Work and enthalpy

When a gas expands against a constant external pressure, p_ex, it pushes the surroundings back and does work:

w = –p_ex ∆V.

The minus sign follows the sign convention: when the gas expands (∆V positive), it does work and loses energy, so w is negative. If a gas expands into a vacuum, there is nothing to push against (p_ex = 0), so w = 0. This is a free expansion.

A reversible change happens through a series of steps so small that the system is always almost at equilibrium; it could be turned back at any moment by a tiny change. For an ideal gas expanding reversibly at constant temperature (isothermally):

w = –2.303 nRT log (V_f / V_i).

A reversible expansion gives the most work that can be got from the change. For an ideal gas at constant temperature, ∆U = 0 (its energy depends only on temperature), so q = –w: all the heat taken in is turned into work.

Most reactions happen in open vessels, at constant pressure. For these we use enthalpy: H = U + pV. At constant pressure, the heat absorbed equals the change in enthalpy: ∆H = q_p. If heat is given out (an exothermic reaction), ∆H is negative; if heat is taken in (endothermic), ∆H is positive.

∆H and ∆U differ only when gases are made or used up:

∆H = ∆U + ∆n_g RT, where ∆n_g = moles of gaseous products – moles of gaseous reactants. Count only gases. For N₂(g) + 3H₂(g) → 2NH₃(g): ∆n_g = 2 – 4 = –2.

Properties are of two kinds. Extensive properties depend on the amount of matter: mass, volume, U, H, heat capacity. Intensive properties do not: temperature, pressure, density. A cup of tea and a pot of the same tea have the same temperature, but the pot has more mass and more energy.