Question

Assuming \(\Delta \mathrm{H}^{\circ}\) remains constant, calculate the equilibrium constant, \(\mathrm{K}\), at \(373^{\circ} \mathrm{K}\), if it equals \(1.6 \times 10^{12}\) at \(298^{\circ} \mathrm{K}\) for the reaction \(2 \mathrm{NO}(\mathrm{g})+\mathrm{O}_{2}(\mathrm{~g}) \rightleftarrows 2 \mathrm{NO}_{2}(\mathrm{~g})\). The standard enthalpy change for this reaction is \(-113 \mathrm{~kJ} / \mathrm{mole}\).

Short answer

The equilibrium constant K at 373 K is approximately \(1.31 \times 10^9\).

Step-by-step solution

Understand the van't Hoff equation

The van't Hoff equation is used to relate the equilibrium constant (K) of a reaction and the temperature (T) of the system, given its standard enthalpy change (ΔH°): \[ \ln\frac{K_{2}}{K_{1}}=-\frac{\Delta H^{\circ}}{R}\left(\frac{1}{T_{2}}-\frac{1}{T_{1}}\right) \] where K1 and K2 are the equilibrium constants at temperatures T1 and T2, respectively, and R is the gas constant (8.314 J/(mol·K)).

Convert temperatures to Kelvin

We are given the temperatures in Celsius. Let's convert them to Kelvin: T1 = 298 + 273.15 = 298 K T2 = 373 + 273.15 = 373 K

Convert ΔH° to J/mol

The standard enthalpy change is given as -113 kJ/mol. Let's convert it to J/mol: ΔH° = -113,000 J/mol

Apply the van't Hoff equation

Now we can plug the values into the van't Hoff equation: \[ \ln\frac{K_{2}}{1.6 \times 10^{12}}=-\frac{-113,000 \textnormal{ J/mol}}{8.314 \textnormal{ J/(mol·K)}}\left(\frac{1}{373 \textnormal{ K}}-\frac{1}{298 \textnormal{ K}}\right) \]

Solve for K2

Finally, we will solve the equation to find the desired equilibrium constant K2: \[ K_2= 1.6 \times 10^{12} \times \exp\left[\frac{-113,000 \textnormal{ J/mol}\left(\frac{1}{373 \textnormal{ K}} - \frac{1}{298 \textnormal{ K}}\right)}{8.314 \textnormal{ J/(mol·K)}}\right] \] After calculating the above expression, we get: K2 ≈ 1.31 × 10^9 Thus, the equilibrium constant K at 373 K is approximately 1.31 × 10^9.

Key concepts

Equilibrium Constant

The equilibrium constant, represented by the symbol K, is a number that expresses the ratio of the concentration of the products to the reactants in a reaction at equilibrium. For a generic reversible reaction \[ aA + bB \leftrightarrow cC + dD \], the equilibrium constant expression is given by \[ K = \frac{{[C]^c[D]^d}}{{[A]^a[B]^b}} \], where \[ [X] \] denotes the concentration of the substance X, and a, b, c, and d indicate the coefficient of each substance in the balanced chemical equation.

An important feature of the equilibrium constant is that it is dimensionless and is only affected by the temperature of the reaction system. If the temperature changes, so will the value of K. This means that predicting the shift in equilibrium with temperature change requires a relation involving both K and T, which is where the van't Hoff equation plays a vital role. Understanding the behavior of K with respect to temperature is essential in chemical engineering and chemistry studies for the optimization of reaction conditions.

Standard Enthalpy Change

Standard enthalpy change, denoted as \( \Delta H^\circ \), is the heat energy change that occurs when a reaction takes place under standard conditions, which is usually at a pressure of 1 bar and a reference temperature, often set at 298 K (25°C). The term 'standard' specifies that all reactants and products are in their standard states. Enthalpy changes are measured in joules per mole (J/mol) or kilojoules per mole (kJ/mol).

When \( \Delta H^\circ \) is negative, the reaction releases heat and is therefore exothermic; when it's positive, the reaction absorbs heat and is endothermic. This concept is crucial in thermodynamics and helps predict whether a reaction will be spontaneous under standard conditions. Moreover, the standard enthalpy change can be used together with the van't Hoff equation to determine the effect of temperature changes on the equilibrium constant.

Temperature Conversion

When solving problems in chemistry, it is often necessary to convert temperature measurements from one unit to another, most commonly from Celsius (°C) to Kelvin (K). Kelvin is the SI unit for temperature, and it is a fundamental requirement to use SI units in scientific calculations to maintain consistency. The conversion formula is \[ T(K) = T(°C) + 273.15 \].

Since the Kelvin scale starts at absolute zero, the lowest possible temperature, there are no negative temperatures in Kelvin, making it a practical scale for scientific work. Temperature conversion is vital in calculations involving the equilibrium constant and the van't Hoff equation because these relationships rely upon the absolute temperature scale provided by the Kelvin measurement. Failing to convert temperatures correctly can result in significant errors in predicting equilibrium constants and understanding reaction behavior.

Gas Constant

The gas constant, symbolized as R, is a physical constant that appears in many fundamental equations in the physical sciences, such as the ideal gas law. Its value is approximately \( 8.314 \, J/(mol\cdot K) \) and it relates the energy scale to the thermodynamic temperature scale. In the context of the van't Hoff equation, R is used as the proportionality factor that combines the change in the equilibrium constant with the change in temperature and standard enthalpy change.

The gas constant is a bridge between the macroscopic and microscopic worlds, connecting variables such as pressure and volume (macroscopic) to temperature and amount of substance (molecules, ions, atoms - microscopic). Understanding the role of the gas constant is essential for anyone studying thermodynamics and reaction kinetics, as it underpins many of the calculations they will encounter.