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 the ideal gas law and derive the necessary equations step by step.

### Step-by-Step Solution:

1. **Understanding the Ideal Gas Law**:
The ideal gas law is given by the equation:
\[ PV = nRT \]
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)

2. **Expressing Moles in Terms of Weight**:
The number of moles \( n \) can be expressed in terms of weight \( w \) and molar mass \( M \):
\[ n = \frac{w}{M} \]
Substituting this into the ideal gas law gives:
\[ PV = \frac{w}{M}RT \]

3. **Rearranging the Equation**:
Rearranging the equation, we get:
\[ P = \frac{wRT}{MV} \]

4. **Introducing Density**:
Density \( d \) is defined as:
\[ d = \frac{w}{V} \]
Therefore, we can express weight \( w \) in terms of density and volume:
\[ w = dV \]
Substituting this into the equation for pressure gives:
\[ P = \frac{dVRT}{MV} \]
Simplifying this, we find:
\[ P = \frac{dRT}{M} \]

5. **Deriving the Relation for Density**:
Rearranging the equation gives us:
\[ PM = dRT \]
Thus, we can express density as:
\[ d = \frac{PM}{RT} \]
This is one of the correct relations.

6. **Exploring Further Relations**:
Now, let's derive another relation. We can square the earlier equation:
\[ (PV) = \frac{w}{M}RT \]
Squaring both sides gives:
\[ P^2V^2 = \left(\frac{w}{M}\right)^2R^2T^2 \]
Rearranging gives:
\[ w^2 = P^2V^2\frac{M^2}{R^2T^2} \]

7. **Final Relation**:
We can also express the number of moles \( n \) in terms of weight, volume, and density:
\[ n = \frac{w}{M} = \frac{w}{dV} \]
Rearranging gives:
\[ w = nMV \]
Thus, we can express \( n \) as:
\[ n = \frac{w}{dV} \]

### Conclusion:
Based on the derivations, the correct relations are:
1. \( d = \frac{PM}{RT} \)
2. \( w^2 = P^2V^2\frac{M^2}{R^2T^2} \)
3. \( n = \frac{w}{dV} \)