The value of the proportionality constant \(R\), can be calculated from the fact that exactly one mole of a gas at exactly 1 atm and at 0 "C \((273 \mathrm{~K})\) has a volume of \(22.414 \mathrm{~L}\). Solution Substituting in the equation: $$ \begin{array}{c} P V=n R T \text { or } R=\frac{P V}{n T} \\ R=\frac{(1 a t m)(22.414 L)}{(1 m o l e)(273 K)}=0.082057 L \text { atm } m o l^{-1} K^{-1} \end{array} $$

The universal gas constant, denoted as 'R', is a fundamental parameter in thermodynamics that appears in the ideal gas law. It's called 'universal' because it has the same value for all ideal gases, irrespective of their chemical identity or properties. This constant is crucial because it relates the energy scale to the gas scale when dealing with molecular gases.

R's value is determined through experimental measurements. To remember the significance of 'R', think of it as a conversion factor that balances the gas equation. It ensures that when we measure pressure (P), volume (V), and temperature (T) in their respective SI units, we attain consistent, reliable results in chemical calculations involving gases.

To calculate the gas constant 'R', we must refer to a known set of conditions for a gas. A common approach is to use standard temperature and pressure conditions, where the volume of one mole of an ideal gas is known. This volume is about 22.414 liters at 0°C (273.15 K) and 1 atmosphere of pressure.

The calculation uses the Ideal Gas Law, rearranged to solve for R: \( R = \frac{PV}{nT} \). We substitute the measured values into this equation. With these standardized conditions (P=1 atm, V=22.414 L, n=1 mole, T=273 K), we can solve for R. The calculated R is consistent and reproducible, allowing chemists and physicists to carry out reliable gas-related calculations.

The Ideal Gas Law is encapsulated in the formula \(PV=nRT\), a cornerstone in the study of thermodynamics and chemistry. Each variable represents a physical property of a gas:

• \(P\) stands for pressure, the force exerted by the gas per unit area.
• \(V\) denotes volume, the space occupied by the gas.
• \(n\) represents the number of moles, a measure of the amount of substance.
• \(R\) is the already discussed universal gas constant.
• \(T\) is the temperature, measured in Kelvin.

The equation states that the product of pressure and volume of a fixed amount of an ideal gas is directly proportional to its temperature. This relationship is vital for explaining how changes in temperature affect gas pressure and volume, which in turn is crucial for various applications, from industrial processes to predicting weather patterns.