Confirm that the units of n R T are those of energy for each...

Confirm that the units of $n R T$ are those of energy for each value of $R:$ (a) $8.31 \mathrm{J} / \mathrm{mol} \cdot \mathrm{K},$ (b) $1.99 \mathrm{cal} / \mathrm{mol} \cdot \mathrm{K},$ and $(\mathrm{c}) 0.0821 \mathrm{L} \cdot \mathrm{atm} / \mathrm{mol} \cdot \mathrm{K}$.

Key Concepts

Dimensional Analysis

Dimensional analysis is a technique used to verify that the units on both sides of an equation are consistent. In this context, it involves checking that when the number of moles, the gas constant, and the temperature are multiplied together, the resulting units are equivalent to those of energy.

Gas Constant (R)

The gas constant, denoted by R, is a fundamental constant in thermodynamics that appears in the ideal gas law. It can be expressed in various units, such as joules per mole-kelvin, calories per mole-kelvin, or liter-atmosphere per mole-kelvin, all of which, when used properly, yield energy units upon multiplication with the number of moles and temperature.

Energy Units and Unit Consistency

Energy is a measurable physical quantity that can be represented in multiple unit systems. Confirming that the product nRT has the units of energy involves ensuring that the different forms of R, when combined with the appropriate units for moles and temperature, consistently produce a unit of energy, whether it be joules, calories, or liter-atmospheres.

Ideal Gas Law

The ideal gas law, expressed as PV = nRT, connects pressure, volume, number of moles, and temperature of an ideal gas. In the expression nRT, understanding the role of each variable through the lens of the ideal gas law helps in confirming that the product of these quantities results in energy, reinforcing the internal consistency within the law.

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