Chapter 4.2: Thermal Physics
Welcome to Thermal Physics! In this chapter of your CCEA A2 Physics course (Unit A2 1), we are going to look closely at what heat really is on a microscopic level. We will explore how energy changes the temperature and physical state of matter, dive into the classic Gas Laws, and uncover how the random, microscopic collisions of tiny particles give rise to large-scale properties like pressure and temperature.
Don't worry if thermal concepts have felt a bit abstract in the past. We will break every single idea down step-by-step with clear analogies, key definitions, and exam tips to help you secure top marks!
1. Internal Energy, Heat Capacity, and Latent Heat
What is Internal Energy?
Everything around you is made of particles (atoms or molecules) in continuous motion. Because of this, every system contains energy inside it.
Internal Energy (\(U\)) is defined as the sum of the randomly distributed kinetic and potential energies of all the molecules or particles in a system.
Let's break that down into two parts:
1. Kinetic Energy (\(E_k\)): Arises from the motion of the particles (vibrational, translational, or rotational). An increase in temperature directly increases the mean kinetic energy of the particles.
2. Potential Energy (\(E_p\)): Arises from the electrostatic intermolecular forces holding particles together. When you change the distance between particles (such as melting a solid into a liquid, or boiling a liquid into a gas), you change their potential energy.
Analogy: Imagine a classroom full of students. The kinetic energy is how fast they are running around, while the potential energy is how far apart their desks are pulled against the springs connecting them.
What Happens During a Change of Phase?
Have you ever noticed that when pure ice melts at \(0\ ^\circ\text{C}\), the temperature stays at \(0\ ^\circ\text{C}\) until every bit of ice has turned into water? Why does this happen?
During a change of phase or state, temperature remains constant because the thermal energy supplied does not increase the kinetic energy of the particles. Instead, the energy goes into changing the electrostatic potential energy by breaking or weakening the intermolecular bonds holding the particles together.
Specific Heat Capacity (\(c\))
Specific Heat Capacity is defined as the energy required per unit mass to raise the temperature of a material by \(1\text{ K}\) (or \(1\ ^\circ\text{C}\)).
The equation to calculate heat transfer during a temperature change is:
\(\Delta Q = mc\Delta\theta\) (or \(Q = mc\Delta T\))
Where:
• \(\Delta Q\) or \(Q\) = thermal energy transferred (\(\text{J}\))
• \(m\) = mass of the material (\(\text{kg}\))
• \(c\) = specific heat capacity (\(\text{J}\cdot\text{kg}^{-1}\cdot\text{K}^{-1}\) or \(\text{J}\cdot\text{kg}^{-1}\cdot^\circ\text{C}^{-1}\))
• \(\Delta\theta\) or \(\Delta T\) = temperature change (\(\text{K}\) or \(^\circ\text{C}\))
Specific Latent Heat (\(l\) or \(L\))
Specific Latent Heat is defined as the energy required to change the state of \(1\text{ kg}\) of a substance at constant temperature.
The equation for phase change is:
\(Q = ml\)
Where:
• \(Q\) = thermal energy supplied or released (\(\text{J}\))
• \(m\) = mass of substance changing state (\(\text{kg}\))
• \(l\) = specific latent heat of the substance (\(\text{J}\cdot\text{kg}^{-1}\))
There are two types of specific latent heat you need to know:
• Specific Latent Heat of Fusion: Energy required to change \(1\text{ kg}\) of a substance from solid to liquid (or liquid to solid) at constant temperature.
• Specific Latent Heat of Vaporisation: Energy required to change \(1\text{ kg}\) of a substance from liquid to gas (or gas to liquid) at constant temperature.
Key Takeaway
• Use \(\Delta Q = mc\Delta\theta\) when there is a change in temperature with no change in state.
• Use \(Q = ml\) when there is a change in state at a constant temperature.
2. Gas Laws and the Absolute Temperature Scale
The Absolute Temperature Scale (Kelvin)
In everyday life, we use degrees Celsius (\(^\circ\text{C}\)), but in physics, all gas law calculations require the thermodynamic temperature scale measured in Kelvin (\(\text{K}\)).
Absolute Zero (\(0\text{ K}\) or \(-273.15\ ^\circ\text{C}\)): The lowest theoretical temperature possible, where molecular kinetic energy is at an absolute minimum, and the pressure and volume of an ideal gas extrapolate to zero.
To convert from Celsius to Kelvin:
\(T(\text{K}) = \theta(^\circ\text{C}) + 273.15\) (or use \(+273\) for standard calculations)
The Three Experimental Gas Laws
Each of the three experimental gas laws describes how an ideal gas behaves when one variable is kept constant:
1. Boyle’s Law (Constant Temperature):
For a fixed mass of gas at constant temperature, pressure is inversely proportional to volume.
\(p \propto \frac{1}{V}\) or \(pV = \text{constant}\) (i.e. \(p_1V_1 = p_2V_2\))
2. Charles’s Law (Constant Pressure):
For a fixed mass of gas at constant pressure, volume is directly proportional to absolute (Kelvin) temperature.
\(V \propto T\) or \(\frac{V}{T} = \text{constant}\) (i.e. \(\frac{V_1}{T_1} = \frac{V_2}{T_2}\))
3. Pressure Law / Gay-Lussac's Law (Constant Volume):
For a fixed mass of gas at constant volume, pressure is directly proportional to absolute (Kelvin) temperature.
\(p \propto T\) or \(\frac{p}{T} = \text{constant}\) (i.e. \(\frac{p_1}{T_1} = \frac{p_2}{T_2}\))
Key Takeaway
Whenever you are solving gas law equations, always convert temperatures to Kelvin first!
3. The Ideal Gas Equation
By combining Boyle's Law, Charles's Law, and the Pressure Law, we obtain the unified Ideal Gas Equation. Depending on whether you are working with moles or individual molecules, you can write it in two ways:
A. Molar Form
\(pV = nRT\)
Where:
• \(p\) = pressure (\(\text{Pa}\) or \(\text{N}\cdot\text{m}^{-2}\))
• \(V\) = volume (\(\text{m}^3\))
• \(n\) = number of moles of gas (\(\text{mol}\)), where \(n = \frac{m}{M_m}\) (\(m\) is mass of gas, \(M_m\) is molar mass)
• \(R\) = molar gas constant \(= 8.31\text{ J}\cdot\text{K}^{-1}\cdot\text{mol}^{-1}\)
• \(T\) = absolute temperature (\(\text{K}\))
B. Molecular Form
\(pV = NkT\)
Where:
• \(N\) = total number of molecules (\(N = nN_A\))
• \(N_A\) = Avogadro constant \(= 6.02 \times 10^{23}\text{ mol}^{-1}\)
• \(k\) = Boltzmann constant \(= \frac{R}{N_A} = 1.38 \times 10^{-23}\text{ J}\cdot\text{K}^{-1}\)
• \(T\) = absolute temperature (\(\text{K}\))
Memory Aid for Equations:
• Use \(n\) (moles) with \(R\) (Molar gas constant) \(\rightarrow pV = nRT\)
• Use \(N\) (molecules) with \(k\) (Boltzmann constant) \(\rightarrow pV = NkT\)
4. Kinetic Theory of Gases
Assumptions of the Kinetic Theory Model
To model an ideal gas mathematically, physicists make five core assumptions about the particles:
1. The gas consists of a very large number of identical particles/molecules moving in continuous random motion.
2. The volume of the molecules themselves is negligible compared to the total volume occupied by the gas.
3. Collisions between molecules and with the walls of the container are perfectly elastic (kinetic energy is conserved).
4. The duration of collisions is negligible compared to the time between collisions.
5. There are negligible intermolecular forces between molecules, except during instantaneous collisions.
The Kinetic Theory Pressure Equation
By applying Newton's laws of motion to the collisions of molecules against the walls of a container, we obtain the kinetic theory pressure equation:
\(pV = \frac{1}{3}Nm\langle c^2 \rangle\)
Where:
• \(p\) = pressure (\(\text{Pa}\))
• \(V\) = volume of container (\(\text{m}^3\))
• \(N\) = number of molecules
• \(m\) = mass of a single molecule (\(\text{kg}\))
• \(\langle c^2 \rangle\) = mean square speed of the molecules (\(\text{m}^2\cdot\text{s}^{-2}\))
Note on Root-Mean-Square Speed: Taking the square root of the mean square speed gives the root-mean-square speed: \(c_{\text{rms}} = \sqrt{\langle c^2 \rangle}\).
Average Molecular Kinetic Energy
Now, let's link the macroscopic temperature of a gas with the microscopic motion of its molecules! We can equate the two expressions for \(pV\):
\(pV = NkT\) and \(pV = \frac{1}{3}Nm\langle c^2 \rangle\)
Setting them equal gives:
\(NkT = \frac{1}{3}Nm\langle c^2 \rangle\)
Cancelling \(N\) from both sides:
\(kT = \frac{1}{3}m\langle c^2 \rangle\)
Multiply both sides by \(\frac{3}{2}\) to reveal the kinetic energy term (\(\frac{1}{2}m\langle c^2 \rangle\)):
\(\frac{1}{2}m\langle c^2 \rangle = \frac{3}{2}kT = \frac{3RT}{2N_A}\)
Crucial Conclusion: The mean translational kinetic energy of an ideal gas molecule is directly proportional to the absolute temperature (\(T\)) of the gas!
Key Takeaway
Temperature is simply a measure of the average kinetic energy of the particles. If you double the Kelvin temperature of an ideal gas, you double the mean kinetic energy of its particles.
5. Common Pitfalls and Exam Tips
Examiner reports frequently highlight the following common mistakes. Make sure to avoid them:
• Celsius vs Kelvin: Forgetting to convert temperatures to the Kelvin scale before using \(pV = nRT\) or any gas law is the most common reason for lost marks. Always check: Is my temperature in \(\text{K}\)?
• Mixing up Constants: Don't pair \(R\) with \(N\) or \(k\) with \(n\). Remember: \(n \leftrightarrow R\) and \(N \leftrightarrow k\).
• Mean Square Speed vs Root-Mean-Square Speed: Be careful with notation. \(\langle c^2 \rangle\) is the average of the squared speeds, while \(c_{\text{rms}} = \sqrt{\langle c^2 \rangle}\). When substituting into \(\frac{1}{3}Nm\langle c^2 \rangle\), ensure you use the mean square speed directly.
• Using the Wrong Formula for Phase Changes: Never use \(\Delta Q = mc\Delta\theta\) during a phase change because \(\Delta\theta = 0\). Always use \(Q = ml\).
• Calorimetry Losses: In experimental questions on specific heat capacity, remember that heat energy can be lost to the surroundings or absorbed by the container/calorimeter, leading to calculated values of \(c\) that are higher than the true theoretical value.
Quick Summary Checklist
Before moving on, make sure you can:
• Define internal energy, specific heat capacity, and specific latent heat.
• Explain why temperature remains constant during a change of phase.
• State and apply Boyle's Law, Charles's Law, and the Pressure Law.
• Use both forms of the ideal gas equation: \(pV = nRT\) and \(pV = NkT\).
• State the 5 assumptions of the kinetic theory of gases.
• Use \(pV = \frac{1}{3}Nm\langle c^2 \rangle\) and relate average kinetic energy to absolute temperature via \(\frac{1}{2}m\langle c^2 \rangle = \frac{3}{2}kT\).