Van der Waal equation for `CH_(4)` at low pressure is

Options

A. `PV = RT – Pb`
B. `PV = RT- (a)/(V)`
C. `PV= RT+(a)/(V)`
D. `PV = RT + Pb`

Explanation

To derive the Van der Waals equation for methane (CH₄) at low pressure, we start with the general form of the Van der Waals equation for real gases. Here’s a step-by-step breakdown of the solution:

### Step 1: Write the General Van der Waals Equation
The Van der Waals equation for one mole of a gas is given by:

\[ \left( P + \frac{a}{V_m^2} \right) (V_m - b) = RT \]

Where:
- \( P \) = pressure of the gas
- \( V_m \) = molar volume of the gas
- \( a \) = Van der Waals constant related to the attraction between molecules
- \( b \) = Van der Waals constant related to the volume occupied by the gas molecules
- \( R \) = universal gas constant
- \( T \) = temperature in Kelvin

### Step 2: Simplify for One Mole
For one mole of gas (\( n = 1 \)), we can simplify the equation to:

\[ \left( P + \frac{a}{V_m^2} \right) (V_m - b) = RT \]

### Step 3: Consider Low Pressure
At low pressure, the volume \( V_m \) becomes very large compared to the constants \( a \) and \( b \). Thus, we can neglect the term \( b \) in the equation because the volume available for the gas molecules to move freely is much larger than the volume occupied by the molecules themselves.

### Step 4: Neglect the Volume Exclusion
Neglecting \( b \), the equation simplifies to:

\[ P + \frac{a}{V_m^2} \approx \frac{RT}{V_m} \]

### Step 5: Rearranging the Equation
Rearranging the equation gives us:

\[ PV_m + a = RT \]

### Step 6: Expressing in Terms of Pressure
Now, we can express the pressure \( P \):

\[ PV_m = RT - a \]

### Step 7: Final Form of the Equation
Thus, the Van der Waals equation for CH₄ at low pressure can be expressed as:

\[ P = \frac{RT - a}{V_m} \]

This is the final form of the Van der Waals equation for methane at low pressure.

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