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Equation of State

What is an Equation of State?

Think of a gas as a crowd of people in a room. The pressure (p) is how hard they push against the walls, the volume (V) is the size of the room, and the temperature (T) tells us how energetic they are. The equation of state links these three quantities together.

Ideal Gas Law – The Classic Formula

For an ideal gas we use the familiar formula:

\$pV = nRT\$

p – pressure (Pa)

V – volume (m³)

n – amount of substance (moles)

R – ideal gas constant ≈ 8.314 J mol⁻¹ K⁻¹

T – temperature (K)

Molecular Form – Counting Molecules

If we want to talk about individual molecules instead of moles, we replace n with N (number of molecules) and R with k (Boltzmann constant ≈ 1.38×10⁻²³ J K⁻¹):

\$pV = NkT\$

The two forms are equivalent because n = N/NA where NA is Avogadro’s number.

Quick Reference Table

Symbol	Meaning	Units
p	Pressure	Pa (N m⁻²)
V	Volume	m³
n	Moles of gas	mol
N	Number of molecules	dimensionless
R	Ideal gas constant	J mol⁻¹ K⁻¹
k	Boltzmann constant	J K⁻¹
T	Temperature	K

Real‑World Example

📚 Example: 1 mol of an ideal gas at 273 K (0 °C) and 101 kPa (1 atm) occupies:

Use the ideal gas law: \$pV = nRT\$

Insert values: \$V = \dfrac{nRT}{p} = \dfrac{1\times8.314\times273}{101000} \approx 0.0224\ \text{m}^3\$

Convert to litres: \$0.0224\ \text{m}^3 = 22.4\ \text{L}\$

So 1 mol of any ideal gas occupies 22.4 L at STP. This is a handy “rule of thumb” for quick calculations. 🌬️

Exam Tips Box

💡 Quick Tips for the Exam

Always check units – pressure in Pa, volume in m³, temperature in K.

Remember that n = N/N_A if you need to switch between moles and molecules.

When given p and T, you can find V if n is known, or vice‑versa.

Use the conversion factor 1 atm = 101 325 Pa if the problem uses atmospheres.

For quick estimates, 1 mol at STP ≈ 22.4 L.

recall and use the equation of state for an ideal gas expressed as pV = nRT, where n = amount of substance (number of moles) and as pV = NkT, where N = number of molecules