Consider the reaction, `N_(2) + 3H_(2) rarr 2NH_(3)` carried out at constant temperature and pressure. If `Delta H and Delta U` are the enthalpy and internal energy changes for the reaction, which of the following expressions is true?

Options

A. `DeltaH gt Delta U`
B. `DeltaH lt Delta U`
C. `DeltaH = Delta U`
D. `DeltaH = 0`

Explanation

To solve the problem regarding the reaction \( N_2 + 3H_2 \rightarrow 2NH_3 \) and the relationship between the changes in enthalpy (\( \Delta H \)) and internal energy (\( \Delta U \)), we will follow these steps:

### Step 1: Understand the Definitions
- **Enthalpy (\( \Delta H \))**: It is the heat content of a system at constant pressure. It accounts for the internal energy and the work done by the system.
- **Internal Energy (\( \Delta U \))**: It is the total energy contained within the system, including kinetic and potential energy.

### Step 2: Use the Relationship Between \( \Delta H \) and \( \Delta U \)
The relationship between enthalpy change and internal energy change at constant temperature and pressure can be expressed by the equation:
\[ \Delta H = \Delta U + \Delta n_g RT \]
where:
- \( \Delta n_g \) = change in the number of moles of gas (products - reactants)
- \( R \) = universal gas constant (approximately \( 8.314 \, \text{J/mol·K} \))
- \( T \) = temperature in Kelvin

### Step 3: Calculate \( \Delta n_g \)
For the given reaction:
- Reactants: \( N_2 + 3H_2 \) → Total = 4 moles of gas
- Products: \( 2NH_3 \) → Total = 2 moles of gas

Now, calculate \( \Delta n_g \):
\[ \Delta n_g = \text{(moles of gaseous products)} - \text{(moles of gaseous reactants)} = 2 - 4 = -2 \]

### Step 4: Substitute \( \Delta n_g \) into the Equation
Now substituting \( \Delta n_g \) into the relationship:
\[ \Delta H = \Delta U + (-2)RT \]
This simplifies to:
\[ \Delta H = \Delta U - 2RT \]

### Step 5: Rearranging the Equation
Rearranging the equation gives:
\[ \Delta U = \Delta H + 2RT \]
Since \( 2RT \) is a positive term (as both \( R \) and \( T \) are positive), we can conclude that:
\[ \Delta U > \Delta H \]

### Conclusion
Thus, the correct expression is:
\[ \Delta H < \Delta U \]