At low pressure, the van der Waal's equation become :
(a)`PV_(m)=RT`
(b)`P(V_(m)-b)=RT`
(c)`(P+(a)/(V_(M)^(2)))V_(m)=RT`
(d)`P=(RT)/(V_(m))+(a)/(V_(m)^(2))`

Options

A. `PV_(m)=RT`
B. `P(V_(m)-b)=RT`
C. `(P+(a)/(V_(M)^(2)))V_(m)=RT`
D. `P=(RT)/(V_(m))+(a)/(V_(m)^(2))`

Explanation

To solve the question regarding the van der Waals equation at low pressure, we will follow these steps:

### Step 1: Write down the van der Waals equation
The van der Waals equation is given by:
\[ P + \frac{a}{V_m^2}(V_m - b) = RT \]
where:
- \( P \) = pressure
- \( V_m \) = molar volume
- \( a \) = measure of the attractive forces between particles
- \( b \) = volume occupied by the particles themselves
- \( R \) = universal gas constant
- \( T \) = temperature

### Step 2: Consider the conditions at low pressure
At low pressure, the volume of the gas is large, which means that the excluded volume \( b \) becomes negligible. Thus, we can simplify the equation by ignoring the term \( b \).

### Step 3: Simplify the equation
Neglecting \( b \), the equation simplifies to:
\[ P + \frac{a}{V_m^2}V_m = RT \]
This can be rearranged to:
\[ P + \frac{a}{V_m} = RT \]

### Step 4: Rearranging to isolate \( P \)
To isolate \( P \), we can rearrange the equation:
\[ P = RT - \frac{a}{V_m} \]

### Step 5: Identify the correct option
Now, let's analyze the options provided:
- (a) \( PV_m = RT \)
- (b) \( P(V_m - b) = RT \)
- (c) \( (P + \frac{a}{V_m^2})V_m = RT \)
- (d) \( P = \frac{RT}{V_m} + \frac{a}{V_m^2} \)

From our simplification, we see that option (c) matches our derived equation at low pressure.

### Final Answer
The correct answer is:
**(c) \( (P + \frac{a}{V_m^2})V_m = RT \)**

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