Final answer:
The ideal gas law, PV=nRT, when analyzed unit by unit, reveals that its components can be expressed in energy units, specifically joules (J). The pressure multiplied by volume (PV) results in units of energy, and when matched with the correct unit of the ideal gas constant (R), the equation aligns perfectly with the concept of energy.
Step-by-step explanation:
The ideal gas law, PV=nRT, relates several different physical quantities: pressure (P), volume (V), the number of moles (n), the ideal gas constant (R), and temperature (T). The equation shows the relationship between these variables when we consider a quantity of ideal gas.
To show that the units of PV=nRT correspond to those of energy, we have to look at each component of the equation:
- Pressure (P) is often measured in atmospheres (atm) but can be converted to Pascals (Pa), where 1 atm = 101325 Pa. Pressure is force per unit area; Pa is equivalent to newtons per square meter (N/m²).
- Volume (V) is measured in liters (L) but can be converted to cubic meters (m³), where 1 L = 0.001 m³.
- The number of moles (n) is a dimensionless unit representing the amount of substance.
- The temperature (T) is measured in Kelvin (K).
- The ideal gas constant (R) has various units, but commonly it's J/(K·mol) where J is Joules, K is Kelvin, and mol is moles. In this case, R will have the units of energy per mole per temperature.
By substituting the unit equivalents into the equation PV=nRT, we can see that the resulting units are indeed joules (J), the unit for energy (kg·m²/s²).
Caution: When performing such calculations, it is crucial to ensure that the units of R are compatible with the units of P, V, n, and T.