Which of the following expressions is correct ?

Options

A. `M = ((rho)/(p)) RT`
B. `M = ((p)/(rho)) RT`
C. `M = ((1)/(p)) RT`
D. `M = (p) RT`

Explanation

To determine which expression is correct among the given options, we will derive the relationship between pressure, density, temperature, and molecular mass using the Ideal Gas Law.

### Step-by-Step Solution:

1. **Start with the Ideal Gas Law:**
The Ideal Gas Law is represented as:
\[ PV = nRT \]
where:
- \( P \) = Pressure
- \( V \) = Volume
- \( n \) = Number of moles
- \( R \) = Universal gas constant
- \( T \) = Temperature

**Hint:** Recall that the Ideal Gas Law relates pressure, volume, temperature, and number of moles.

2. **Express Number of Moles:**
The number of moles \( n \) can be expressed in terms of mass \( m \) and molecular mass \( M \):
\[ n = \frac{m}{M} \]

**Hint:** Remember that moles can be calculated from mass and molecular mass.

3. **Substitute for Number of Moles in the Ideal Gas Law:**
Substitute \( n \) in the Ideal Gas Law:
\[ PV = \left(\frac{m}{M}\right)RT \]

**Hint:** This step involves replacing \( n \) with its expression in terms of mass and molecular mass.

4. **Rearranging the Equation:**
Rearranging the equation gives:
\[ PM = \frac{mRT}{V} \]

**Hint:** Isolate \( M \) on one side to express it in terms of other variables.

5. **Express Mass per Volume:**
The term \( \frac{m}{V} \) is the definition of density \( \rho \):
\[ \rho = \frac{m}{V} \]

**Hint:** Density is mass divided by volume, a key concept in fluid mechanics.

6. **Substituting Density into the Equation:**
Substitute \( \rho \) back into the equation:
\[ PM = \rho RT \]

**Hint:** This shows how pressure, density, and temperature are related through molecular mass.

7. **Solving for Molecular Mass:**
Rearranging the equation to solve for \( M \):
\[ M = \frac{\rho RT}{P} \]

**Hint:** This final form expresses molecular mass in terms of density, temperature, and pressure.

8. **Final Expression:**
If we denote density as \( \rho \), the final expression becomes:
\[ M = \frac{\rho RT}{P} \]

**Hint:** This is the expression we derived, and it should match one of the options provided.

### Conclusion:
From the derivation, we find that the correct expression for molecular mass \( M \) is:
\[ M = \frac{\rho RT}{P} \]
Thus, the correct answer is option A (or option 1), as it matches our derived expression.