First Law of Thermodynamics & Specific Heat Capacities

The First Law of Thermodynamics is the macroscopic generalization of the law of conservation of energy applied to thermodynamic systems. It establishes that heat and mechanical work are mutually convertible forms of energy, and energy can neither be created nor destroyed.

1. Mathematical Formulation of the First Law

When a quantity of heat ΔQ is supplied to a thermodynamic system, it is utilized in two distinct ways:

1. To increase the internal energy of the system by ΔU.
2. To perform external work ΔW on the surroundings.

In differential form for an infinitesimal quasi-static process:

2. Molar Specific Heat Capacities of Gases

For solids and liquids, thermal expansion upon heating is negligible, so specific heat capacity has a unique value. For gases, however, the amount of heat required to raise the temperature of 1 mole by 1 K depends heavily on how the volume and pressure are allowed to change during heating:

Depending on the process, C can range from −∞ to +∞. Two principal molar specific heats are defined:

• Molar Heat Capacity at Constant Volume (C_v): The amount of heat required to raise the temperature of 1 mole of gas by 1 K while keeping volume constant. Since dV = 0, dW = 0, we have:
  C_v = (1 / n) (dU / dT)   ⇒   dU = n C_v dT
• Molar Heat Capacity at Constant Pressure (C_p): The amount of heat required to raise the temperature of 1 mole of gas by 1 K at constant pressure. Here, heat supplied increases internal energy AND does expansion work against constant pressure. Thus, C_p > C_v always.

3. Derivation of Mayer's Relation (C_p − C_v = R)

Consider 1 mole (n = 1) of an ideal gas governed by the equation of state P V = R T:

1. At constant volume: dQ_v = dU = C_v dT.
2. At constant pressure: dQ_p = C_p dT. From the first law:
   dQp = dU + P dV = Cv dT + P dV
3. Differentiating the ideal gas equation P V = R T at constant pressure P:
   P dV = R dT
4. Substituting P dV into the first law equation:
   Cp dT = Cv dT + R dT
5. Dividing throughout by dT yields Mayer's equation:

4. Degrees of Freedom (f) & Law of Equipartition of Energy

The degrees of freedom (f) of a molecule is the total number of independent coordinates or quadratic terms required to specify its position and configuration in space. According to the Law of Equipartition of Energy, the average kinetic energy associated with each degree of freedom per molecule in thermal equilibrium at temperature T is (1/2) k_B T, or (1/2) R T per mole.

Total internal energy of 1 mole of gas: U = (f / 2) R T.

Differentiating with respect to T:

5. Comparison Table for Monoatomic, Diatomic & Polyatomic Gases

6. Mixture of Non-Reacting Ideal Gases

When n₁ moles of a gas with molar heat capacity C_v1 are mixed with n₂ moles of a gas with molar heat capacity C_v2: