrium states. For an ideal gas, the change in entropy is the same as for isothermal expansion where all heat is converted to work:
  For an ideal monatomic gas, the entropy as a function of the internal energy , volume , and number of moles  is given by the Sackur–Tetrode equation:
  In this expression  is the particle mass and  Planck's constant. For a monatomic ideal gas , with  the molar heat capacity at constant volume. 
  A second way to evaluate the entropy change is to choose a route from the initial state to the final state where all the intermediate states are in equilibrium. Such a route can only be realized in the limit where the changes happen infinitely slowly. Such routes are also referred to as quasistatic routes. In some books one demands that a quasistatic route has to be reversible, here we don't add this extra condition. The net entropy change from the initial state to the final state is independent of the particular choice of the quasistatic route, as the entropy is a function of state.
  Here is how we can effect the quasistatic route. Instead of letting the gas undergo a free expansion in which the volume is doubled, a free expansion is allowed in which the volume expands by a very small amount . After thermal equilibrium is reached, we then let the gas undergo another free expansion by  and wait until thermal equilibrium is reached. We repeat this until the volume has been doubled. In the limit  to zero, this becomes an ideal quasistatic process, albeit an irreversible one. Now, according to the fundamental thermodynamic relation, we have:
  As this equation relates changes in thermodynamic state variables, it is valid for any quasistatic change, regardless of whether it is irreversible or reversible. For the above defined path we have that  and thus , and hence the increase in entropy for the Joule expansion is
  A third way to compute the entropy change involves a route consisting of reversible adiabatic expansion followed by heating. We first let the system undergo a reversible adiabatic expansion in which the volume is doubled. During the expansion, the system performs work and the gas temperature goes down, so we have to supply heat to the system equal to the work performed to bring it to the same final state as in case of Joule expansion. 
  During the reversible adiabatic expansion, we have . From the classical expression for the entropy it can be derived that the temperature after the doubling of the volume at constant entropy is given as:
  for the monoatomic ideal gas. Heating the gas up to the initial temperature  increases the entropy by the amount
  We might ask what the work would be if, once the Joule expansion has occurred, the gas is put back into the left-hand side by compressing it. The best method (i.e. the method involving the least work) is that of a reversible isothermal compression, which would take work  given by
  During the Joule expansion the surroundings do not change, i.e. the entropy of the surroundings is constant. Therefore the entropy change of the so-called "universe" is equal to the entropy change of the gas which is .