Which of the following does not represent ideal gas equation?

Options

A. `PV = 1//2 mNu^(2)`
B. `PV = nRT`
C. `P = rho RT//M`
D. `PV = RT`

Explanation

To determine which of the given equations does not represent the ideal gas equation, we need to analyze each option provided. The ideal gas equation is typically expressed as:

\[ PV = nRT \]

Where:
- \( P \) = pressure
- \( V \) = volume
- \( n \) = number of moles of gas
- \( R \) = universal gas constant
- \( T \) = temperature in Kelvin

Let's evaluate each option step by step:

### Step 1: Analyze the first equation
The first equation given is:

\[ PV = \frac{1}{2} m u^2 \]

This equation does not match the ideal gas equation format. The term \( \frac{1}{2} m u^2 \) is related to kinetic energy, not the ideal gas law. Therefore, this equation does not represent the ideal gas equation.

### Step 2: Analyze the second equation
The second equation is:

\[ PV = \frac{1}{3} m u^2 \]

This equation is a variation of the kinetic theory of gases, where it relates pressure, volume, and the average kinetic energy of gas molecules. This equation is valid for ideal gases, as it can be derived from the ideal gas law. Thus, this equation represents the ideal gas equation.

### Step 3: Analyze the third equation
The third equation is:

\[ PV = nRT \]

This is the standard form of the ideal gas equation. Therefore, this equation clearly represents the ideal gas equation.

### Step 4: Analyze the fourth equation
The fourth equation is:

\[ P = \rho \frac{RT}{M} \]

Where \( \rho \) is the density and \( M \) is the molar mass. This equation can be derived from the ideal gas law by substituting \( n \) with \( \frac{m}{M} \) and using the definition of density \( \rho = \frac{m}{V} \). Hence, this equation also represents the ideal gas equation.

### Conclusion
After analyzing all four equations, we find that the only equation that does not represent the ideal gas equation is:

**Answer: Option A: \( PV = \frac{1}{2} m u^2 \)**

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