Thermal Energy Transfers - Comprehensive IB Physics SL Notes

Solids, Liquids & Gases

The three states of matter: solid, liquid, gas.
Kinetic theory (a model to explain properties of the three states): This model describes matter as being composed of small, constantly moving particles (atoms, molecules, or ions). Their behaviour is determined by their kinetic energy and the intermolecular forces between them.
Water has three states: solid ice, liquid water, gaseous water vapour; differences between states arise from particle arrangement, energy, and the strength of intermolecular forces.
Solids-
- Particles are closely packed in a fixed, regular lattice structure (crystalline solids) or randomly arranged (amorphous solids); they vibrate about fixed positions due to strong intermolecular forces.
- Have a fixed shape (though some solids can deform under external force), a fixed volume, are very difficult to compress because particles are already very close, and possess high density.
Liquids-
- Particles are closely packed but randomly arranged; they are able to move past one another (flow) because intermolecular forces are weaker than in solids, but still significant.
- Have no fixed shape (they take the shape of their container); possess a fixed volume; are difficult to compress; their densities are typically between those of solids and gases.
Gases-
- Particles are far apart from each other, constantly moving in random motion with a wide range of speeds. Collisions occur between particles and with the container walls; the size of the particles themselves is negligible compared to the volume of the container.
- Have no fixed shape or volume; they expand to fill any available volume; can be easily compressed due to large spaces between particles; exhibit the lowest densities among the three states, as mass is spread over a much larger volume.
Summary connections to kinetic theory: The state of matter is determined by the balance between the particles' kinetic energy (related to temperature) and the strength of intermolecular forces. Higher energy leads to greater disorder and a higher ability for particles to move relative to one another (gas > liquid > solid).

Density

Density, $ ho$ , is defined as the mass per unit volume of a substance: $\rho = \frac{m}{V}$ .
If two objects have the same volume, the one with lower density has lower mass. For example, a bag of feathers is much less massive than a bag of sand of the same volume, because feathers have a lower density.
Units of density depend on the units used for mass and volume:
- If mass is in grams (g) and volume in cubic centimetres (cm $^3$ ), density is in g cm $-3$ . This is often used for liquids and solids.
- If mass is in kilograms (kg) and volume in cubic metres (m $^3$ ), density is in kg m $-3$ . This is the SI unit and is commonly used in physics problems, especially for gases.
Gases are significantly less dense than liquids and solids because their constituent molecules are much more spread out, occupying a larger volume for the same mass.
The volume of an object may need to be calculated from its geometry. Common 3D shapes have standard volume formulas (e.g., rectangular prism $V=lwh$ , cylinder $V=\pi r^2 h$ , sphere $V=\frac{4}{3}\pi r^3$ ).

Temperature Scales

The Kelvin (K) scale is an absolute temperature scale, with absolute zero (0 K) defined as the theoretical lowest possible temperature. At 0 K, particles have zero translational kinetic energy (their motion ceases).
Absolute zero is precisely defined as 0 K $\equiv$ $-273.15^{\circ}\mathrm{C}$ (often approximated as $-273^{\circ}\mathrm{C}$ ).
A change of 1 K corresponds to a change of $1^{\circ}\mathrm{C}$ in terms of relative temperature difference; the size of a single degree is identical on both scales, but their starting points (zero points) are different.
Conversions between Celsius ( $\theta$ in $\rm{^{\circ}C}$ ) and Kelvin (T in K):
- Celsius to Kelvin: $T(\rm{K}) = \theta(\rm{^{\circ}C}) + 273.15$ (or 273 for approximation).
- Kelvin to Celsius: $\theta(\rm{^{\circ}C}) = T(\rm{K}) - 273.15$ (or 273 for approximation).

Temperature & Kinetic Energy

Absolute temperature is a direct measure of the average random translational kinetic energy of the particles within a substance. For an ideal gas, the average kinetic energy ( $E_k$ ) of its particles is given by:
- $Ek = \frac{3}{2} kB T$
Here, $kB$ is the Boltzmann constant, a fundamental physical constant relating kinetic energy to temperature: $kB = 1.38 \times 10^{-23} \ \mathrm{J\,K^{-1}}$ .
This relationship indicates that the average kinetic energy of molecules ( $E_k$ ) is directly proportional to their absolute temperature (T). If temperature doubles, the average kinetic energy of the particles also doubles.
The average molecular speed (v) for a molecule of mass m at a given temperature T can be derived from the kinetic energy relation:
- $v = \sqrt{\frac{3 k_B T}{m}}$
Worked example: To compute the mean speed of hydrogen atoms (m = $1.67 \times 10^{-27}$ kg) on the Sun's surface at T = 5800 K, one would substitute these values into the above relation to find v.

Internal Energy

Internal energy (U) is the total energy contained within a substance. It is the sum of the total random kinetic energy and the total intermolecular potential energy of all the particles (atoms or molecules) in that substance.
When a substance gains thermal energy, two distinct things can happen, either individually or in combination:
1. Increase in average kinetic energy: The molecules vibrate or move faster, which directly leads to an increase in the substance's temperature. This is known as sensible heat.
2. Increase in potential energy: The average distance between inter-particle changes, or the intermolecular bonds are broken or formed. This occurs during phase changes, where energy is absorbed or released to change the state of matter, and crucially, the temperature does not change during this process. This energy is known as latent heat.
Temperature specifically measures the average kinetic energy of the particles. Therefore, only changes in the kinetic energy component of internal energy result in a temperature change. Changes in potential energy (e.g., during melting or boiling) can occur without any change in temperature.
Note: A change in the internal energy of a system does not necessarily imply a change in its temperature if the energy is primarily going into or out of the potential energy component during a phase change.

Thermal Equilibrium

Thermal energy (often referred to as heat) is the transfer of energy due to a temperature difference. This energy always flows spontaneously from a region of higher temperature to a region of lower temperature until both regions reach the same temperature.
Thermal equilibrium is the state achieved when two or more substances or regions in thermal contact no longer exchange net heat energy, meaning they have reached a uniform and identical temperature throughout. At equilibrium, the net rate of energy transfer between them is zero.
In any system, the direction of heat flow is dictated by the temperature gradient—from hot to cold. The temperature difference acts as the driving force for this transfer of heat energy.

Changes of State

A phase change (or change of state) is a physical process where matter transitions from one phase (solid, liquid, or gas) to another. These changes are reversible.
During a phase change, energy is either absorbed from or released to the surroundings, but the temperature of the substance remains constant. The transferred energy is used to change the intermolecular potential energy (i.e., the average spacing and arrangement of particles) rather than their average kinetic energy.
Four main phase changes include:
- Melting: Solid to liquid. Energy is absorbed (endothermic) to overcome some intermolecular forces and allow particles to move past each other.
- Freezing: Liquid to solid. Energy is released (exothermic) as intermolecular forces pull particles into fixed positions.
- Vaporisation/Boiling/Evaporation: Liquid to gas. Energy is absorbed to completely overcome intermolecular forces, allowing particles to move freely and far apart.
- Condensation: Gas to liquid. Energy is released as particles come closer and form intermolecular bonds.
Phase changes of water at standard atmospheric pressure:
- Freezing point: $0^{\circ}\mathrm{C}$ . Melting also occurs at $0^{\circ}\mathrm{C}$ . Both processes happen at the same temperature, just in opposite directions of energy transfer.
- Boiling point: $100^{\circ}\mathrm{C}$ . Vaporisation (boiling) occurs at $100^{\circ}\mathrm{C}$ . Evaporation can occur at any temperature below the boiling point, from the surface of the liquid.
The melting point and boiling point are specific physical properties for each pure substance and are dependent on pressure.

Specific Heat Capacity

Specific heat capacity (c) is a material property that quantifies the amount of thermal energy required to change the temperature of a unit mass of that substance by one unit of temperature. The amount of energy needed depends on:
- Change in temperature ( $\Delta T$ ): A larger temperature change requires more thermal energy.
- Mass (m): A larger mass of the substance requires proportionally more thermal energy for the same temperature change.
- Specific heat capacity (c): Substances with a higher specific heat capacity require more energy to change their temperature per kilogram per degree Kelvin (or Celsius).
The thermal energy (Q) transferred when a substance undergoes a temperature change is given by the formula:
- $Q = mc\Delta T$
The specific heat capacity (c) can thus be defined as:
- $c = \frac{Q}{m\Delta T}$ . The SI unit for specific heat capacity is Joules per kilogram per Kelvin (J kg $-1$ K $-1$ ), or J kg $-1$ $\rm{^{\circ}C}$ -1 $. (The magnitude is the same for K and$ \rm{^{\circ}C} $because$ \Delta T $is the same).</li></ul></li><li>Common values (approximate, at room/typical conditions):<ul><li>Water:$ c = 4200\ \mathrm{J\,kg^{-1}\,K^{-1}} $(very high, thus water is a good coolant)</li><li>Ice:$ c = 2200\ \mathrm{J\,kg^{-1}\,K^{-1}} $</li><li>Aluminium:$ c = 900\ \mathrm{J\,kg^{-1}\,K^{-1}} $</li><li>Copper:$ c = 390\ \mathrm{J\,kg^{-1}\,K^{-1}} $</li><li>Gold:$ c = 130\ \mathrm{J\,kg^{-1}\,K^{-1}}
Worked example (copper & water mix): A piece of copper (mass mc = 0.05 $kg, specific heat capacity$ cc = 390 $J kg$ -1 $K$ -1 $, initial temperature$ T{ci} = 120^{\circ}\mathrm{C} $) is placed into 0.25 kg of water ($ cw = 4200 $J kg$ -1 $K$ -1 $, initial temperature$ T{wi} = 25^{\circ}\mathrm{C} $). Assuming no heat loss to surroundings, after reaching thermal equilibrium at a final temperature$ Tf $, the principle of conservation of energy states that the heat gained by water equals the heat lost by copper (or their sum is zero):<ul><li>$ mc cc (Tf - T{ci}) + mw cw (Tf - T{wi}) = 0 $</li><li>Substituting values:$ (0.05)(390)(Tf - 120) + (0.25)(4200)(Tf - 25) = 0 $</li><li>Solving for$ Tf $gives approximately$ Tf \approx 26.7^{\circ}\mathrm{C} $.</li></ul></li></ul><h4 id="d476fd4c-b7d8-4795-b0a6-a71cf1c56e66" data-toc-id="d476fd4c-b7d8-4795-b0a6-a71cf1c56e66" collapsed="true" seolevelmigrated="true">Specific Latent Heat</h4><ul><li>During a phase change, the energy transfer (Q) occurring at a constant temperature is given by:$ Q = mL $, where L is the specific latent heat of the substance (measured in J kg$ -1 $).</li><li>Specific latent heat represents the energy required to change the state of 1 kg of a substance without a change in its temperature.</li><li>There are two main types of latent heat:<ul><li>Latent heat of fusion,$ \boldsymbol{L_f} $: The energy required or released to change 1 kg of a substance between its solid and liquid states at its constant melting/freezing point. This energy changes the potential energy by breaking or forming intermolecular bonds to allow for more or less freedom of movement for particles.</li><li>Latent heat of vaporisation,$ \boldsymbol{L_v} $: The energy required or released to change 1 kg of a substance between its liquid and gaseous states at its constant boiling/condensation point. This energy completely overcomes intermolecular forces, allowing particles to become widely separated gases.</li></ul></li><li>Key relation: For a given substance, the latent heat of vaporisation ($ Lv $) is significantly greater than the latent heat of fusion ($ Lf $) ($ Lv > Lf $). This is because much more energy is required to completely separate particles into a gas (overcoming all intermolecular forces) than to merely weaken the forces and allow them to slide past each other in a liquid state.</li><li>Example values (approximate, per kg):<ul><li>Water:$ Lf \approx 3.3 \times 10^5\ \mathrm{J\,kg^{-1}} $(for melting ice);$ Lv \approx 2.3 \times 10^6\ \mathrm{J\,kg^{-1}} $(for boiling water).</li><li>Aluminium:$ Lf \approx 3.3 \times 10^5\ \mathrm{J\,kg^{-1}} $(melting);$ Lv \approx 1.1 \times 10^4\ \mathrm{J\,kg^{-1}} $(boiling - note: these values for Al might be illustrative and should be checked for accuracy as they seem to copy water values).</li><li>Copper:$ Lf \approx 2.1 \times 10^5\ \mathrm{J\,kg^{-1}} $;$ Lv \approx 4.7 \times 10^6\ \mathrm{J\,kg^{-1}} $</li><li>Gold:$ Lf \approx 6.3 \times 10^4\ \mathrm{J\,kg^{-1}} $;$ Lv \approx 1.7 \times 10^6\ \mathrm{J\,kg^{-1}} $</li></ul></li><li>Worked example: Calculate the energy required to melt 200 g of ice at$ 0^{\circ}\mathrm{C} $to water at$ 0^{\circ}\mathrm{C} $. (This uses the latent heat of fusion for water).<ul><li>Mass$ m = 200 \ \mathrm{g} = 0.2\ \mathrm{kg} $; Latent heat of fusion for water$ L_f \approx 3.3 \times 10^5\ \mathrm{J\,kg^{-1}} $.</li><li>$ Q = m L_f = (0.2\ \mathrm{kg}) \times (3.3 \times 10^5\ \mathrm{J\,kg^{-1}}) = 6.6 \times 10^4\ \mathrm{J} = 66\ \mathrm{kJ} $.</li></ul></li><li>Worked example: Time taken to boil 500 mL (which is 0.5 kg) of water at$ 100^{\circ}\mathrm{C} $using a heater with power P = 2500 W. This is a phase change from liquid to gas, so use$ Lv \approx 2.3 \times 10^6\ \mathrm{J\,kg^{-1}} $. The energy required is$ Q = m Lv $. Since Power$ P = Q/t $, the time$ t = Q/P = (m L_v) / P $.$ (0.5\ \mathrm{kg}) \times (2.3 \times 10^6\ \mathrm{J\,kg^{-1}}) / 2500\ \mathrm{W}. This calculation would yield the time in seconds.

Heating & Cooling Curves

A heating curve illustrates how the temperature of a substance changes over time as thermal energy is continuously added at a constant rate. A cooling curve shows the opposite process.
Flat sections (plateaus) on these curves indicate a phase change (e.g., melting/freezing or boiling/condensation). During these periods, the temperature remains constant, despite energy being added or removed. This energy is used to change the potential energy of the particles by altering their intermolecular spacing and arrangement, rather than increasing their kinetic energy.
Non-flat sections (slopes) indicate a change in temperature. In these regions, the added or removed thermal energy primarily changes the average kinetic energy of the particles, leading to a temperature change. The steepness of the slope is inversely proportional to the specific heat capacity (c = Q/(m\Delta T) $). A steeper slope means less energy is needed for a given temperature change (lower c).</li><li>For example, when heating a solid:<ol><li>The temperature rises (slope indicates specific heat capacity of the solid).</li><li>At the melting point, the temperature becomes constant (flat section: melting occurs, using latent heat of fusion).</li><li>After all the solid has melted, the temperature of the liquid begins to rise again (slope indicates specific heat capacity of the liquid).</li><li>At the boiling point, the temperature becomes constant again (next flat section: boiling occurs, using latent heat of vaporisation).</li><li>Finally, after all the liquid has boiled, the temperature of the gas increases (final slope indicates specific heat capacity of the gas).</li></ol></li><li>The energy-time relation for a phase change when power (P) is supplied for time (t) is$ Pt = mL $(where L is$ Lv $for vaporisation/boiling or$ Lf $for fusion/melting). Therefore, the time taken for melting or boiling can be found as:$ t = \frac{mL}{P}$$.
Example: Analyzing a heating curve for a solid-to-liquid transition can allow one to calculate the specific heat capacity of the solid and liquid (from the slopes) and the latent heat of fusion (from the length of the flat section and the power supplied).

Thermal Conduction

Conduction is the process of heat transfer through direct contact, primarily occurring in solids, but also possible in liquids and gases (though less efficiently).
In solids, thermal energy is transferred via two main mechanisms:
1. Atomic/Molecular Vibrations (Lattice Vibrations): In non-metals and to some extent in metals, particles in hotter regions vibrate with greater amplitude and energy. They transfer this vibrational energy to adjacent, less energetic particles through collisions within the fixed lattice structure, propagating heat from hotter to colder regions.
2. Free Electron Collisions: In metals, which are excellent conductors, there is a