The pressure of an ideal gas is directly proportional to

Options

A. total kinetic energy
B. translational kinetic energy
C. rotational kinetic energy
D. vibrational kinetic energy

Explanation

To solve the question regarding the relationship between the pressure of an ideal gas and its properties, we can follow these steps:

### Step 1: Understand the Kinetic Theory of Gases
The kinetic theory of gases relates the macroscopic properties of gases (like pressure and temperature) to the microscopic behavior of gas molecules. According to this theory, the pressure exerted by a gas is due to the collisions of gas molecules with the walls of the container.

### Step 2: Recall the Expression for Kinetic Energy
The translational kinetic energy (KE) of a gas molecule can be expressed as:
\[ KE = \frac{1}{2} m v_{\text{rms}}^2 \]
where \( m \) is the mass of the gas molecule and \( v_{\text{rms}} \) is the root mean square velocity of the gas molecules.

### Step 3: Relate RMS Velocity to Pressure
The root mean square velocity can be related to pressure and density using the equation:
\[ v_{\text{rms}} = \sqrt{\frac{3P}{\rho}} \]
where \( P \) is the pressure and \( \rho \) is the density of the gas.

### Step 4: Substitute Density
Density \( \rho \) can be expressed as:
\[ \rho = \frac{m}{V} \]
where \( V \) is the volume of the gas. Substituting this into the equation for \( v_{\text{rms}} \):
\[ v_{\text{rms}} = \sqrt{\frac{3P V}{m}} \]

### Step 5: Substitute \( v_{\text{rms}} \) into Kinetic Energy
Now, substituting \( v_{\text{rms}} \) back into the kinetic energy equation:
\[ KE = \frac{1}{2} m \left(\sqrt{\frac{3P V}{m}}\right)^2 \]
This simplifies to:
\[ KE = \frac{1}{2} m \cdot \frac{3PV}{m} = \frac{3PV}{2} \]

### Step 6: Express Pressure in Terms of Kinetic Energy
From the equation \( KE = \frac{3PV}{2} \), we can express pressure \( P \) in terms of kinetic energy:
\[ P = \frac{2KE}{3V} \]

### Step 7: Conclusion
From the derived equation, we can conclude that the pressure \( P \) of an ideal gas is directly proportional to the total kinetic energy of the gas molecules. Therefore, the answer is that the pressure of an ideal gas is directly proportional to the total kinetic energy.

### Final Answer
The pressure of an ideal gas is directly proportional to the total kinetic energy of the gas molecules.

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