The Ideal Gas Law Derivations: Videos & Practice Problems

The ideal gas law is a fundamental equation in chemistry that relates pressure (P), volume (V), temperature (T), and the number of moles (n) of a gas. It is expressed as $P\cdot V=n\cdot R\cdot T$, where R is the ideal gas constant. This equation can be rearranged to solve for any one variable when the others are known. For example, to find pressure, rearrange to $P=\frac{\mathrm{nRT}}{V}$. Similarly, volume can be found by $V=\frac{\mathrm{nRT}}{P}$. These rearrangements are essential when solving problems involving changes in gas conditions or when two sets of variables are given.

When a gas undergoes a change in conditions, such as pressure, volume, or temperature, the ideal gas law can be applied to both initial and final states. By setting up the equation for each state, $P\cdot V=n\cdot R\cdot T$, and assuming the amount of gas (n) and R remain constant, you can derive the combined gas law: $\frac{\mathrm{P\_1}}{\mathrm{T\_1}}\cdot \mathrm{V\_1}=\frac{\mathrm{P\_2}}{\mathrm{T\_2}}\cdot \mathrm{V\_2}$. This allows you to solve for an unknown variable when two sets of conditions are given, making it easier to analyze gas behavior during changes.

The combined gas law is a useful equation that relates pressure, volume, and temperature of a gas when the amount of gas is constant. It is derived by combining Boyle's, Charles's, and Gay-Lussac's laws, all of which are special cases of the ideal gas law. Starting from the ideal gas law $P\cdot V=n\cdot R\cdot T$, and assuming n and R are constant, we get $\frac{P}{T}\cdot V=\; constant$. Comparing two states, the combined gas law is $\frac{\mathrm{P\_1}}{\mathrm{T\_1}}\cdot \mathrm{V\_1}=\frac{\mathrm{P\_2}}{\mathrm{T\_2}}\cdot \mathrm{V\_2}$. This equation helps solve problems involving changes in gas conditions.

To calculate the number of moles (n) of a gas, you can rearrange the ideal gas law $P\cdot V=n\cdot R\cdot T$ to solve for n: $n=\frac{P}{\cdot }/\mathrm{RT}$. Here, P is pressure, V is volume, R is the ideal gas constant, and T is temperature in Kelvin. By plugging in the known values, you can find the amount of gas in moles. This is particularly useful in stoichiometry and gas reaction calculations.

Temperature must be in Kelvin when using the ideal gas law because the law is based on absolute temperature, which starts at absolute zero (0 K). Kelvin is an absolute scale where 0 K represents the point at which particles have minimal kinetic energy. Using Celsius or Fahrenheit would lead to incorrect results because these scales do not start at absolute zero. For example, the ideal gas law $P\cdot V=n\cdot R\cdot T$ requires T in Kelvin to maintain proportionality between variables. Always convert Celsius to Kelvin by adding 273.15 before calculations.