Calculate the entropy of ideal mixing when 2 m oles of `N_(2)`, 3 moles of `H_(2) and 2` moles of `NH_(3)` are mixed at constant temperature, assuming no chemical reaction is occurring.

Explanation

To calculate the entropy of ideal mixing for the given gases, we will follow these steps:

### Step 1: Identify the number of moles of each component
We have:
- \( N_2 \): 2 moles
- \( H_2 \): 3 moles
- \( NH_3 \): 2 moles

### Step 2: Calculate the total number of moles
Total moles \( N_{total} \) can be calculated as:
\[ N_{total} = N_{N_2} + N_{H_2} + N_{NH_3} = 2 + 3 + 2 = 7 \text{ moles} \]

### Step 3: Calculate the mole fraction of each component
The mole fraction \( X_i \) for each component is calculated using the formula:
\[ X_i = \frac{N_i}{N_{total}} \]
Calculating for each component:
- For \( N_2 \):
\[ X_{N_2} = \frac{2}{7} \]
- For \( H_2 \):
\[ X_{H_2} = \frac{3}{7} \]
- For \( NH_3 \):
\[ X_{NH_3} = \frac{2}{7} \]

### Step 4: Use the formula for entropy change of ideal mixing
The formula for the entropy change \( \Delta S \) for ideal mixing is given by:
\[ \Delta S = -R \sum (N_i \log X_i) \]
Where \( R \) is the universal gas constant, approximately \( 8.314 \, \text{J/mol·K} \).

### Step 5: Substitute the values into the formula
Now substituting the values:
\[ \Delta S = -R \left( N_{N_2} \log X_{N_2} + N_{H_2} \log X_{H_2} + N_{NH_3} \log X_{NH_3} \right) \]
Substituting the values:
\[ \Delta S = -8.314 \left( 2 \log \frac{2}{7} + 3 \log \frac{3}{7} + 2 \log \frac{2}{7} \right) \]

### Step 6: Calculate each term
Calculating each logarithm:
- \( \log \frac{2}{7} \)
- \( \log \frac{3}{7} \)

Using a calculator:
- \( \log \frac{2}{7} \approx -0.3567 \)
- \( \log \frac{3}{7} \approx -0.1552 \)

Now substituting these values back:
\[ \Delta S = -8.314 \left( 2 \times (-0.3567) + 3 \times (-0.1552) + 2 \times (-0.3567) \right) \]
Calculating:
\[ \Delta S = -8.314 \left( -0.7134 - 0.4656 - 0.7134 \right) \]
\[ \Delta S = -8.314 \left( -1.8924 \right) \]
\[ \Delta S \approx 15.73 \, \text{J/K} \]

### Final Result
The entropy of ideal mixing for the given gases is approximately:
\[ \Delta S \approx 15.73 \, \text{J/K} \]