0.5 moles of gas A and x moles of gas B exert a pressure of 200 Pa in a container of volume `10 m^(3)` at 1000K. Given R is the gas constant in `jk^(-1)` ,x is :

Options

A. `(4+R)/(2R)`
B. `(2R)/(4-R)`
C. `(2R)/(4+R)`
D. `(4-R)/(2R)`

Explanation

To solve the problem, we will use the ideal gas equation, which is given by:

\[ PV = nRT \]

Where:
- \( P \) = Pressure (in Pascals)
- \( V \) = Volume (in cubic meters)
- \( n \) = Total number of moles of gas
- \( R \) = Gas constant (in Joules per Kelvin)
- \( T \) = Temperature (in Kelvin)

### Step-by-step Solution:

1. **Identify the given values**:
- Pressure, \( P = 200 \, \text{Pa} \)
- Volume, \( V = 10 \, \text{m}^3 \)
- Temperature, \( T = 1000 \, \text{K} \)
- Moles of gas A, \( n_A = 0.5 \, \text{moles} \)
- Moles of gas B, \( n_B = x \, \text{moles} \)

2. **Calculate the total number of moles**:
\[ n = n_A + n_B = 0.5 + x \]

3. **Substitute the known values into the ideal gas equation**:
\[ PV = nRT \]
Substituting the known values:
\[ 200 \times 10 = (0.5 + x) R \times 1000 \]

4. **Simplify the equation**:
\[ 2000 = (0.5 + x) R \times 1000 \]
Dividing both sides by 1000:
\[ 2 = (0.5 + x) R \]

5. **Rearranging the equation to isolate \( x \)**:
\[ 0.5 + x = \frac{2}{R} \]
Therefore,
\[ x = \frac{2}{R} - 0.5 \]

6. **Express \( x \) in a simplified form**:
\[ x = \frac{2 - 0.5R}{R} \]
This can also be written as:
\[ x = \frac{4 - R}{2R} \]

### Final Answer:
Thus, the value of \( x \) is:
\[ x = \frac{4 - R}{2R} \]