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 solve the problem, we need to derive the correct expression for molar mass using the ideal gas equation and the definition of density. Here’s a step-by-step breakdown:

### Step 1: Start with the Ideal Gas Equation
The ideal gas equation is given by:
\[ PV = nRT \]
where:
- \( P \) = pressure
- \( V \) = volume
- \( n \) = number of moles
- \( R \) = ideal gas constant
- \( T \) = temperature in Kelvin

### Step 2: Express the Number of Moles
The number of moles \( n \) can be expressed in terms of mass \( m \) and molar mass \( M \):
\[ n = \frac{m}{M} \]
Substituting this into the ideal gas equation gives:
\[ PV = \frac{m}{M} RT \]

### Step 3: Rearrange the Equation
Rearranging the equation to isolate pressure \( P \):
\[ P = \frac{mRT}{MV} \]

### Step 4: Introduce Density
Density \( \rho \) is defined as:
\[ \rho = \frac{m}{V} \]
Thus, we can express mass \( m \) in terms of density and volume:
\[ m = \rho V \]
Substituting this into the pressure equation gives:
\[ P = \frac{\rho V RT}{MV} \]

### Step 5: Simplify the Expression
The volume \( V \) cancels out:
\[ P = \frac{\rho RT}{M} \]

### Step 6: Solve for Molar Mass
Now, rearranging the equation to solve for molar mass \( M \):
\[ M = \frac{\rho RT}{P} \]

### Conclusion
The correct expression for molar mass \( M \) is:
\[ M = \frac{\rho RT}{P} \]

### Final Answer
Thus, the correct option is:
**Option 1: \( M = \frac{\rho RT}{P} \)**

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