Question

The correct value of the gas constant \({ }^{\prime} R\) is close to (a) \(0.082\) litre-atmosphere \(\mathrm{K}\) (b) \(0.082\) litre-atmosphere \(\mathrm{K}^{-1} \mathrm{~mol}^{-1}\) (c) \(0.082\) litre-atmosphere \(^{-1} \mathrm{~K} \mathrm{~mol}^{-1}\) (d) \(0.082\) litre \(^{-1}\) atmosphere \(^{-1} \mathrm{~K} \mathrm{~mol}\).

Short answer

The correct value for the gas constant R is option (b) 0.082 litre-atmosphere K^{-1} mol^{-1}.

Step-by-step solution

Identify the Correct Units for Gas Constant

Recall the ideal gas law, which is given by the equation PV = nRT, where P is the pressure, V is the volume, n is the number of moles, T is the temperature, and R is the gas constant. The units of R should be such that when multiplied with the number of moles (n) and the temperature (T), it gives a product with units of pressure (P) times volume (V).

Assess Unit Consistency for Each Option

The pressure is measured in atmospheres, volume in litres, temperature in Kelvin (K), and the amount of substance in moles (mol). The gas constant should have units that yield atmospheres multiplied by litres when moles and Kelvin are taken into account.

Determine the Correct Option

Option (b) '0.082 litre-atmosphere K^{-1} mol^{-1}' correctly implies that per Kelvin per mole, the pressure-volume product is 0.082 litre-atmosphere. The other options either do not include all the necessary units or incorrectly imply direct proportionality or inverse proportionality where it does not exist.

Key concepts

Understanding the Ideal Gas Law

At the core of many chemical and physical studies is an elegant equation known as the ideal gas law, expressed as PV = nRT. This law provides a relationship between four fundamental physical properties: pressure (P), volume (V), number of moles of gas (n), and temperature (T), with R being the gas constant. Crucial to the law is the assumption that the gas behaves 'ideally', meaning its molecules are point particles with no volume or interactions, and energy is purely kinetic. While no gas is truly ideal, the law approximates behavior for most gases under standard conditions.

Each variable in the equation is significant. Pressure is the force applied by the gas per unit area and is influenced by how frequently and forcefully the gas molecules collide with the walls of their container. Volume refers to the space the gas occupies. The number of moles (n) relays the quantity of gas particles present. Finally, temperature measures the average kinetic energy of the gas particles. All these variables are intertwined, and changing one impacts the others—consistent with states of matter principles.

When solving problems involving the ideal gas law, it's crucial to think about what you know and what you need to find out, making this principle a cornerstone in problem-solving within chemistry.

Units of Measurement in Gas Laws

Units are the language through which we communicate quantities in science, enabling us to understand the magnitude and measure of what we observe. In the context of gas laws, unit consistency is paramount to obtain meaningful results. Pressure may be expressed in atmospheres, pascals, or torr among others, while volume can be conveyed in liters or cubic meters. Temperature is typically measured in Kelvin because it is an absolute scale, starting at absolute zero, where there is no particle motion.

An important aspect of units in the context of the ideal gas law is ensuring that they are compatible across the equation. The gas constant R has various unit expressions to match the pressure and volume units utilized. For instance, if pressure is measured in atmospheres and volume in liters, R is commonly expressed as liters · atmosphere per mole · Kelvin (litre-atmosphere K^-1 mol^-1). Ensuring correct units when calculating with the ideal gas law prevents errors and provides accurate and consistent results. The subtleties in unit conversion often pose challenges, but mastering them is essential for proficiency in chemistry and physics alike.

Stoichiometry and Gas Laws

Stoichiometry is the mathematical relationship between the quantities of reactants and products in a chemical reaction. It stems from the law of conservation of mass and the concept that matter is neither created nor destroyed in a reaction. When dealing with gases, the ideal gas law becomes a powerful tool in stoichiometric calculations.

The key advantage of applying the ideal gas law in stoichiometry lies in its ability to bridge molar quantities of gases with conditions of pressure, volume, and temperature. By rearranging the law to solve for n, the number of moles, one can link quantitative gas conditions to the mole ratios derived from balanced chemical equations. This integration allows for calculation of how much reactant is needed or how much product is produced under specific conditions. Furthermore, since gases occupy space, their volumes are directly related to the quantities involved in reactions, making the ideal gas law an indispensable aspect of any gas-related stoichiometric calculation. Familiarity with both stoichiometry and the ideal gas law thus empowers students with a broader understanding of chemical reactions in gaseous states.