KE per unit volume is E. The pressure exerted by the gas is given by

Options

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

Explanation

To find the pressure exerted by the gas in terms of the kinetic energy per unit volume (E), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the relationship between pressure, density, and velocity:**
The formula for pressure (P) in terms of density (ρ) and velocity (V) is given by:
\[ P = \frac{1}{3} \rho V^2 \]

2. **Expressing density in terms of mass and volume:**
The density (ρ) can be expressed as:
\[ \rho = \frac{m}{V} \]
where \(m\) is the mass of the gas and \(V\) is its volume.

3. **Substituting density into the pressure equation:**
Substituting the expression for density into the pressure formula gives:
\[ P = \frac{1}{3} \left(\frac{m}{V}\right) V^2 \]

4. **Simplifying the equation:**
This simplifies to:
\[ P = \frac{1}{3} \frac{m V^2}{V} = \frac{1}{3} m V \]

5. **Introducing kinetic energy:**
The kinetic energy (KE) of the gas can be expressed as:
\[ KE = \frac{1}{2} m V^2 \]
Therefore, we can express \(m V^2\) in terms of kinetic energy:
\[ m V^2 = 2 KE \]

6. **Substituting kinetic energy into the pressure equation:**
Now substituting \(m V^2\) back into the pressure equation gives:
\[ P = \frac{1}{3} \cdot 2 KE = \frac{2}{3} KE \]

7. **Expressing kinetic energy per unit volume:**
The kinetic energy per unit volume (E) is defined as:
\[ E = \frac{KE}{V} \]
Thus, we can express kinetic energy (KE) in terms of E:
\[ KE = E \cdot V \]

8. **Final substitution:**
Substituting \(KE\) back into the pressure equation:
\[ P = \frac{2}{3} (E \cdot V) \]

9. **Finding pressure in terms of E:**
Since we are looking for pressure in terms of E, we can write:
\[ P = \frac{2}{3} E \]

### Final Answer:
Thus, the pressure exerted by the gas in terms of the kinetic energy per unit volume (E) is:
\[ P = \frac{2}{3} E \]