For an ideal gas, pressure `(p)` and interal energy `(E )` per unit volume are related as

Options

A. `p=(2)/(3)E`
B. `p=(3)/(2)E`
C. `p=(1)/(2)E`
D. `p=2E`

Explanation

To solve the problem of relating pressure \( p \) and internal energy \( E \) per unit volume for an ideal gas, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Kinetic Theory of Gases**:
The mean kinetic energy of an ideal gas per mole is given by the formula:
\[ E = \frac{3}{2} nRT \]
where:
- \( E \) is the internal energy,
- \( n \) is the number of moles,
- \( R \) is the universal gas constant,
- \( T \) is the temperature in Kelvin.

2. **Use the Ideal Gas Law**:
The ideal gas law states:
\[ PV = nRT \]
For a unit volume (i.e., \( V = 1 \)), this simplifies to:
\[ P = nRT \]

3. **Substitute the Ideal Gas Law into the Kinetic Energy Equation**:
From the ideal gas law, we have \( nRT = P \). We can substitute this into the equation for internal energy:
\[ E = \frac{3}{2} nRT \implies E = \frac{3}{2} P \]

4. **Express Internal Energy per Unit Volume**:
Since we are looking for the internal energy per unit volume, we can express it as:
\[ E = \frac{3}{2} P \]
This shows the relationship between pressure and internal energy per unit volume for an ideal gas.

5. **Final Relation**:
Thus, the relationship between pressure \( p \) and internal energy \( E \) per unit volume is:
\[ p = \frac{2}{3} E \]

### Conclusion:
The relationship between pressure and internal energy per unit volume for an ideal gas is given by:
\[ p = \frac{2}{3} E \]