Equations for an ideal gas is :

Options

A. ` PV = RT `
B. ` PV ^(gamma ) = ` constant
C. ` C_p- C_V = R `
D. None of these.

Explanation

To solve the question regarding the equation for an ideal gas, we will derive and explain the ideal gas law step by step.

### Step-by-Step Solution:

1. **Understanding the Ideal Gas Law**:
The ideal gas law is a fundamental equation in thermodynamics that relates the pressure (P), volume (V), temperature (T), and the number of moles (n) of an ideal gas. The equation is given by:
\[ PV = nRT \]
where:
- \( P \) = pressure of the gas,
- \( V \) = volume of the gas,
- \( n \) = number of moles of the gas,
- \( R \) = universal gas constant (approximately \( 8.314 \, \text{J/(mol K)} \)),
- \( T \) = absolute temperature of the gas in Kelvin.

2. **Special Case for One Mole**:
If we consider the case where there is 1 mole of gas (\( n = 1 \)), the equation simplifies to:
\[ PV = RT \]
This is a specific case of the ideal gas law for one mole of an ideal gas.

3. **Understanding the Conditions**:
The ideal gas law assumes that the gas behaves ideally, which means:
- The gas particles do not interact with each other except during elastic collisions.
- The volume of the gas particles themselves is negligible compared to the volume of the container.
- The gas follows the law at relatively high temperatures and low pressures.

4. **Other Equations**:
The video also mentions another equation related to specific heat capacities:
\[ C_P - C_V = R \]
This equation is valid for processes involving ideal gases but is not the ideal gas law itself. It relates the heat capacities at constant pressure (\( C_P \)) and constant volume (\( C_V \)).

5. **Conclusion**:
The ideal gas equation is indeed \( PV = nRT \) or \( PV = RT \) for one mole of gas. Any other equations mentioned, such as \( C_P - C_V = R \), are valid under specific conditions but do not represent the ideal gas law.

### Final Answer:
The equation for an ideal gas is:
\[ PV = nRT \]
For one mole of gas, it simplifies to:
\[ PV = RT \]