According to the ideal gas law: (a) If you measured \(P, V, n\), and \(T\) for any gas sample and then calculated the quantity \(P V / n T\), what would be the units and numerical value of the result? (b) If you measured \(P, V, n\), and \(T\) for any gas sample and then calculated \(P V / n R T\), what would be the units and numerical value of the result?

To find the units and numerical value of \(PV / nT\), let's first simplify the expression using the ideal gas law equation. We already know that the ideal gas law equation is: \[PV = nRT\] Now, substitute the ideal gas law equation into the given expression: \[\frac{PV}{nT} = \frac{nRT}{nT}\] The \(n\) and \(T\) values can be canceled out: \[\frac{PV}{nT} = R\] Now that we have simplified the expression to the universal gas constant \(R\), we can now determine the units and numerical value for \(PV / nT\). The units of the universal gas constant \(R\) are Joules per mole Kelvin (J/mol.K). Therefore, the units for \(PV / nT\) are J/mol.K. The numerical value of the universal gas constant \(R\) is approximately 8.31 J/mol.K. Hence, the numerical value for \(PV / nT\) is approximately 8.31.

To find the units and numerical value of \(PV / nRT\), let's first simplify the expression using the ideal gas law equation. We already know that the ideal gas law equation is: \[PV = nRT\] Now, substitute the ideal gas law equation into the given expression: \[\frac{PV}{nRT} = \frac{nRT}{nRT}\] The entire expression can be canceled out, thus leaving a dimensionless value with no units: \[\frac{PV}{nRT} = 1\] So, for part (b), the units of the result will be dimensionless or without any units, and the numerical value of the result will be 1.