How many moles of \(\mathrm{O}_{2}\) are contained in \(5.25 \mathrm{~L}\) at \(26^{\circ} \mathrm{C}\) and \(1.2 \mathrm{~atm}\) ?

The concept of 'moles of a gas' is fundamental to understanding gas behavior and the stoichiometry involved in chemical reactions. A mole is a unit of measurement used in chemistry to express amounts of a substance. One mole is equivalent to Avogadro's number, which is approximately 6.022 × 10²³ particles (atoms, molecules, ions, etc.).

In the context of gases, a mole refers to the amount of gas particles that would occupy 22.4 liters at standard temperature and pressure (STP), which are 0°C and 1 atmosphere of pressure. However, when conditions deviate from STP, we need to apply the Ideal Gas Law to find out how many moles we're dealing with under new conditions of temperature and pressure.

The Ideal Gas Law, denoted as PV=nRT, is an equation that relates the pressure (P), volume (V), number of moles (n), and temperature (T) of a gas, with R being the gas constant. This law is a powerful tool in chemistry for it allows us to predict how a gas will behave under different conditions.

If we know any three of the variables in the equation, we can solve for the fourth. For instance, if we are given the pressure, volume, and temperature of a gas, we can easily calculate the number of moles present in the gas. In the case of the exercise example, we are given the oxygen gas volume, temperature, and pressure and asked to solve for the number of moles.

Temperature calculations in gas laws require that we use the Kelvin scale. This absolute temperature scale starts at absolute zero, the theoretically lowest possible temperature. To convert from Celsius to Kelvin, we add 273.15 to the Celsius temperature.

Why use Kelvin? The Kelvin scale ensures that there are no negative values in our calculations, which could invalidate the relations established by gas laws. Degrees Celsius can be negative, which would not make sense in the context of calculations involving temperatures of gases because the volume or pressure can't be negative.

The gas constant R is a proportionality constant that appears in the Ideal Gas Law and is crucial for calculations involving gaseous substances. Its value depends on the units used for pressure, volume, and temperature. In the given solution, R is given as 0.0821 L atm mol⁻¹ K⁻¹, which is appropriate for pressure in atmospheres, volume in liters, and temperature in Kelvin.

This constant is derived from experimental measurements and essentially encapsulates the behavior of an 'ideal' gas, which is a hypothetical gas that perfectly follows the Ideal Gas Law. In reality, no gas is truly ideal, but many gases behave closely enough to ideal that the law is very useful for predicting their behavior in a variety of conditions.