Rewrite the ideal gas law solving for \(n\). Also show how all units cancel to leave you with just units of moles.

The ideal gas law equation is given as: \(PV = nRT\) Step 2: Rearrange the equation to solve for n

To solve for n, we need to isolate n on one side of the equation. Divide both sides of the equation by RT: \[n = \frac{PV}{RT}\] Step 3: Show cancellation of units

Now, let's consider the units in the equation while ensuring they cancel out to give us just units of moles. The units of pressure (P) can be pascal (Pa) or atmospheres (atm), volume (V) is usually in liters (L) or cubic meters (m³), the gas constant (R) can have different values depending on the units: 8.314 J/(mol·K) or 0.08206 L·atm/(mol·K) and temperature (T) is in Kelvin (K). Expressing the ideal gas law in SI units: \[\frac{Pa \cdot m^3}{(\frac{J}{mol\cdot K}) \cdot K} = mol\] We need to show that J(Pa·m³) cancels out to leave the moles unit. 1 J (joule) is the standard unit of energy and can be defined as: \[1\,J = 1\,Pa \cdot m^3\] With this in mind, the units in the equation become: \[\frac{Pa \cdot m^3}{(\frac{Pa \cdot m^3}{mol\cdot K}) \cdot K} = mol\] As we can see, the units of Pa and m³ in both the numerator and denominator cancel each other out, leaving us with the unit of moles: \[\frac{1}{\frac{1}{mol}} = mol\] Therefore, we have successfully shown that the ideal gas law solved for n and all units cancel out to leave us with just units of moles.