10.1 Ideal gas law and its applications

Ideal Gas Law

Ideal gas law fundamentals

The ideal gas law connects four key properties of a gas: pressure, volume, temperature, and the amount of gas present. It assumes the gas behaves "ideally," meaning it follows the assumptions of the kinetic molecular theory.

An ideal gas is a hypothetical gas whose particles:

• Have negligible volume compared to the container they occupy
• Exert no attractive or repulsive forces on each other
• Undergo perfectly elastic collisions (no energy lost)
• Have average kinetic energy directly proportional to absolute temperature

The ideal gas law equation is:

$PV = nRT$

where:

• $P$ = pressure (in pascals, Pa, or atmospheres, atm)
• $V$ = volume (in cubic meters, $m^3$, or liters, L)
• $n$ = amount of gas (in moles)
• $R$ = ideal gas constant
• $T$ = absolute temperature (in Kelvin, K)

The value of $R$ depends on which units you're using. If pressure is in pascals and volume in cubic meters, use $R = 8.314 \text{ J/(mol·K)}$. If pressure is in atmospheres and volume in liters, use $R = 0.08206 \text{ L·atm/(mol·K)}$. Matching $R$ to your units is one of the most common places students make mistakes.

Problem-solving with the ideal gas law

You can rearrange $PV = nRT$ to isolate whichever variable you need:

1. Solving for pressure: $P = \frac{nRT}{V}$
2. Solving for volume: $V = \frac{nRT}{P}$
3. Solving for moles of gas: $n = \frac{PV}{RT}$
4. Solving for temperature: $T = \frac{PV}{nR}$

Steps for solving any ideal gas law problem:

1. Identify which variable you're solving for and which three are given.
2. Convert temperature to Kelvin if it's given in Celsius: $T_K = T_C + 273.15$. The ideal gas law only works with absolute temperature.
3. Check that your units for $P$, $V$, and $T$ are consistent with the value of $R$ you're using.
4. Plug in values and solve.

Quick example: Find the volume of 2.0 mol of gas at 1.0 atm and 25°C.

• Convert temperature: $T = 25 + 273.15 = 298.15 \text{ K}$
• Use $R = 0.08206 \text{ L·atm/(mol·K)}$ since pressure is in atm
• $V = \frac{nRT}{P} = \frac{(2.0)(0.08206)(298.15)}{1.0} = 48.9 \text{ L}$

Assumptions and limitations of the ideal gas law

The ideal gas law works well under many conditions, but it rests on simplifying assumptions that don't always hold.

The key assumptions are:

• Gas particles take up zero volume (they're treated as point masses)
• There are no intermolecular forces between particles
• All collisions are perfectly elastic
• Average kinetic energy scales linearly with absolute temperature

Where the law breaks down:

• High pressures: Gas particles get forced close together, so their actual volume matters and intermolecular forces become significant. For example, at pressures above ~100 atm, most gases show noticeable deviation.
• Low temperatures: As a gas cools toward its condensation point, attractive forces between molecules become stronger relative to their kinetic energy, pulling behavior away from the ideal model.
• Near phase transitions: Right before a gas condenses into a liquid, the ideal gas law becomes quite inaccurate because intermolecular attractions dominate.

A useful rule of thumb: gases behave most ideally at high temperatures and low pressures, where particles are far apart and moving fast.

Real-world applications of the ideal gas law

Calculating gas density under specific conditions is a direct application. Since density $\rho = \frac{m}{V}$ and $m = nM$ (where $M$ is the molar mass), you can substitute into the ideal gas law to get:

$\rho = \frac{PM}{RT}$

This tells you that gas density increases with pressure and decreases with temperature, which makes physical sense.

Determining the molar mass of an unknown gas works in reverse. If you measure the mass, pressure, volume, and temperature of a gas sample, you can find $n$ from the ideal gas law and then calculate $M = \frac{m}{n}$.

Engineering applications include analyzing gases in internal combustion engines. During the compression stroke, the volume of the gas in the cylinder decreases rapidly. The ideal gas law lets you estimate the resulting pressure increase, which is central to engine design.

Atmospheric science uses the ideal gas law to relate pressure and altitude. Atmospheric pressure drops with increasing altitude because there's less air above you. This relationship allows rough altitude estimates from pressure readings, which is how altimeters in aircraft work.