`Delta H` and `Delta U` for the reaction,
`2S_((s))+3O_(2(g)) to 2SO_(3(g))`, are related as _______ .

Options

A. `Delta H = Delta U-1.5 RT`
B. `Delta H=Delta U-0.5 RT`
C. `Delta H=Delta U-RT`
D. `Delta H=Delta U+1.5 RT`

Answer

To determine the relationship between \(\Delta H\) (enthalpy change) and \(\Delta U\) (internal energy change) for the reaction

\[ 2S_{(s)} + 3O_{2(g)} \rightarrow 2SO_{3(g)}, \]

we can follow these steps:

### Step 1: Understand the First Law of Thermodynamics
The first law of thermodynamics states that the change in internal energy (\(\Delta U\)) is related to the change in enthalpy (\(\Delta H\)) and the work done by the system. The relationship can be expressed as:

\[ \Delta U = \Delta H - \text{Work done} \]

### Step 2: Identify the Work Done
In a chemical reaction, the work done is often related to the change in volume of gases. At constant pressure, the work done can be expressed as:

\[ \text{Work done} = P \Delta V \]

### Step 3: Use the Ideal Gas Law
According to the ideal gas law, \(PV = nRT\). Thus, when we consider the change in volume (\(\Delta V\)), we can express it in terms of the change in the number of moles of gas (\(\Delta n\)):

\[ \Delta (PV) = \Delta nRT \]

### Step 4: Calculate \(\Delta n\)
For the given reaction, we need to calculate \(\Delta n\), which is the difference between the number of moles of gaseous products and the number of moles of gaseous reactants.

- Products: \(2SO_{3(g)}\) contributes 2 moles.
- Reactants: \(3O_{2(g)}\) contributes 3 moles.

Thus,

\[ \Delta n = \text{moles of products} - \text{moles of reactants} = 2 - 3 = -1 \]

### Step 5: Substitute \(\Delta n\) into the Work Done Expression
Now we can substitute \(\Delta n\) into the expression for work done:

\[ \text{Work done} = P \Delta V = -\Delta nRT = -(-1)RT = RT \]

### Step 6: Substitute Work Done into the First Law Equation
Now we can substitute the expression for work done back into the first law equation:

\[ \Delta U = \Delta H - RT \]

### Step 7: Rearranging the Equation
Rearranging this equation gives us the relationship between \(\Delta H\) and \(\Delta U\):

\[ \Delta H = \Delta U + RT \]

### Conclusion
Thus, the relationship between \(\Delta H\) and \(\Delta U\) for the reaction is:

\[ \Delta H = \Delta U + RT \]