Thermodynamics

In thermodynamics, the system is the specific part of the universe we choose to study (for example, the gas inside a cylinder, or the reactants in a flask). Everything else in the universe that can exchange energy or matter with the system is called the surroundings. The system and surroundings are separated by a real or imaginary boundary.

Systems are classified into three types based on exchange of matter and energy with the surroundings:

• Open system: exchanges both matter and energy with the surroundings, e.g. hot tea in an open cup, or an open beaker of a reaction mixture.
• Closed system: exchanges only energy, not matter, e.g. a sealed test tube of reactants placed in a water bath, or gas in a closed, conducting cylinder fitted with a piston.
• Isolated system: exchanges neither matter nor energy, e.g. a reaction carried out in an ideal, perfectly insulated thermos flask.

The condition of a system at a given time, described by macroscopic variables such as pressure (p), volume (V), temperature (T) and amount (n), is called its state. Properties whose value depends only on the initial and final state of the system, and not on the path followed, are called state functions (e.g. internal energy U, enthalpy H, entropy S, Gibbs energy G). Quantities such as heat (q) and work (w) depend on the path taken and are called path functions. Properties are also classed as extensive (depend on the amount of matter present, e.g. volume, mass, internal energy) or intensive (independent of amount, e.g. temperature, pressure, density, molar volume).

Worked example: Classify the following — (i) tea kept in a closed, perfectly insulated thermos flask, (ii) hot water in an open beaker, (iii) a gas enclosed in a sealed but thermally conducting cylinder. Since (i) exchanges no heat and no matter with the surroundings, it is an isolated system. In (ii), both water vapour (matter) and heat can escape to the air, so it is an open system. In (iii), heat can pass through the conducting walls but no matter can leave the sealed cylinder, so it is a closed system.

The internal energy (U) of a system is the sum of all forms of kinetic and potential energy of its constituent particles. U is a state function; only its change, ΔU, between two states can be measured. Internal energy can change by transfer of heat (q), by work (w), or both.

The first law of thermodynamics (a statement of conservation of energy) states that the energy of an isolated system is constant, and gives: ΔU = q + w. Using the IUPAC sign convention, heat absorbed by the system is positive (q > 0) and heat lost by the system is negative; work done on the system is positive (w > 0) and work done by the system is negative.

For a gas expanding against a constant external pressure p_ext, the work done on the system is w = −p_extΔV (negative for expansion, positive for compression). If a gas expands into a vacuum (p_ext = 0), no work is done: w = 0 (free expansion). For an isothermal reversible expansion of an ideal gas from volume V₁ to V₂ at temperature T, w = −2.303 nRT log₁₀(V₂/V₁); since U of an ideal gas depends only on T, ΔU = 0 for any isothermal process, so q = −w. In an adiabatic process, q = 0, so ΔU = w exactly.

Worked example: One mole of an ideal gas expands isothermally and reversibly at 300 K from 10 L to 20 L (R = 8.314 J K^-1 mol^-1). Find w and q.

w = −2.303 × 1 × 8.314 × 300 × log₁₀(20/10) = −2.303 × 8.314 × 300 × 0.301 ≈ −1729 J.

Since the process is isothermal, ΔU = 0, so q = ΔU − w = 0 − (−1729) = +1729 J. The gas absorbs about 1.73 kJ of heat, all of which is used to do 1.73 kJ of work on the surroundings.

Most reactions are carried out in open vessels at constant atmospheric pressure, where some of the heat exchanged goes into pV-work. To handle this conveniently, we define enthalpy: H = U + pV. Enthalpy is a state function. At constant pressure, ΔH = ΔU + pΔV, and the heat exchanged at constant pressure equals ΔH (q_p = ΔH), while the heat exchanged at constant volume equals ΔU (q_v = ΔU). For a reaction involving ideal gases, ΔH = ΔU + Δn_gRT, where Δn_g is the difference between moles of gaseous products and gaseous reactants.

The heat capacity (C) of a system is the heat required to raise its temperature by one degree: C = q/ΔT. The specific heat capacity (s) is heat capacity per unit mass: s = q/(m·ΔT), so q = m·s·ΔT. The molar heat capacity is heat capacity per mole. For an ideal gas, the molar heat capacities at constant pressure and constant volume are related by C_p − C_v = R.

Calorimetry is the experimental measurement of heat changes. A bomb calorimeter is a sealed, rigid, insulated vessel — since volume is constant, no pV-work is done, and the heat measured directly gives ΔU (q_v = ΔU). A simple calorimeter open to constant atmospheric pressure measures q_p = ΔH directly.

Worked example: The internal energy change for the combustion of methane, CH₄(g) + 2O₂(g) → CO₂(g) + 2H₂O(l), is ΔU° = −742.7 kJ mol^-1 at 298 K. Find ΔH°.

Δn_g = (moles gaseous products) − (moles gaseous reactants) = 1 − (1 + 2) = −2.

ΔH° = ΔU° + Δn_gRT = −742.7 + (−2)(8.314 × 10^-3 kJ K^-1mol^-1)(298 K) = −742.7 − 4.96 = −747.7 kJ mol^-1.

The standard enthalpy of reaction (ΔrH°) is the enthalpy change of a reaction when all reactants and products are in their standard states (usually 1 bar pressure, specified temperature, pure form). Important named enthalpy changes include: standard enthalpy of formation (ΔfH°) — enthalpy change when 1 mole of a compound is formed from its elements in their most stable states (ΔfH° of an element in its standard state is zero); standard enthalpy of combustion (ΔcH°) — enthalpy change when 1 mole of a substance is completely burnt in excess oxygen; enthalpy of atomization — energy to convert 1 mole of a substance into gaseous atoms; and bond enthalpy — energy needed to break one mole of a particular bond in the gaseous state.

Hess's Law of Constant Heat Summation states that the total enthalpy change for a reaction is the same whether it takes place in a single step or in a series of steps, because enthalpy is a state function. This allows thermochemical equations to be added, subtracted, or multiplied algebraically to obtain the enthalpy change of a reaction that is difficult to measure directly. In particular, ΔrH° = ΣΔfH°(products) − ΣΔfH°(reactants), and for bond-enthalpy calculations, ΔrH° = Σ(bond enthalpies broken) − Σ(bond enthalpies formed).

Worked example: Given C(graphite) + ½O₂(g) → CO(g), ΔH₁ = −110.5 kJ mol^-1, and CO(g) + ½O₂(g) → CO₂(g), ΔH₂ = −283.0 kJ mol^-1, find ΔH for C(graphite) + O₂(g) → CO₂(g).

Adding the two given equations, CO cancels out, leaving the target equation directly.

ΔH = ΔH₁ + ΔH₂ = −110.5 + (−283.0) = −393.5 kJ mol^-1. This matches the known standard enthalpy of formation of CO₂(g), confirming that the two-step path and the direct path give the same total enthalpy change, as Hess's Law predicts.

A spontaneous process is one that occurs on its own, without any continuous external intervention, once initiated (e.g. heat flowing from hot to cold, a gas expanding to fill a vacuum, ice melting above 0°C). Spontaneity is a natural tendency, not a statement about the rate of the process — a spontaneous process may still be fast or extremely slow. Enthalpy change alone cannot decide spontaneity, because many spontaneous processes are endothermic (e.g. melting of ice above 0°C, dissolution of ammonium chloride in water).

The missing factor is entropy (S), a state function that measures the degree of randomness or disorder of a system; entropy generally increases on melting, vaporisation, dissolution, and whenever the number of gas moles increases. For a reversible process at temperature T, the entropy change is ΔS = q_rev/T. The second law of thermodynamics states that for any spontaneous process, the total entropy of the system plus surroundings always increases: ΔS_total = ΔS_sys + ΔS_surr > 0. At equilibrium, ΔS_total = 0. At constant temperature and pressure, the entropy change of the surroundings is related to the enthalpy change of the system by ΔS_surr = −ΔH_sys/T.

Worked example: The enthalpy of vaporisation of water is 40.7 kJ mol^-1 at its boiling point of 373 K. Calculate the entropy of vaporisation.

ΔS_vap = ΔH_vap/T_b = 40700 J mol^-1 / 373 K ≈ 109.1 J K^-1mol^-1. The large positive value reflects the big increase in disorder when liquid water turns into a much more randomly arranged vapour.

Checking ΔS_total (system + surroundings) is often inconvenient because it requires tracking the surroundings. J. W. Gibbs defined a state function, Gibbs energy: G = H − TS, that depends only on the system. At constant temperature and pressure, ΔG = ΔH − TΔS. This follows because ΔS_total = ΔS_sys − ΔH_sys/T, and multiplying by −T gives −TΔS_total = ΔH_sys − TΔS_sys = ΔG_sys, so ΔG_sys = −TΔS_total.

Since T is always positive (Kelvin scale), the sign of ΔG is opposite to that of ΔS_total, giving a purely system-based criterion: if ΔG < 0, the process is spontaneous; if ΔG = 0, the system is at equilibrium; if ΔG > 0, the process is non-spontaneous (the reverse process is spontaneous). Combining the signs of ΔH and ΔS shows four cases: (i) ΔH < 0, ΔS > 0 — spontaneous at all temperatures; (ii) ΔH > 0, ΔS < 0 — non-spontaneous at all temperatures; (iii) ΔH < 0, ΔS < 0 — spontaneous only below a certain temperature; (iv) ΔH > 0, ΔS > 0 — spontaneous only above a certain temperature. Gibbs energy is also linked to the equilibrium constant K by ΔG° = −RT ln K = −2.303 RT log₁₀K.

Worked example: For the reaction N₂(g) + 3H₂(g) → 2NH₃(g), ΔH° = −92.4 kJ mol^-1 and ΔS° = −198.3 J K^-1mol^-1. Is the reaction spontaneous at 298 K?

ΔG° = ΔH° − TΔS° = −92400 J mol^-1 − (298 K)(−198.3 J K^-1mol^-1) = −92400 + 59093.4 = −33306.6 J mol^-1 ≈ −33.3 kJ mol^-1.

Since ΔG° is negative, the reaction is spontaneous at 298 K, even though ΔS° is negative (unfavourable), because the large negative ΔH° dominates at this temperature.