Thermodynamics

The relationship between the heat capacities at constant volume ( ) and constant pressure ( ) for an ideal gas is defined as: This is a fundamental result for ideal gases and can be derived using thermodynamic principles. Here's a brief explanation:Energy Transfer at Constant Pressure:For a process occurring at constant pressure, the heat added or removed ( ) is related to the change in enthalpy ( ): Energy Transfer at Constant Volume:For a process occurring at constant volume, the heat added or removed ( ) is related to the change in internal energy ( ): Relationship at Molecular Level:For one mole of an ideal gas, the relationship between the change in enthalpy ( ) and change in internal energy ( ) is: For an ideal gas, by the equation of state ( ), this becomes: Deriving the Relationship:Substituting the expressions for and : Simplifying for the heat capacities, you get: This derivation illuminates a key aspect of heat capacities and is crucial for understanding thermodynamic processes in gases.

ΔₐH⁰ is the mean bond enthalpy of a C-H bond.This is false. ΔₐH⁰ represents the total enthalpy change for breaking all four C-H bonds in one mole of CH₄, converting it into carbon and hydrogen atoms in the gas phase. The mean bond enthalpy for a single C-H bond can be calculated as: All four C-H bonds in CH₄ are identical in bond length and energy.This is true. Methane (CH₄) has a tetrahedral structure with four equivalent C-H bonds. These bonds are identical in length and energy due to the molecule's symmetrical shape.The energy required to break individual C-H bonds in successive steps is different.This is true. In a polyatomic molecule like CH₄, the energy required to break each successive C-H bond can vary slightly. This variation results from changes in the electronic environment as bonds are broken.Mean C-H bond enthalpies differ slightly from compound to compound.This is true. The mean bond enthalpy can differ based on the specific chemical environment and molecular structure in different compounds.

To determine the spontaneity, we need to consider the Gibbs free energy change for the reaction. The Gibbs free energy change is given by the equation:$ \Delta H -T -\Delta S \Delta H<0 (\Delta S) (C(g) D(g)) (A(s)) (\Delta S>0) \Delta H \Delta S \Delta G=\Delta H-T \Delta S$ will be negative at all temperatures.Therefore, the reaction will be spontaneous at all temperatures.