A gaseous mixture of 2 moles of A, 3 moles of B, 5 moles of C and 10 moles of D is contained in a vessel. Assuming that gases are ideal and the partial pressure of C is 1.5 atm, total pressure is

Options

A. `3 atm`
B. `6 atm`
C. `9 atm`
D. `15 atm`

Answer

To solve the problem, we need to find the total pressure of a gaseous mixture given the partial pressure of one of the gases and the number of moles of each gas in the mixture. Here’s a step-by-step solution:

### Step 1: Identify the number of moles of each gas
We are given:
- Moles of A (Na) = 2
- Moles of B (Nb) = 3
- Moles of C (Nc) = 5
- Moles of D (Nd) = 10

### Step 2: Calculate the total number of moles in the mixture
The total number of moles (N_total) is the sum of the moles of all gases:
\[ N_{total} = N_a + N_b + N_c + N_d \]
\[ N_{total} = 2 + 3 + 5 + 10 = 20 \]

### Step 3: Calculate the mole fraction of gas C
The mole fraction (X_C) of gas C is calculated using the formula:
\[ X_C = \frac{N_C}{N_{total}} \]
Substituting the values:
\[ X_C = \frac{5}{20} = \frac{1}{4} \]

### Step 4: Use the relationship between partial pressure and mole fraction
We know that the partial pressure of a gas in a mixture is given by:
\[ P_C = P_t \times X_C \]
Where:
- \( P_C \) is the partial pressure of gas C (given as 1.5 atm)
- \( P_t \) is the total pressure
- \( X_C \) is the mole fraction of gas C (calculated as \( \frac{1}{4} \))

### Step 5: Rearrange the equation to solve for total pressure
Rearranging the equation gives:
\[ P_t = \frac{P_C}{X_C} \]
Substituting the known values:
\[ P_t = \frac{1.5 \text{ atm}}{\frac{1}{4}} \]
\[ P_t = 1.5 \text{ atm} \times 4 = 6 \text{ atm} \]

### Final Answer
The total pressure of the gaseous mixture is **6 atm**.

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