Welcome to Ideal Gases! Your Journey into Thermal Physics
Hello future Physicists! This chapter, part of the wider "Thermal Physics" section, might seem abstract, but it’s actually about understanding something you interact with every day: air.
We are moving from studying solids and liquids (where particles stick together) to gases (where particles fly freely). We will learn how pressure, volume, and temperature are mathematically linked for an Ideal Gas—a simplified model that perfectly describes real gases under most conditions. Understanding this foundation is essential for everything from calculating rocket thrust to designing refrigeration systems!
Section 1: The Macroscopic View – Gas Laws (Empirical Relationships)
1.1 Temperature is Key: Absolute Zero
When dealing with gases, we must always use the absolute temperature scale, which is measured in Kelvin (K).
The concept of zero degrees Celsius (\(0 \text{ °C}\)) is arbitrary, but Absolute Zero (\(0 \text{ K}\)) is the theoretically lowest possible temperature where particles have minimum (zero classical) kinetic energy.
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Conversion: \(T (\text{K}) = \theta (\text{°C}) + 273.15\)
- Absolute Zero corresponds to \(-273.15 \text{ °C}\).
🔥 Common Mistake Alert!
If you use temperature in Celsius in gas calculations, your answer will be wildly wrong. Get into the habit of converting to Kelvin immediately!
1.2 The Experimental Gas Laws (3.11.3)
These three laws are empirical—they are based purely on what we observe experimentally when we keep one variable constant. (This is the focus of Required Practical 9).
A. Boyle's Law (Constant Temperature)
If you squeeze a balloon (decrease the volume), the pressure inside increases, provided the temperature and the amount of gas remain fixed.
The pressure is inversely proportional to the volume: \[p \propto \frac{1}{V} \quad \text{or} \quad pV = \text{constant}\]
For two states (1 and 2): \[p_1 V_1 = p_2 V_2\]
B. Charles's Law (Constant Pressure)
If you heat a gas, it expands. (Think about a hot air balloon). If the pressure and amount of gas are kept constant, the volume increases proportionally with absolute temperature.
The volume is directly proportional to the absolute temperature (in K): \[V \propto T \quad \text{or} \quad \frac{V}{T} = \text{constant}\]
C. The Pressure Law (Constant Volume)
If you heat a gas trapped in a rigid container (like a pressure cooker), the pressure dramatically increases.
The pressure is directly proportional to the absolute temperature (in K): \[p \propto T \quad \text{or} \quad \frac{p}{T} = \text{constant}\]
Quick Review: Empirical Gas Laws
These three laws combine into the Combined Gas Law: \[\frac{p_1 V_1}{T_1} = \frac{p_2 V_2}{T_2}\]
Section 2: The Ideal Gas Equation (3.11.3)
2.1 Unifying the Variables
The Ideal Gas Equation links the three variables (\(p\), \(V\), \(T\)) with the amount of substance. We have two main versions, depending on whether we use moles or individual molecules.
Version 1: Using Moles (\(n\)) and the Molar Gas Constant (\(R\))
The primary form of the Ideal Gas Equation for n moles of gas is: \[\mathbf{pV = nRT}\]
- \(p\): Pressure (Pa, Pascals)
- \(V\): Volume (\(\text{m}^3\), cubic metres)
- \(T\): Absolute Temperature (K, Kelvin)
- \(n\): Number of moles (mol)
- \(R\): Molar Gas Constant (\(8.31 \text{ J mol}^{-1} \text{K}^{-1}\))
Did you know? \(R\) is a universal constant—it’s the same for any gas, provided it behaves ideally.
Connecting Moles, Mass, and Molecules
We often need to convert between the amount of substance:
- Molar Mass: The mass of one mole of a substance (units \(\text{g mol}^{-1}\) or \(\text{kg mol}^{-1}\)). If the total mass of gas is \(M\), then \(n = M / \text{Molar Mass}\).
- Avogadro Constant (\(N_A\)): The number of particles (atoms or molecules) in one mole. \[N_A = 6.02 \times 10^{23} \text{ mol}^{-1}\]
Version 2: Using Molecules (\(N\)) and the Boltzmann Constant (\(k\))
If we have \(N\) individual molecules, we use the Boltzmann constant, \(k\).
Since \(N = n N_A\), we can substitute \(n = N/N_A\) into the original equation: \[p V = \left(\frac{N}{N_A}\right) R T\]
The term \(R/N_A\) is defined as the Boltzmann Constant (\(k\)): \[k = \frac{R}{N_A} \approx 1.38 \times 10^{-23} \text{ J K}^{-1}\]
This gives the second, equally important form: \[\mathbf{pV = NkT}\]
Analogy: Using \(R\) is like calculating the cost of 10 dozen eggs. Using \(k\) is like calculating the cost of 120 individual eggs. They represent the same physics, just different counting methods!
2.2 Work Done by a Gas (3.11.3)
In thermodynamics, when a gas expands, it pushes back the surroundings (like a piston) and does work.
If the expansion occurs at a constant pressure (\(p\)), and the volume changes by \(\Delta V\), the work done by the gas is: \[\mathbf{W = p \Delta V}\]
Where:
- \(W\) is the work done (J).
- \(p\) is the constant external pressure (Pa).
- \(\Delta V\) is the change in volume (\(\text{m}^3\)).
If the gas expands, \(\Delta V\) is positive, and the gas does positive work. If the gas is compressed, \(\Delta V\) is negative, meaning work is done on the gas.
Key Takeaway (Section 2)
The Ideal Gas Equation (\(pV = nRT\) or \(pV = NkT\)) is the single most important equation in this topic, allowing us to calculate any variable if we know the others. Remember that temperature MUST be in Kelvin.
Section 3: The Microscopic View – Kinetic Theory (3.11.4)
The Kinetic Theory of Gases moves from "what happens" (empirical laws) to "why it happens" (a theoretical model based on particle motion).
3.1 Evidence for Particle Motion: Brownian Motion (3.11.4)
The idea that gas is made of tiny, moving particles isn't just theory—it has evidence.
- Brownian Motion: The random, erratic movement of small particles (like smoke or pollen grains) suspended in a fluid (gas or liquid).
- This motion is caused by the invisible, random bombardment of the surrounding, much smaller, fast-moving air or water molecules.
- This observation is crucial evidence for the existence of atoms and molecules and their continuous, random motion.
3.2 Assumptions of the Ideal Gas Model (3.11.4)
To derive the ideal gas equation from first principles (the kinetic theory model), we must make several simplifying assumptions about the gas molecules:
- The gas consists of a large number of identical particles (atoms or molecules).
- The particles are in rapid, random motion.
- Particles have negligible volume compared to the volume of the container. (The molecules are tiny point masses).
- Interactions only occur during perfectly elastic collisions between particles and with the container walls. (No energy is lost, and there are no intermolecular forces between collisions).
- The duration of collisions is negligible compared to the time between collisions.
💡 Why are these important?
Real gases behave ideally only when they are dilute (low pressure) and hot. Under high pressure or low temperature, assumptions 3 and 4 break down, as the volume of the particles themselves and the attractive forces between them become significant.
3.3 Relating Pressure, Mass, and Speed (3.11.4)
Pressure arises from the force exerted when gas molecules collide with the walls and change their momentum. By applying Newton's laws and using the ideal gas assumptions, we arrive at the full kinetic theory equation:
\[\mathbf{pV = \frac{1}{3} Nm \langle c_{rms} \rangle^2}\]
Where:
- \(p\), \(V\), \(N\) are Pressure, Volume, and the total Number of molecules.
- \(m\): Mass of one single gas molecule (kg).
- \(\mathbf{\langle c_{rms} \rangle}\): The root-mean-square speed (\(\text{m s}^{-1}\)).
Understanding Root-Mean-Square Speed (\(c_{rms}\))
Since gas molecules move at many different speeds, we need an average speed. We cannot simply use the arithmetic average because velocity is a vector (some particles move left, some right, cancelling out the average).
The root-mean-square speed is calculated by:
1. Squaring all the speeds (\(c^2\)).
2. Finding the Mean (average) of these squared speeds (\(\langle c^2 \rangle\)).
3. Taking the Square Root of the mean (\(\sqrt{\langle c^2 \rangle}\)).
This gives the \(\langle c_{rms} \rangle\), which is a measure of the typical speed of the particles.
Key Takeaway (Section 3)
The Kinetic Theory Equation is the theoretical link between the macroscopic quantities (\(p, V\)) and the microscopic properties (\(N, m, c_{rms}\)).
Section 4: Temperature and Energy (3.11.4)
4.1 The Kinetic Interpretation of Temperature
We have two equations for \(pV\):
1. The Ideal Gas Equation: \(pV = NkT\)
2. The Kinetic Theory Equation: \(pV = \frac{1}{3} Nm \langle c_{rms} \rangle^2\)
Since both equal \(pV\), we can equate them: \[NkT = \frac{1}{3} Nm \langle c_{rms} \rangle^2\]
We can cancel \(N\) and rearrange this to find the expression for the average translational kinetic energy of a single molecule, \(\frac{1}{2} m \langle c_{rms} \rangle^2\):
\[\mathbf{\frac{1}{2} m \langle c_{rms} \rangle^2 = \frac{3}{2} kT}\]
This is one of the most fundamental results in thermal physics! It shows that the absolute temperature (T) is directly proportional to the average kinetic energy of the gas molecules.
- If \(T\) doubles (in Kelvin), the average translational kinetic energy doubles.
- Note that this relationship does not depend on the mass (\(m\)) or type of gas. At the same temperature, all ideal gases have the same average kinetic energy per molecule.
4.2 Internal Energy of an Ideal Gas (3.11.4)
Remember from Section 3.11.1 that Internal Energy (\(U\)) is the sum of the kinetic and potential energies of the randomly distributed particles.
For an Ideal Gas, one of our key assumptions (Assumption 4 in 3.2) was that there are no intermolecular forces. This means:
- The potential energy between molecules is zero.
- Therefore, the Internal Energy (\(U\)) of an ideal gas is entirely the total random kinetic energy of its atoms.
Since the average kinetic energy per particle is \(\frac{3}{2} kT\), the total internal energy for \(N\) particles is: \[U = N \left(\frac{3}{2} kT\right) = \frac{3}{2} NkT\]
Or, using moles (\(NkT = nRT\)): \[U = \frac{3}{2} nRT\]
This confirms a crucial idea: Changing the temperature is the only way to change the internal energy of a fixed mass of an ideal gas.
Final Key Takeaway
We use two models: the Empirical Gas Laws (based on observation, relating \(p, V, T\)) and the Kinetic Theory Model (based on theoretical assumptions, explaining why gases behave this way). The true strength of physics is that these two models lead to consistent mathematical results!