The vander waal's equation of state reduces itself to the idea gas equation at

Options

A. high pressure and low temperature
B. low pressure and low temperature
C. low pressure and high temperature
D. high pressure and high temperature

Answer

To determine the conditions under which the Van der Waals equation of state reduces to the ideal gas equation, we can follow these steps:

### Step 1: Understand the Van der Waals Equation
The Van der Waals equation is given by:
\[ \left(P + a \frac{n^2}{V^2}\right)(V - nb) = nRT \]
where:
- \(P\) = pressure
- \(V\) = volume
- \(n\) = number of moles
- \(R\) = universal gas constant
- \(T\) = temperature
- \(a\) and \(b\) are constants specific to the gas.

### Step 2: Identify the Ideal Gas Equation
The ideal gas equation is:
\[ PV = nRT \]

### Step 3: Analyze the Conditions for Reduction
To reduce the Van der Waals equation to the ideal gas equation, we need to consider the terms \(a\) and \(b\):
- The term \(a \frac{n^2}{V^2}\) accounts for the attractive forces between gas molecules.
- The term \(nb\) accounts for the volume occupied by the gas molecules themselves.

For the Van der Waals equation to approximate the ideal gas equation, the effects of these terms must become negligible.

### Step 4: Conditions for Negligibility
1. **High Temperature**: At high temperatures, the kinetic energy of the gas molecules is much greater than the attractive forces, making the \(a\) term negligible.
2. **Low Pressure**: At low pressures, the volume \(V\) becomes much larger than \(nb\), making the \(b\) term negligible.

### Step 5: Conclusion
Thus, the Van der Waals equation reduces to the ideal gas equation under the conditions of **high temperature and low pressure**.

### Final Answer
The Van der Waals equation of state reduces itself to the ideal gas equation at **high temperature and low pressure**.

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