Equation of state

1. Overview

The equation of state for an ideal gas is a fundamental relationship that links the macroscopic physical properties of a gas: pressure ($p$), volume ($V$), and thermodynamic temperature ($T$). It serves as a mathematical model to describe how a gas behaves under varying conditions. An ideal gas is a theoretical construct that perfectly obeys this relationship across all ranges of temperature and pressure. In reality, while no gas is truly "ideal," most real gases (such as hydrogen, helium, and nitrogen) behave very much like an ideal gas at low pressures and high temperatures (well above their boiling points), where intermolecular forces are negligible.

Key Definitions

• Ideal Gas: A gas that obeys the equation $pV \propto T$ (or $pV = nRT$) for all values of pressure, volume, and thermodynamic temperature, where $T$ is the thermodynamic temperature in Kelvin.
• Mole ($n$): The SI unit for the amount of substance. One mole is defined as the amount of substance containing exactly $6.02214076 \times 10^{23}$ elementary entities (atoms, molecules, or ions).
• Avogadro Constant ($N_A$): The number of elementary entities per mole of substance, equal to $6.02 \times 10^{23} \text{ mol}^{-1}$.
• Thermodynamic Temperature ($T$): A temperature scale that does not depend on the properties of any particular substance. It is measured in Kelvin (K) and starts at absolute zero ($0 \text{ K}$), the temperature at which the internal energy of a system is at its minimum.
• Boltzmann Constant ($k$): A fundamental physical constant that relates the average kinetic energy of particles in a gas with the thermodynamic temperature. It is defined as the molar gas constant per molecule ($k = R / N_A$).
• Molar Gas Constant ($R$): The constant of proportionality in the ideal gas equation when the amount of substance is expressed in moles, equal to $8.31 \text{ J K}^{-1} \text{ mol}^{-1}$.

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Content

3.1 The Ideal Gas Concept

The ideal gas law is the culmination of three experimental gas laws discovered over centuries:

1. Boyle’s Law: $pV = \text{constant}$ (at constant $T$)
2. Charles’s Law: $V/T = \text{constant}$ (at constant $p$)
3. Gay-Lussac’s (Pressure) Law: $p/T = \text{constant}$ (at constant $V$)

When these are combined for a fixed mass of gas, we derive the relationship: $$\frac{pV}{T} = \text{constant}$$

An ideal gas is defined as a gas that strictly follows this relationship. In the kinetic theory of gases, this implies that:

• The volume of the gas molecules themselves is negligible compared to the volume of the container.
• There are no intermolecular forces (forces of attraction or repulsion) between the molecules, except during collisions.
• All internal energy is in the form of random kinetic energy.

3.2 The Molar Ideal Gas Equation

When we consider the amount of gas in terms of moles ($n$), the constant of proportionality is the Molar Gas Constant ($R$).

$$pV = nRT$$

• $p$: Pressure in Pascals (Pa) or $\text{N m}^{-2}$.
• $V$: Volume in cubic metres ($\text{m}^3$).
• $n$: Amount of substance in moles (mol).
• $R$: Molar gas constant ($8.31 \text{ J K}^{-1} \text{ mol}^{-1}$).
• $T$: Thermodynamic temperature in Kelvin (K).

3.3 The Molecular Ideal Gas Equation

In many physics contexts, it is more useful to consider the total number of molecules ($N$) rather than the number of moles. For this, we use the Boltzmann constant ($k$).

$$pV = NkT$$

• $N$: Total number of molecules (a dimensionless integer).
• $k$: Boltzmann constant ($1.38 \times 10^{-23} \text{ J K}^{-1}$).

This form of the equation is particularly useful when linking macroscopic measurements ($p, V, T$) to the microscopic behavior of individual particles in the kinetic theory of gases.

3.4 Relationship between $R$, $k$, and $N_A$

The constants $R$ and $k$ are fundamentally linked by the Avogadro constant. $R$ represents the gas constant for a "mole's worth" of particles, while $k$ represents the gas constant for a "single particle."

Derivation:

1. The number of molecules $N$ is equal to the number of moles $n$ multiplied by the number of molecules per mole $N_A$: $$N = n N_A$$
2. Substitute $N$ into the molecular equation: $$pV = (n N_A) kT$$
3. Compare this to the molar equation $pV = nRT$: $$nRT = n N_A kT$$
4. Cancel $n$ from both sides: $$RT = N_A kT$$
5. Rearrange to define the Boltzmann constant: $$k = \frac{R}{N_A}$$

3.5 The Thermodynamic (Kelvin) Temperature Scale

The ideal gas equation only functions if temperature is measured on an absolute scale. The Kelvin scale is used because $T = 0 \text{ K}$ (absolute zero) corresponds to the state where $pV = 0$ (theoretically, the volume or pressure of an ideal gas would vanish).

• Conversion: $T \text{ (K)} = \theta \text{ (}^\circ\text{C)} + 273.15$
• In most A-Level problems, using 273 is acceptable unless the data provided uses decimals.
• Crucial Note: A change in temperature of $1 \text{ K}$ is identical to a change of $1^\circ\text{C}$. However, the absolute values used in $pV=nRT$ must always be in Kelvin.

3.6 The Ratio Method for Changing States

If a fixed amount of gas ($n$ is constant) undergoes a change from state 1 to state 2, we can use the ratio: $$\frac{p_1 V_1}{T_1} = \frac{p_2 V_2}{T_2}$$ This is often the most efficient way to solve exam problems where a gas is compressed, expanded, or heated without any gas escaping.

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3.7 Worked Examples

Worked Example 1 — Calculating Molar Quantity

A high-pressure cylinder has a volume of $0.050 \text{ m}^3$ and contains Oxygen gas at a pressure of $2.0 \times 10^7 \text{ Pa}$ at a room temperature of $21^\circ\text{C}$. Calculate the amount of Oxygen in moles.

1. Identify and Convert Units:
   • $p = 2.0 \times 10^7 \text{ Pa}$
   • $V = 0.050 \text{ m}^3$
   • $T = 21 + 273 = 294 \text{ K}$
2. Select Equation: $pV = nRT \rightarrow n = \frac{pV}{RT}$
3. Substitution: $$n = \frac{(2.0 \times 10^7) \times 0.050}{8.31 \times 294}$$ $$n = \frac{1,000,000}{2443.14}$$
4. Final Answer: $n = 409.3 \dots \approx 410 \text{ mol}$ (to 2 sig figs, matching the input data).

Worked Example 2 — The Ratio Method (Weather Balloon)

A weather balloon is filled with $15 \text{ m}^3$ of Helium at the Earth's surface, where the pressure is $101 \text{ kPa}$ and the temperature is $25^\circ\text{C}$. The balloon rises to an altitude where the pressure is $30 \text{ kPa}$ and the temperature is $-40^\circ\text{C}$. Calculate the new volume of the balloon.

1. Identify Variables:
   • State 1: $p_1 = 101 \times 10^3 \text{ Pa}$, $V_1 = 15 \text{ m}^3$, $T_1 = 25 + 273 = 298 \text{ K}$
   • State 2: $p_2 = 30 \times 10^3 \text{ Pa}$, $V_2 = ?$, $T_2 = -40 + 273 = 233 \text{ K}$
2. Set up Ratio: $\frac{p_1 V_1}{T_1} = \frac{p_2 V_2}{T_2}$
3. Rearrange for $V_2$: $$V_2 = \frac{p_1 V_1 T_2}{T_1 p_2}$$
4. Substitution: $$V_2 = \frac{(101 \times 10^3) \times 15 \times 233}{298 \times (30 \times 10^3)}$$ $$V_2 = \frac{352,995,000}{8,940,000}$$
5. Final Answer: $V_2 = 39.48 \dots \approx 39 \text{ m}^3$ (to 2 sig figs).

Worked Example 3 — Molecular Calculations

A small glass bulb of volume $250 \text{ cm}^3$ is evacuated to a "high vacuum" pressure of $1.0 \times 10^{-7} \text{ Pa}$ at $20^\circ\text{C}$. Calculate the number of molecules remaining in the bulb.

1. Convert Units:
   • $V = 250 \text{ cm}^3 = 250 \times (10^{-2})^3 \text{ m}^3 = 2.5 \times 10^{-4} \text{ m}^3$
   • $T = 20 + 273 = 293 \text{ K}$
2. Select Equation: $pV = NkT \rightarrow N = \frac{pV}{kT}$
3. Substitution: $$N = \frac{(1.0 \times 10^{-7}) \times (2.5 \times 10^{-4})}{(1.38 \times 10^{-23}) \times 293}$$ $$N = \frac{2.5 \times 10^{-11}}{4.0434 \times 10^{-21}}$$
4. Final Answer: $N = 6,182,915 \approx 6.2 \times 10^6 \text{ molecules}$. (Note: Even in a "vacuum," millions of molecules remain!)

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Key Equations

Equation	Description	Data Sheet?
$pV = nRT$	Ideal gas equation (molar form)	Yes
$pV = NkT$	Ideal gas equation (molecular form)	Yes
$k = \frac{R}{N_A}$	Definition of Boltzmann constant	Yes
$n = \frac{N}{N_A}$	Relationship between moles and molecules	No
$n = \frac{m}{M}$	Moles = mass / molar mass	No
$T = \theta + 273.15$	Celsius to Kelvin conversion	Yes

Constants to use:

• $R = 8.31 \text{ J K}^{-1} \text{ mol}^{-1}$
• $k = 1.38 \times 10^{-23} \text{ J K}^{-1}$
• $N_A = 6.02 \times 10^{23} \text{ mol}^{-1}$

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Common Mistakes to Avoid

• ❌ Wrong: Using Volume in $\text{cm}^3$, $\text{dm}^3$, or Litres.
  • ✓ Right: Always convert to $\text{m}^3$.
  • $1 \text{ cm}^3 = 10^{-6} \text{ m}^3$
  • $1 \text{ dm}^3 = 1 \text{ Litre} = 10^{-3} \text{ m}^3$
• ❌ Wrong: Using Temperature in Celsius ($^\circ\text{C}$).
  • ✓ Right: Always convert to Kelvin (K). Even if the question gives a temperature change, if you are using $pV=nRT$, the absolute value must be Kelvin.
• ❌ Wrong: Confusing $n$ and $N$.
  • ✓ Right: $n$ is for moles (small number, usually $0.1$ to $100$). $N$ is for molecules (huge number, usually $10^{20}$ or higher).
• ❌ Wrong: Using the wrong constant with the wrong quantity (e.g., $pV = NRT$).
  • ✓ Right: Remember the "M" rule: Moles goes with the Molar gas constant ($n$ with $R$).
• ❌ Wrong: Forgetting that $p$ must be in Pascals, not kPa or MPa.
  • ✓ Right: $100 \text{ kPa} = 100,000 \text{ Pa}$.

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Exam Tips

1. The "Ideal Gas" Definition: If asked to define an ideal gas, you must mention that it obeys $pV \propto T$ and specify that $T$ is the thermodynamic temperature. Simply saying "it follows $pV=nRT$" is often not enough for full marks without defining $T$.
2. Unit Check First: Before starting any calculation, write down $p, V, n, R, T$ and convert them all to SI units immediately. This prevents "silly" errors later in the algebra.
3. Mass and Moles: Be prepared to link this topic to chemistry. If a question gives you the mass of a gas and its molar mass ($M$), use $n = m/M$ to find the number of moles before using $pV=nRT$.
4. Standard Form: Gas law problems often involve very large ($N_A$) or very small ($k$) numbers. Use the scientific mode on your calculator and double-check your powers of 10.
5. Graphing:
   • A graph of $pV$ against $T$ (in K) is a straight line through the origin.
   • A graph of $p$ against $V$ (at constant $T$) is an isothermal hyperbola ($p \propto 1/V$).
   • A graph of $p$ against $1/V$ is a straight line through the origin.
6. Significant Figures: Look at the values provided in the question. If the pressure is $2.0 \text{ Pa}$ (2 sig figs) and volume is $1.55 \text{ m}^3$ (3 sig figs), provide your final answer to 2 sig figs.