B1-2. Internal Energy, State Changes and Heat Capacity

1. Internal Energy

The internal energy of a substance is the total energy stored within it due to its microscopic particles.

Definition of Internal Energy ($U$):

Internal energy is the sum of the total random kinetic energy and the total intermolecular potential energy of the particles in a substance.

  • Kinetic Energy ($E_k$): Arises from particle motion. It is directly related to the absolute temperature.
  • Potential Energy ($E_p$): Arises from intermolecular forces between particles. It is related to the phase (state) of the substance.

2. Thermal Equilibrium

Thermal energy naturally flows from a region of higher temperature to a region of lower temperature. Thermal Equilibrium is reached when two objects transfer thermal energy at equal rates, resulting in zero net heat transfer. They now share the exact same temperature.

3. The Heating Curve & Changes of State

When heating a substance, the thermal energy supplied either increases the temperature (by increasing kinetic energy) OR changes the state (by increasing potential energy to break bonds), but never both at the same time.

Time (Thermal Energy Added) Temperature (T) Solid Melting Liquid Boiling Gas Ek increases Ep increases Ek increases Ep increases M.P. B.P.

4. Specific Heat Capacity

Different substances require different amounts of energy to change their temperature. This mathematically describes the red, slanted portions of the heating curve above.

Specific Heat Capacity ($c$):

The energy required to raise the temperature of $1$ kg of a substance by $1$ K without a change of state.

$$Q = m c \Delta T$$
  • $Q$: Thermal energy transferred (J)
  • $m$: Mass of the substance (kg)
  • $c$: Specific heat capacity (J kg⁻¹ K⁻¹)
  • $\Delta T$: Change in temperature (K)

5. Specific Latent Heat

When a substance changes state (the blue, flat parts of the heating curve), its temperature doesn't change. The energy supplied is entirely used to break intermolecular bonds.

Specific Latent Heat ($L$):

The amount of thermal energy required to change the state of $1$ kg of a substance at a constant temperature.

$$Q = m L$$
  • $Q$: Thermal energy transferred during the phase change (J)
  • $m$: Mass changing state (kg)
  • $L$: Specific latent heat (J kg⁻¹)

Note: The Latent Heat of Vaporisation (boiling) is typically much larger than the Latent Heat of Fusion (melting) because turning a liquid into a gas requires separating the molecules completely, which takes significantly more energy than just loosening them to form a liquid.

Example 1: Melting Ice

Problem: Determine the energy needed to melt $200$ g of ice at $0^\circ\text{C}$. The specific latent heat of fusion of water is $3.3 \times 10^5$ J kg⁻¹.

Solution:

  • Step 1: Identify the process. We are going from solid to liquid at a constant temperature, so we use the Latent Heat equation.
  • Step 2: Convert mass to standard SI units. $m = 200$ g = $0.2$ kg.
  • Step 3: Calculate. $Q = m L_f = (0.2) \times (3.3 \times 10^5) = \mathbf{66,000}$ J (or $66$ kJ).

Example 2: Boiling Water with a Heater

Problem: Energy is supplied to a heater at a rate of $2500$ W. Determine the time taken, in minutes, to completely boil away $500$ ml of water that is already at $100^\circ\text{C}$. The specific latent heat of vaporisation of water is $2.3 \times 10^6$ J kg⁻¹. (Ignore energy losses to the surroundings).

Solution:

  • Step 1: Identify the knowns. Power $P = 2500$ W (Joules per second). Mass $m = 0.5$ kg (since 1 liter of water has a mass of 1 kg). $L_v = 2.3 \times 10^6$ J kg⁻¹.
  • Step 2: Equate the energy equations. The total energy supplied by the heater ($E = P \times t$) must equal the energy needed to boil the water ($Q = m L_v$).
    $$P \times t = m \times L_v$$
  • Step 3: Rearrange for time ($t$) and solve.
    $$t = \dfrac{m L_v}{P} = \dfrac{(0.5)(2.3 \times 10^6)}{2500} = \dfrac{1,150,000}{2500} = 460 \text{ seconds}$$
  • Step 4: Convert to minutes. $\dfrac{460}{60} = \mathbf{7.67}$ minutes.