Which of the following relation (s) is `//` are correct ?
Where `P = ` pressure ( in atm )
V = volume ( in litre )
T= temperature ( in K )
R = gas constant ( 0.0821 atm L`K^(-1)` ` m o l ^(-1)` )
n = mole
d = density `g//L`
M = molecular weight ( in g )
w = weight ( in g )

Options

A. `PV = ( R T )/( d)`
B. `d = ( P M )/( R T )`
C. `(PV ^(2) Md)/( R T) = w^(2)`
D. `n = ( w^(2))/( V M d )`

Answer

To determine which of the given relations are correct, we will analyze each option using the ideal gas law and the definitions of density and molecular weight.

### Step 1: Analyze the Ideal Gas Law
The ideal gas law is given by:
\[ PV = nRT \]
Where:
- \( P \) = Pressure (in atm)
- \( V \) = Volume (in liters)
- \( n \) = Number of moles
- \( R \) = Gas constant (0.0821 atm L K\(^{-1}\) mol\(^{-1}\))
- \( T \) = Temperature (in K)

### Step 2: Relation of Density
Density (\( d \)) is defined as:
\[ d = \frac{m}{V} \]
Where:
- \( m \) = mass (in grams)
- \( V \) = volume (in liters)

We can also express the number of moles (\( n \)) in terms of mass and molecular weight (\( M \)):
\[ n = \frac{m}{M} \]

### Step 3: Evaluate Each Option

**Option 1: \( PV = \frac{RT}{d} \)**

Rearranging the ideal gas law:
\[ PV = nRT \]
Substituting \( n = \frac{m}{M} \):
\[ PV = \frac{m}{M}RT \]

Using the definition of density:
\[ m = dV \]
Substituting this into the equation:
\[ PV = \frac{dV}{M}RT \]
Cancelling \( V \) (assuming \( V \neq 0 \)):
\[ P = \frac{dRT}{M} \]
This does not simplify to \( PV = \frac{RT}{d} \), so **Option 1 is incorrect**.

**Option 2: \( d = \frac{PM}{RT} \)**

From the rearranged ideal gas law:
\[ PV = nRT \]
Substituting \( n = \frac{m}{M} \):
\[ PV = \frac{m}{M}RT \]
Using \( d = \frac{m}{V} \):
\[ m = dV \]
Substituting:
\[ PV = \frac{dV}{M}RT \]
Cancelling \( V \):
\[ P = \frac{dRT}{M} \]
Rearranging gives:
\[ d = \frac{PM}{RT} \]
Thus, **Option 2 is correct**.

**Option 3: \( PV^2 \frac{M d}{RT} = w^2 \)**

Starting from the ideal gas law:
\[ PV = nRT \]
Substituting \( n = \frac{w}{M} \):
\[ PV = \frac{w}{M}RT \]
Rearranging gives:
\[ w = \frac{PVM}{RT} \]
Now squaring both sides:
\[ w^2 = \left(\frac{PVM}{RT}\right)^2 \]
This does not simplify to \( PV^2 \frac{M d}{RT} \), so **Option 3 is incorrect**.

**Option 4: \( n = \frac{w^2}{V M d} \)**

Using \( n = \frac{w}{M} \):
\[ n = \frac{w}{M} \]
Substituting \( w = dV \):
\[ n = \frac{dV}{M} \]
Now, squaring gives:
\[ n^2 = \frac{d^2 V^2}{M^2} \]
This does not simplify to \( \frac{w^2}{V M d} \), so **Option 4 is incorrect**.

### Conclusion
The correct relations are:
- **Option 2**: \( d = \frac{PM}{RT} \)