The Universal Gas Constant, symbolized as \(R\), is a fundamental proportionality factor that connects the macroscopic properties of gases. This constant is a single, fixed value in physics and chemistry, but its numerical expression changes depending on the units used for pressure, volume, and temperature measurements. It provides the necessary link to relate the amount of a gaseous substance to its measurable physical conditions.
The Relationship Between Pressure, Volume, and Temperature
The theoretical foundation for calculating \(R\) is the Ideal Gas Law, expressed by the equation \(PV = nRT\). This single mathematical expression combines the empirical observations from earlier gas laws, showing how the four main variables that define a gas sample are interconnected. The equation describes the behavior of an “ideal gas,” a theoretical concept where particles have no volume and no intermolecular forces, which serves as an excellent approximation for most real gases under ordinary conditions.
In the formula, \(P\) represents the absolute pressure of the gas, and \(V\) is the volume it occupies, while \(n\) stands for the amount of substance measured in moles. The variable \(T\) is the absolute temperature, which must always be expressed in Kelvin for the relationship to hold true. The symbol \(R\) is the Universal Gas Constant, which acts as the necessary scaling factor to ensure the equation is dimensionally balanced.
Establishing the Inputs: Standard Conditions
To determine a constant, scientists must use fixed, known reference points, which are provided by a set of conditions known as Standard Temperature and Pressure (STP). For the purpose of calculating \(R\), the historical definition of STP is typically used, which sets the standard temperature at 0 degrees Celsius, or \(273.15 \text{ K}\). The standard pressure is set at one atmosphere (\(1 \text{ atm}\)).
These conditions establish a fixed volume for a specific amount of gas. A mole (\(n = 1 \text{ mol}\)) of any ideal gas under these STP conditions occupies approximately \(22.4 \text{ liters}\) (\(V = 22.4 \text{ L}\)), a value known as the molar volume. By using these four known values—\(P\), \(V\), \(n\), and \(T\)—the Universal Gas Constant can be mathematically isolated and calculated.
Step-by-Step Derivation of the Constant
The calculation begins by rearranging the Ideal Gas Law equation, \(PV = nRT\), to solve for \(R\). Dividing both sides of the equation by \(nT\) yields the expression \(R = \frac{PV}{nT}\). This allows the constant to be determined by plugging in the established numerical values for one mole of gas at STP.
The values substituted into the equation are \(P = 1.0 \text{ atm}\), \(V = 22.4 \text{ L}\), \(n = 1.0 \text{ mol}\), and \(T = 273.15 \text{ K}\).
Performing the multiplication of the numerator, \(P \times V\), gives \(22.4 \text{ L} \cdot \text{atm}\). The denominator, \(n \times T\), results in \(273.15 \text{ mol} \cdot \text{K}\).
Dividing the two products, \(\frac{22.4 \text{ L} \cdot \text{atm}}{273.15 \text{ mol} \cdot \text{K}}\), results in a value of approximately \(0.08206\). This numerical result is the value of the Universal Gas Constant, \(R\), when expressed in the units of \(\text{L} \cdot \text{atm}/\text{mol} \cdot \text{K}\). The resulting units directly reflect the constant’s role as a proportionality factor.
Interpreting the Constant Across Different Units
The numerical value of the Universal Gas Constant changes based on the units of measurement used for the other variables. The value calculated using atmospheres and liters, \(R \approx 0.08206 \text{ L} \cdot \text{atm}/\text{mol} \cdot \text{K}\), is commonly used in chemistry problems where pressure is given in atmospheres. The SI unit value of \(R\) is most frequently encountered in physics and engineering applications.
The SI value is \(R \approx 8.314 \text{ J}/\text{mol} \cdot \text{K}\). This different number arises because the product of pressure and volume, \(P \times V\), has the physical dimensions of energy, which is measured in Joules (\(\text{J}\)) in the SI system. To calculate this value, pressure must be in Pascals (\(\text{Pa}\)) and volume in cubic meters (\(\text{m}^3\)). Understanding that the constant represents energy per mole per degree of temperature helps clarify why the numerical value changes.