For the reversible reaction: \(\mathrm{N}_{2}(\mathrm{~g})\) \(+3 \mathrm{H}_{2}(\mathrm{~g}) \rightleftharpoons 2 \mathrm{NH}_{3}(\mathrm{~g})\) at \(500^{\circ} \mathrm{C}\), the value of \(K_{\mathrm{p}}\) is \(1.44 \times 10^{-5}\) when partial pressure is measured in atmospheres. The corresponding value of \(K_{c}\), with concentration in mole litre \(^{-1}\), is (a) \(\frac{1.44 \times 10^{-5}}{(0.082 \times 500)^{-2}}\) (b) \(\frac{1.44 \times 10^{-5}}{(8.314 \times 773)^{-2}}\) (c) \(\frac{1.44 \times 10^{-5}}{(0.082 \times 773)^{2}}\) (d) \(\frac{1.44 \times 10^{-5}}{(0.082 \times 773)^{-2}}\)

Understanding the relationship between equilibrium constants expressed in terms of concentrations (\( K_c \)) and partial pressures (\( K_p \)) is crucial in chemical equilibrium. This relationship is especially important when considering reactions involving gases.

The equilibrium constant in terms of concentration, \( K_c \) , pertains to the concentrations of the reactants and products in moles per liter. On the other hand, \( K_p \) corresponds to the partial pressures of the gases involved and is used when the reaction occurs in a gas phase. The key to relating \( K_c \)$ and \( K_p \) lies in the following equation: \[ K_p = K_c(RT)^{\Delta n} \. \]This equation shows that \( K_p \) is equal to \( K_c \) multiplied by the gas constant \( R \) times the temperature in Kelvin \( T \) all raised to the power of the change in moles of gaseous species, \( \Delta n \) .
To apply this relationship, one must account for the change in the number of moles of gas (\( \Delta n \) ) before and after the reaction. For instance, in our example, \( \Delta n \) was determined by subtracting the number of moles of reactant gases from that of product gases. A negative \( \Delta n \) signifies that there is a decrease in the total number of moles of gas as the reaction proceeds to equilibrium.

The ideal gas constant, often denoted as \( R \) , is a fundamental constant in chemistry used to connect various units involved in gas-related equations.

The value of the ideal gas constant can vary depending on the units used in a given calculation. For problems involving gas pressures in atmospheres and volumes in liters, the value of \( R \) is commonly used as \( 0.0821 \; atm \cdot L/mol \cdot K \) . However, when dealing with energy, \( R \) might be expressed as \( 8.314 \; J/mol \cdot K \) . These values are not interchangeable and must be used in context with the proper units for pressure, volume, and temperature.
It is important to select the correct \( R \) value when working with the relationship between \( K_c \) and \( K_p \) as using an incorrect \( R \) value will lead to a wrong calculation of the equilibrium constants.

When performing calculations in chemistry, particularly those related to gases, temperatures need to be in the absolute temperature scale, which is Kelvin (K).

The Celsius scale is commonly used in daily life, but it is critical to convert to the Kelvin scale to avoid negative temperatures and to ensure proper application of laws like the Ideal Gas Law. To convert degrees Celsius to Kelvin, you add 273.15 to the Celsius temperature:
\[ T(K) = T(\degree C) + 273.15 \. \]For example, in our problem, the temperature given was \( 500\degree C \). By applying the conversion, the temperature in Kelvin is \( 773.15 K \). Rounding off to three significant figures, usually acceptable in chemistry, gives us \( 773 K \), which is the value used in further calculations.

The mole concept is a bridge between the microscopic world of atoms and molecules and the macroscopic world we observe.

A mole is defined as the quantity that contains as many elementary entities (atoms, molecules, ions, etc.) as there are atoms in 12 grams of carbon-12, which is approximately \( 6.022 \times 10^{23} \) entities, known as Avogadro's number. This concept allows chemists to count particles by weighing macroscopic amounts of material.
In chemical reactions, balancing equations according to the mole concept ensures stoichiometric relationships are maintained, facilitating the prediction of product amounts from given reactant quantities. Additionally, the mole concept helps to establish relationships between the mass of substances and their corresponding gas volumes in reactions.