Question

If \(\Delta H\) is the change in cathalpy and \(\Delta U\) is the change in internal cnergy, then (1) \(\Delta H\) is always less than \(\Delta U\) (2) \(\Delta H\) is always greatur than \(\Delta U\) (3) \(\Delta H<\Delta U\) only if the number of moles of gascous products is greater than the number of moles of gascous reactants (4) \(\Delta H<\Delta U\) only if the number of moles of gascous products is less than the number of moles of gascous reactants

Short answer

Option (4) is correct.

Step-by-step solution

Understand the Relationship

Recall the relation between enthalpy change ( \( \Delta H \)) and internal energy change ( \( \Delta U \)). The formula is: \[ \Delta H = \Delta U + \Delta (PV) \] Since \( PV = nRT \), we can write \[ \Delta H = \Delta U + \Delta nRT \] where \( \Delta n \) is the change in the number of moles of gas.

Analyze \( \Delta n \)

From \( \Delta H = \Delta U + \Delta nRT \), the term \( \Delta nRT \) depends on the change in the number of moles of gaseous substances. If \( \Delta n > 0 \), \( \Delta H \) will be greater than \( \Delta U \). If \( \Delta n < 0 \), \( \Delta H \) will be less than \( \Delta U \).

Relate to Given Options

The scenarios given in the options can now be evaluated: \( \Delta H < \Delta U \) when \( \Delta n < 0 \), which means the number of moles of gaseous products is less than the number of moles of gaseous reactants. This corresponds to Option (4).

Key concepts

enthalpy change (ΔH)

Enthalpy change, denoted as \( \Delta H \), refers to the total heat content change within a system during a reaction.
It's an essential concept in thermodynamics.
When a chemical reaction occurs, energy is either absorbed or released as heat.
This is what we measure as \( \Delta H \).
For example, in exothermic reactions, heat is released, making \( \Delta H \) negative.
Conversely, endothermic reactions absorb heat, resulting in a positive \( \Delta H \).
To understand \( \Delta H \), remember it's about the heat exchange between the system and its environment.
This is crucial for predicting reaction behavior and process efficiency.

internal energy change (ΔU)

Internal energy change, symbolized as \( \Delta U \), indicates the change in total energy contained within a system.
This includes both kinetic and potential energy of all molecules in the system.
Unlike enthalpy, \( \Delta U \) doesn't account for energy exchange related to volume changes since it focuses solely on internal components.
It's a fundamental idea in the first law of thermodynamics, which states: \( \Delta U = q + w \), where \( q \) is heat added to the system and \( w \) is work done on the system.
Thus, \( \Delta U \) covers both heat transfer and work done, making it broader than just the heat content change.

ideal gas law (PV=nRT)

The ideal gas law, expressed as \[ PV = nRT \], bridges the physical properties of gases.
Here, \( P \) stands for pressure, \( V \) is volume, \( n \) represents moles of gas, \( R \) is the gas constant (8.314 J/(mol·K)), and \( T \) denotes temperature in Kelvin.
It's a crucial equation for predicting and understanding gas behavior.
By using this formula, one can derive relationships between variables affecting a gas, useful in various scientific fields.
This equation particularly plays a role in calculating the enthalpy changes involving gases, where \( \Delta (PV) = \Delta (nRT) \).
This latter concept is vital in chemical reactions where gas volumes change.

number of moles of gas

The number of moles of gas, symbolized as \( n \), represents the quantity of gas present, measured in moles.
A mole corresponds to Avogadro's number, about 6.022 x 10^23 particles.
In chemical reactions, tracking the moles is crucial since it impacts the overall reaction dynamics and enthalpy changes.
For reactions involving gases, changes in the moles of gas (\