The relations between various variables of gaseous substances are given along with their formulae . which one is wrong?

Options

A. Density and molar mass : `M = (dRT)/(P)`
B. Universal gas constant , P , V , T : `R = (PV)/(nT)`
C. Volume and pressure : `V_(2) = (P_(2)V_(1))/(P_(1))`
D. Volume and temperature : `V_(2)= (V_(1) T_(2))/(T_(1))`

Explanation

To determine which relationship among the given options is incorrect, we will analyze each statement one by one based on the principles of gas laws.

### Step 1: Analyze the relationship between density and molar mass
1. Start with the ideal gas equation:
\[ PV = nRT \]
where \( n \) (number of moles) can be expressed as \( \frac{m}{M} \) (mass/molar mass).

2. Substitute \( n \) in the ideal gas equation:
\[ PV = \frac{m}{M}RT \]

3. Rearranging gives:
\[ M = \frac{mRT}{PV} \]

4. Recognize that density \( D \) is defined as:
\[ D = \frac{m}{V} \]

5. Substitute \( m \) in terms of density:
\[ m = DV \]

6. Substitute this back into the equation for molar mass:
\[ M = \frac{D \cdot V \cdot RT}{PV} \]

7. Simplifying gives:
\[ M = \frac{D \cdot RT}{P} \]

This relationship is correct.

### Step 2: Analyze the relationship for the universal gas constant \( R \)
1. From the ideal gas equation:
\[ PV = nRT \]

2. Rearranging for \( R \) gives:
\[ R = \frac{PV}{nT} \]

This relationship is also correct.

### Step 3: Analyze the relationship between volume and pressure (Boyle's Law)
1. Boyle's Law states:
\[ P_1V_1 = P_2V_2 \]

2. Rearranging to find \( V_2 \):
\[ V_2 = \frac{P_1V_1}{P_2} \]

3. The statement given is:
\[ V_2 = \frac{P_2V_1}{P_1} \]

This is incorrect because it should be \( V_2 = \frac{P_1V_1}{P_2} \).

### Step 4: Analyze the relationship between volume and temperature (Charles's Law)
1. Charles's Law states:
\[ \frac{V_1}{T_1} = \frac{V_2}{T_2} \]

2. Rearranging gives:
\[ V_2 = \frac{V_1T_2}{T_1} \]

This relationship is correct.

### Conclusion
The incorrect relationship is the one relating volume and pressure, specifically:
\[ V_2 = \frac{P_2V_1}{P_1} \]
This should correctly be:
\[ V_2 = \frac{P_1V_1}{P_2} \]