Question

\((\Delta \mathrm{H}-\Delta \mathrm{U})\) for the formation of carbon monoxide (CO) from its elements at \(298 \mathrm{~K}\) is: (a) \(-1238.78 \mathrm{~J} \mathrm{~mol}^{-1}\) (b) \(1238.78 \mathrm{~J} \mathrm{~mol}^{-1}\) (c) \(-2477.57 \mathrm{~J} \mathrm{~mol}^{-1}\) (d) \(2477.57 \mathrm{~J} \mathrm{~mol}^{-1}\)

Short answer

The correct answer is (b) \(1238.78 \mathrm{~J} \mathrm{~mol}^{-1}\).

Step-by-step solution

Write the Reaction for Formation of CO

The reaction for the formation of carbon monoxide (CO) from its elements is: \( \text{C}(s) + \frac{1}{2} \text{O}_2(g) \rightarrow \text{CO}(g) \). This is the balanced equation for the reaction.

Apply the Relationship Between ∆H and ∆U

For reactions involving gases, the relationship between enthalpy change \( (\Delta H) \) and internal energy change \( (\Delta U) \) is given by the equation: \( \Delta H = \Delta U + \Delta n_g RT \), where \( \Delta n_g \) is the change in number of moles of gas, \( R \) is the ideal gas constant (8.314 J/mol·K), and \( T \) is the temperature in Kelvin (298 K in this case).

Calculate Δng

For the reaction \( \text{C}(s) + \frac{1}{2} \text{O}_2(g) \rightarrow \text{CO}(g) \), the change in the number of moles of gas \( \Delta n_g \) is: \( 1 - \frac{1}{2} = \frac{1}{2} \).

Substitute Values into the Equation

Substitute the given values into the equation: \( \Delta \mathrm{H} - \Delta \mathrm{U} = \Delta n_g RT \). This gives: \( \Delta H - \Delta U = \frac{1}{2} \times 8.314 \times 298 \approx 1238.78 \mathrm{~J} \mathrm{~mol}^{-1} \).

Identify the Correct Option

The calculated value of \( \Delta \mathrm{H} - \Delta \mathrm{U} \) is \( 1238.78 \mathrm{~J} \mathrm{~mol}^{-1} \). This corresponds to option (b).

Key concepts

Relationship between ΔH and ΔU

To understand the relationship between enthalpy change (H) and internal energy change (U), we need to delve into the fundamental principles of thermodynamics. Enthalpy and internal energy are state functions that describe the energy content of a system. However, they are affected differently during reactions involving gases.
When a chemical reaction occurs, the enthalpy change (H) accounts for both heat exchange and any work done due to volume change in the system. In contrast, the internal energy change (U) considers only the energy directly associated with the particles within the system.
For reactions involving gases, the change in enthalpy and internal energy is related by the equation:

• H = U + n_g RT

Here, n_g represents the change in the number of moles of gas, while R is the ideal gas constant, and T is the temperature in Kelvin. This equation highlights how the difference between H and U is determined by the moles of gas evolved or consumed in the reaction at a given temperature.

Ideal Gas Constant

The ideal gas constant, denoted as R, is a fundamental parameter in thermodynamics and physical chemistry. It's used in the ideal gas law equation, which describes the relationship among pressure, volume, and temperature in a gaseous system. The ideal gas constant has a value of 8.314 J/mol K. This specific value helps us to conveniently translate energy units into caloric terms, which is essential in understanding changes in thermodynamic functions during chemical reactions. In the context of reactions involving gases, R plays a crucial role in connecting changes in energy terms by influencing the calculation of the differences between the enthalpy and internal energy. For many calculations, such as those involving enthalpy changes in gas reactions, R provides the necessary link between mole-based chemistry and energy-related thermodynamic calculations. Thus, the ideal gas constant is a key connector, balancing different units and measurements within chemical and physical equations.

Formation of Carbon Monoxide

The formation of carbon monoxide (CO) from its elements is a classic example in thermodynamics, helping illustrate the principles of enthalpy and internal energy. This reaction can be expressed by the balanced equation: \[ \text{C}(s) + \frac{1}{2} \text{O}_2(g) \rightarrow \text{CO}(g) \]In this reaction, a solid carbon source reacts with a fractional amount of oxygen gas to produce carbon monoxide gas. This is a synthesis reaction where a compound forms from simpler substances. Understanding this reaction involves calculating the change in the number of moles of gas (ng), which in this case, changes from 0.5 moles of oxygen gas to 1 mole of CO gas. This difference is directly used in calculations to determine the difference between H and U for the reaction.Additionally, carbon monoxide plays a role in various industries and environmental scenarios, often being a focus in discussions about air quality and combustion processes. Hence, understanding its formation is fundamental in both theoretical and applied chemistry contexts.