Thermodynamics

1. Fundamentals and the First Law

Thermodynamics is the study of energy transformations. In JEE/NEET, we focus on macroscopic properties and state functions. A State Function (like $U$, $H$, $S$, $G$) depends only on the current state, not the path taken.

The First Law

Energy cannot be created or destroyed, only transformed:

• Work ($w$): For expansion against constant external pressure, $w = -P_{ext}\Delta V$. For reversible isothermal expansion, $w = -2.303nRT \log(V_2/V_1)$.
• Enthalpy ($H$): Defined as $H = U + PV$. At constant pressure, $\Delta H = q_p$.

2. The Second Law and Spontaneity

The universe tends toward disorder. Entropy ($S$) is a measure of this randomness. For a spontaneous process, $\Delta S_{total} = \Delta S_{sys} + \Delta S_{surr} > 0$.

Gibbs Free Energy ($G$)

The ultimate criterion for spontaneity at constant $T$ and $P$:

• $\Delta G < 0$: Spontaneous (Exergonic).
• $\Delta G > 0$: Non-spontaneous (Endergonic).
• $\Delta G = 0$: Equilibrium.

3. Relating $\Delta H$ and $\Delta U$

For reactions involving gases, $\Delta H = \Delta U + \Delta n_g RT$. For $N_2 + 3H_2 \to 2NH_3$, $\Delta n_g = -2$, so $\Delta H < \Delta U$. For reactions with only solids and liquids, $\Delta H \approx \Delta U$.

4. Hess's Law and Enthalpies of Reaction

Because $H$ is a state function, the enthalpy change of a reaction is the same whether it happens in one step or many. Thermochemical equations can be added and subtracted like algebra.

• $\Delta_r H^\circ = \sum \Delta_f H^\circ(products) - \sum \Delta_f H^\circ(reactants)$
• $\Delta_r H^\circ = \sum$ bond enthalpies (bonds broken) $- \sum$ bond enthalpies (bonds formed), for gas-phase reactions.
• $\Delta_f H^\circ$ of an element in its standard state is zero (graphite, not diamond, for carbon).

5. Entropy Changes

• $\Delta S = \frac{q_{rev}}{T}$. For a phase change at its transition temperature, $\Delta S = \frac{\Delta H}{T}$.
• Entropy increases on melting, vapourisation, dissolving most solids, and when $\Delta n_g > 0$.
• Third law: the entropy of a perfect crystal at 0 K is zero.

When is a reaction spontaneous?

• $\Delta H < 0$, $\Delta S > 0$: spontaneous at all temperatures.
• $\Delta H > 0$, $\Delta S < 0$: never spontaneous.
• $\Delta H < 0$, $\Delta S < 0$: spontaneous at low temperature.
• $\Delta H > 0$, $\Delta S > 0$: spontaneous at high temperature, above $T = \Delta H / \Delta S$.

Interactive diagram: open this page in a browser to use it.

6. Work in Different Processes

• Isothermal reversible expansion of an ideal gas: $\Delta U = 0$, so $q = -w$.
• Free expansion (into vacuum): $w = 0$.
• Adiabatic process: $q = 0$, so $\Delta U = w$.
• Extensive properties (mass, volume, $U$, $H$) depend on amount; intensive properties (T, P, density) do not.

Practice questions

1. For an isothermal reversible expansion of an ideal gas:

   1. Δ U = 0
   2. q = 0
   3. w = 0
   4. Δ H > 0
   Answer

   A. For an ideal gas U depends only on T; at constant T, Δ U = 0.

2. A reaction with Δ H > 0 and Δ S > 0 is spontaneous:

   1. At all temperatures
   2. At no temperature
   3. Only at high temperatures
   4. Only at low temperatures
   Answer

   C. Δ G = Δ H - TΔ S becomes negative only when T > Δ H/Δ S.

3. For N₂(g) + 3H₂(g) → 2NH₃(g) at constant temperature:

   1. Δ H = Δ U
   2. Δ H > Δ U
   3. Δ H < Δ U
   4. Δ H = 0
   Answer

   C. Δ H = Δ U + Δ n_g RT with Δ n_g = -2, so Δ H < Δ U.

4. Which of the following is an intensive property?

   1. Volume
   2. Enthalpy
   3. Temperature
   4. Internal energy
   Answer

   C. Temperature does not depend on the amount of substance.

5. The standard enthalpy of formation is zero for:

   1. Diamond
   2. Graphite
   3. O₃(g)
   4. H₂O(l)
   Answer

   B. Graphite is the standard (most stable) state of carbon; diamond and ozone are not standard states.