DP IB Physics: SL

B. The particulate nature of matter

B.3 Gas Laws

DP IB Physics: SL B. The particulate nature of matter B.3 Gas Laws Understandings Students should understand:
a)	Pressure as given by [math]P = \frac{F}{A}[/math] where F is the force exerted perpendicular to the surface
b)	The amount of substance n as given by [math]n = \frac{N}{N_A}[/math] where N is the number of molecules and [math]N_A[/math] is the Avogadro constant
c)	That ideal gases are described in terms of the kinetic theory and constitute a modelled system used to approximate the behavior of real gases
d)	That the ideal gas law equation can be derived from the empirical gas laws for constant pressure, constant volume and constant temperature as given by [math]\frac{PV}{T} = \text{constant}[/math]
e)	The equations governing the behavior of ideal gases as given by [math]PV = Nk_B T \text{ and } PV = nRT[/math]
f)	That the change in momentum of particles due to collisions with a given surface gives rise to pressure in gases and, from that analysis, pressure is related to the average translational speed of molecules as given by [math]P = \frac{1}{3} \rho v^2[/math]
g)	The relationship between the internal energy of an ideal monatomic gas and the number of molecules or amount of substance as given by [math]U = \frac{3}{2} N k_B T \text{ and } U = \frac{3}{2} R n T [/math]
h)	The temperature, pressure and density conditions under which an ideal gas is a good approximation of a real gas.

a) Pressure:
Pressure (P) is defined as the force (F) exerted per unit area (A) when the force acts perpendicularly to the surface. Mathematically, it is given by:
[math]P = \frac{F}{A}[/math]
⇒ Explanation of Pressure
Pressure is a scalar quantity, meaning it has magnitude but no direction.
The SI unit of pressure is the pascal (Pa), where 1 Pa = 1 N/m².
Other Units of Pressure:
– Atmosphere (atm): 1 atm = 101,325 Pa
– Millimeters of Mercury (mmHg): 1 atm ≈ 760 mmHg
– Bar: 1 bar = 10⁵ Pa
Figure 1 Force applies on a surface area
⇒ Pressure Works in Different Contexts
Solids: Pressure depends on how force is distributed over an area. A sharp knife exerts higher pressure than a blunt one because the same force is applied over a smaller area.
Liquids: Pressure increases with depth due to the weight of the fluid above, as given by:
[math]P = ρgh[/math]
Where:
– ρ is the density of the liquid,
– g is gravitational acceleration,
– h is depth.
Gases: Gas molecules collide with container walls, exerting force over the surface area, creating pressure.
b) The Amount of Substance and Avogadro’s Constant
The amount of substance (n) is a measure of the number of particles (atoms, molecules, or ions) in a system. It is related to the number of molecules (N) and Avogadro’s constant (N_A) by:
[math]n = \frac{N}{N_A}[/math]
⇒ Avogadro’s Constant ([math]N_A[/math])
Avogadro’s constant is:
[math]N_A = 6.022 × 10^{23} mol^{-1}[/math]
This means that one mole of any substance contains [math]6.022 × 10^{23}[/math] particles.
⇒ Relationship Between Moles, Mass, and Molar Mass
The number of moles (n) of a substance can also be found using:
[math]n = \frac{m}{M}[/math]
Where:
– m is the mass of the substance,
– M is the molar mass (g/mol).
c) Ideal Gases and the Kinetic Theory of Gases
⇒ The Ideal Gas Model
An ideal gas is a theoretical gas that follows the ideal gas law under all conditions. It assumes:
– Gas molecules are point particles with negligible volume.
– There are no intermolecular forces between gas particles.
– Collisions between molecules are perfectly elastic (no energy loss).
– Gas molecules are in constant random motion.
– The average kinetic energy of gas molecules is proportional to temperature.
⇒ The Ideal Gas Equation
[math]PV = nRT[/math]
Where:
– P = pressure (Pa),
– V = volume (m³),
– n = number of moles,
– R = universal gas constant (314 J/mol.K),
– T = absolute temperature (K).
Figure 2 Kinetic theory of gases
⇒ The Kinetic Theory of Gases
The kinetic theory explains gas behavior using molecular motion and statistical mechanics.
⇒ Assumptions:
1. Random Motion: Gas molecules move in random directions with a range of speeds.
2. Elastic Collisions: No kinetic energy is lost in molecular collisions.
3. Negligible Volume: Gas molecules occupy very little space compared to the container volume.
4. No Intermolecular Forces: Molecules only interact during collisions.
⇒ Mathematical Formulation:
The kinetic theory leads to the expression for pressure in an ideal gas:
[math]P = \frac{1}{3} \frac{N m \overline{v}^2}{V}[/math]
Where:
– N is the number of molecules,
– m is the mass of one molecule,
– [math]\overline{v}^2[/math]is the mean square velocity of gas molecules,
– V is the volume.
From kinetic theory, the average kinetic energy per molecule is:
[math]\overline{E_k} = \frac{3}{2} k_B T[/math]
Where:
– [math]k_B[/math] = Boltzmann constant (38×10⁻²³ J/K),
– T = absolute temperature (K).
This shows that temperature is directly proportional to kinetic energy.
⇒ Real Gases vs. Ideal Gases
Ideal gases follow the ideal gas law perfectly.
Real gases deviate from ideal behavior at high pressures and low temperatures due to intermolecular forces and non-negligible molecular volume.
The van der Waals equation corrects for these deviations:
[math]\left(P + \frac{a}{V^2}\right)(V – b) = nRT[/math]
Where:
– a accounts for intermolecular forces,
– b accounts for molecular volume.
d) Derivation of the Ideal Gas Law from Empirical Gas Laws
The ideal gas law is derived from three fundamental gas laws:
1. Boyle’s Law (Constant Temperature, T)
[math]PV = \text{constant}[/math]
– As pressure increases, volume decreases proportionally (inverse relationship).
– This is due to more frequent molecular collisions when volume is reduced.
2. Charles’ Law (Constant Pressure, P)
[math]V \propto T \Rightarrow \frac{V}{T} = \text{constant}[/math]
Volume expands as temperature increases because molecules move faster.
3. Gay-Lussac’s Law (Constant Volume, V)
[math]P \propto T \Rightarrow \frac{P}{T} = \text{constant}[/math]
At constant volume, increasing temperature causes pressure to rise due to faster-moving molecules colliding more forcefully.
Figure 3 Gay-Lussac’s Law
⇒ Generalization into the Combined Gas Law
By combining these three laws, we get:
[math]\frac{PV}{T} = \text{constant}[/math]
This equation holds for a fixed amount of gas. If the number of particles changes, we introduce the ideal gas constant R, leading to:
[math]PV = nRT[/math]
Where:
– P = pressure (Pa),
– V = volume (m³),
– n = number of moles,
– R = universal gas constant (314 J/mol·K),
– T = absolute temperature (K).
This is the ideal gas law, which approximates real gas behavior at high temperatures and low pressures (where intermolecular forces are negligible).
e) The Equations Governing Ideal Gases
The ideal gas law can also be expressed in terms of number of molecules rather than moles.
⇒ Using Boltzmann’s Constant
Since the number of moles (n) is related to the number of molecules (N) by Avogadro’s constant ([math]N_A[/math]):
[math]n = \frac{N}{N_A}[/math]
Substituting this into [math]PV = nRT[/math] gives:
[math]PV = \frac{N}{N_A} RT[/math]
Since [math]k_B = \frac{R}{N_A}[/math] (Boltzmann’s constant), we get:
[math]PV = Nk_B T[/math]
Where:
– [math]k_B = 1.38 × 10^{-23} J/K[/math] is the Boltzmann constant,
– N is the number of molecules.
This equation is useful for microscopic analysis of gases.
f) Pressure and Molecular Collisions
Gas pressure arises due to molecular collisions with container walls.
⇒ Momentum Change from Collisions
A single gas molecule of mass mmm moving with velocity [math]v_x[/math] collides elastically with the wall and rebounds with velocity [math]-v_x[/math].
– Initial momentum
[math]p_i = mv_x[/math]
– Final momentum
[math]p_f = -mv_x[/math]
– Change in momentum
[math]∆p = p_f – p_i \\ ∆p = -mv_x – mv_x \\ ∆p = -2mv_x[/math]
If many molecules collide per second, the total force exerted on the wall is proportional to the sum of momentum changes. This leads to the pressure equation:
[math]P = \frac{1}{3} \rho \overline{v^2}[/math]
Where:
– [math]ρ = \frac{mN}{V}[/math] is the density of the gas,
– [math]\overline{v^2}[/math] is the mean square speed of molecules.
This equation shows that pressure is directly related to molecular motion.
Figure 4 Pressure and molecular collusion
g) Internal Energy of an Ideal Monatomic Gas
The internal energy (U) of a system refers to the total kinetic energy of its molecules since an ideal gas has no intermolecular forces (no potential energy)
For a monatomic ideal gas, where each atom has three translational degrees of freedom (motion along x,y,z), the average kinetic energy per molecule is given by the kinetic theory of gases:
[math]\langle E_k \rangle = \frac{3}{2} k_B T[/math]
Since internal energy is the sum of all molecules’ kinetic energies:
[math]U = N \langle E_k \rangle = \frac{3}{2} N k_B T[/math]
Where:
– U = total internal energy (J),
– N = total number of gas molecules,
– k_B = Boltzmann’s constant ([math]1.38 × 10^{-23} J/K[/math]),
– T = absolute temperature (K).
⇒ Alternative Form in Terms of Moles
Since the number of molecules is related to the number of moles by Avogadro’s number N_A:
[math]N = nN_A[/math]
and using the relation [math]R = k_B N_A[/math], we get:
[math]U = \frac{3}{2} nRT[/math]
Where:
– n = number of moles,
– R = universal gas constant (314 J/mol·K).
Thus, the internal energy of an ideal monatomic gas depends only on temperature and not on pressure or volume.
h) Ideal Gas:
An ideal gas assumes:
– No intermolecular forces (only elastic collisions),
– Molecules occupy negligible volume,
– Energy is purely kinetic.
⇒ Low Pressure
At low pressures, intermolecular forces become negligible, making real gases behave like ideal gases.
When pressure is high, molecules are closer together, and attractive forces (Van der Waals forces) cause deviations from ideal behavior.
⇒ High Temperature
At high temperatures, gas molecules move fast enough to overcome intermolecular attractions, ensuring that their behavior aligns with kinetic theory assumptions.
At low temperatures, molecules slow down, intermolecular attractions become significant, and condensation may occur.
⇒ Low Density
A gas behaves ideally when molecular density is low.
At high densities, gas molecules interact more frequently, leading to non-ideal behavior.
⇒ Large Volume
A gas is more ideal when it expands into a large volume because molecular interactions become insignificant.
Ideal Gas Approximation Holds When:
– Pressure is low,
– Temperature is high,
– Density is low,
– Volume is large.
Real gases deviate from ideal behavior under high pressure, low temperature, or high density.
Figure 5 Real gases and Ideal gases
⇒ Deviations from Ideal Gas Behavior
Real gases deviate from ideal gas laws at:
– High pressure (molecules are forced closer together, intermolecular forces matter).
– Low temperature (molecules slow down, attracting each other).
– High density (volume occupied by molecules is no longer negligible).
The Van der Waals equation corrects the ideal gas law by accounting for these effects:
[math]\left( P + \frac{a}{V^2} \right)(V – b) = nRT[/math]
Where:
– a accounts for intermolecular attraction,
– b accounts for finite molecular volume.
⇒ Summary of Ideal Gas Approximations

Condition	Ideal Gas Approximation Holds?
Low pressure	Yes, fewer intermolecular forces
High temperature	Yes, molecules move fast, reducing interactions
Low density	Yes, molecules are far apart
Large volume	Yes, reduces interaction frequency
High pressure	No, intermolecular forces become significant
Low temperature	No, condensation and attraction occur
High density	No, molecular volume is significant

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DP IB Physics: SL

B. The particulate nature of matter

B.3 Gas Laws

DP IB Physics: SL

B. The particulate nature of matter

B.3 Gas Laws

a) Pressure:

b) The Amount of Substance and Avogadro’s Constant

c) Ideal Gases and the Kinetic Theory of Gases

d) Derivation of the Ideal Gas Law from Empirical Gas Laws

e) The Equations Governing Ideal Gases

f) Pressure and Molecular Collisions

g) Internal Energy of an Ideal Monatomic Gas

h) Ideal Gas:

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