heat and temp W_250717_153535

Units of Heat

• Heat = energy transferred because of a temperature difference between two objects.
• Because heat is energy, it is measured with energy units.
  • SI: joule (J)
  • CGS: calorie (cal) with $1\,\text{cal}=4.184\,\text{J}$
  • MKS: kilocalorie (kcal) with $1\,\text{kcal}=4184\,\text{J}=10^3\,\text{cal}$
  • British: British thermal unit (Btu) with $1\,\text{Btu}=1055\,\text{J}$
  • Mechanical (Imperial): foot-pound $1\,\text{ft·lb}=1.356\,\text{J}$
• Definition: 1 kilocalorie is the heat required to raise the temperature of 1 kg of water by 1 K.
• Historical constant: Robert Mayer first stated the mechanical equivalent of heat, $1\,\text{cal}=4.184\,\text{J}$.

Internal Energy

• Internal energy (thermal energy) = sum of total molecular kinetic energy + potential energy inside a substance.
  • Kinetic part is mainly translational; directly linked to temperature.
  • Temperature rise ⇒ increase in average molecular kinetic energy ⇒ increase in internal energy.
  • Potential part comes from intermolecular forces; grows with intermolecular separation.
• Modes of molecular motion:
  • Vibrational, Rotational, Translational.
• Heat = transfer of internal (thermal) energy from one body to another.

Thermal Capacity

• When two objects at different temperatures contact, heat flows from hot → cold until thermal equilibrium.
• Thermal (heat) capacity $C$ of an object:
  $C=\frac{\Delta Q}{\Delta T}$
  • $\Delta Q$: heat added • $\Delta T=T<em>2-T</em>1$
• SI unit: $\text{J K}^{-1}$.
• Dependence: $C=c m$ (directly proportional to mass and the materialʼs specific heat capacity $c$).
• Consequences / applications of high or low $C$:
  • Small $C$ ⇒ object heats/cools easily.
  • Water has very high $C$ (≈4× that of aluminium for equal mass), so it is used for cooling engines, solar-heating reservoirs, and hot-water bags; moderates regional climate.
• Molar thermal capacity (per mole):
  $C_{\text{molar}}=\frac{1}{n}\frac{\Delta Q}{\Delta T}$
  Unit: $\text{J mol}^{-1}\,\text{K}^{-1}$.

Specific Heat Capacity

• Specific heat capacity $c$: heat needed to raise 1 kg of a substance by 1 K.
  $c=\frac{C}{m}=\frac{\Delta Q}{m\,\Delta T}$
• Heat required for any mass $m$ over $\Delta T$:
  $\Delta Q = m c \Delta T$
• SI unit: $\text{J kg}^{-1}\,\text{K}^{-1}$ (others: $\text{kcal kg}^{-1}\,^{\circ}\text{C}^{-1}$, $\text{cal g}^{-1}\,^{\circ}\text{C}^{-1}$).
• Representative $c$ values (J kg⁻¹ K⁻¹):
  • Water 4184 • Ice 2089 • Aluminium 898 • Glass 837 • Copper 385 • Lead 130 • Helium gas 5180 • Hydrogen gas 14 250.
• Climatic moderation: Large lakes absorb huge heat in summer and release it in winter due to waterʼs high $c$.

Law of Heat Exchange & Calorimetry

• Law: In an isolated system, total heat lost = total heat gained.
  $\Delta Q<em>{\text{lost}} = \Delta Q</em>{\text{gained}}$
  For two bodies A (hot) and B (cold):
  $m<em>A c</em>A (T<em>A-T</em>f)=m<em>B c</em>B (T<em>f-T</em>B)$
• Energy-conservation statement for heat processes.
• Calorimeter: insulated vessel (often Cu/Al) + stirrer + thermometer; measures heat exchange.
• Determining an unknown $c$ with calorimeter mass $m<em>c$ and its $cc$:
  $c=\frac{\Delta Q - m<em>c c</em>c \Delta T}{m\,\Delta T}$
• Typical problem types: mixing liquids, condensation of steam, cooling/heating times, etc.

Change of State & Latent Heat

• Phase changes occur abruptly at definite temperatures that depend on pressure.
• Latent heat: energy absorbed/released during phase change at constant T.
  • Specific latent heat $L$ (fusion $Lf$ or vaporization $Lv$): $\Delta Q = m L$ ; unit $\text{J kg}^{-1}$.
• Water heating curve (ice −10 °C → steam 120 °C): heat portions
  1. Warm ice: $m c_{\text{ice}} \Delta T$
  2. Melt: $m L_f$ (0 °C plateau)
  3. Warm water: $m c_w \Delta T$
  4. Vaporize: $m L_v$ (100 °C plateau)
  5. Superheat steam: $m c_{\text{steam}} \Delta T$
• Sublimation: solid → gas directly (e.g., freeze-drying).

Fusion (Melting)

• Melting point = temperature where solid ↔ liquid at given pressure (ice: 0 °C at 1 atm).
• Water/ice: $L_f = 3.335\times10^5\,\text{J kg}^{-1}$.
• Example: heat to melt 5 kg ice at 0 °C: $\Delta Q = 5 L_f =1.67\times10^6\,\text{J}$.
• Mechanical-heat equivalence example: Lead bullet melts on impact. Entire kinetic energy converts to $m L<em>f$ leading to $v=\sqrt{2L</em>f}$ (≈221 m s⁻¹).

Vaporization (Boiling / Condensation)

• Boiling point = temperature where liquid ↔ vapour at given pressure (water: 100 °C at 1 atm).
• Water: $L_v = 2.255\times10^6\,\text{J kg}^{-1}$.
• Steam burns worse than boiling water: extra energy $L_v$ released on condensation.
• Sample: 5 kg water → steam needs $1.13\times10^7\,\text{J}$.
• Condensation liberates same amount of heat; exploited in steam heating systems.

Worked Mixed-Phase Example

• Ice cubes (0.045 kg at −10 °C) in 0.3 kg tea (30 °C): sequential heats (warm ice, melt ice, warm melt) balanced with tea cooling. Outcome: final equilibrium T ≈ 15 °C; all ice melts (since T>0 °C).

Pressure Dependence of Melting & Boiling Points

• Melting point of ice decreases with increased pressure (slope negative on P-T diagram).
  • Ice skating: blade pressure slightly melts surface, forming lubricating water film; refreezes afterward.
• Boiling point increases with pressure.
  • Pressure cooker operates at ≈2 atm so water boils at ≈120 °C ⇒ faster cooking.
• Phase diagram salient points (water):
  • Triple point: all 3 phases coexist.
  • Critical point: liquid & gas densities equal; beyond this water becomes supercritical fluid.

Summary of Key Equations

• Heat-temperature relation: $\Delta Q = m c \Delta T$.
• Thermal capacity: $C = \frac{\Delta Q}{\Delta T} = c m$.
• Molar thermal capacity: $C_{\text{m}} = \frac{1}{n}\frac{\Delta Q}{\Delta T}$