Relation between pressure (P) and energy (E) of a gas is :

Options

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

Explanation

To find the relation between pressure (P) and energy (E) of a gas, we can follow these steps:

### Step 1: Understand the energy of the gas
The energy (E) of an ideal gas can be expressed as:
\[ E = \frac{3}{2} RT \]
where \( R \) is the universal gas constant and \( T \) is the temperature in Kelvin.

### Step 2: Use the ideal gas equation
The ideal gas equation is given by:
\[ PV = nRT \]
For one mole of gas (\( n = 1 \)), this simplifies to:
\[ PV = RT \]
From this equation, we can express \( RT \) in terms of pressure and volume:
\[ RT = PV \]

### Step 3: Substitute \( RT \) in the energy equation
Now, substitute \( RT \) from the ideal gas equation into the energy equation:
\[ E = \frac{3}{2} PV \]

### Step 4: Calculate energy per unit volume
To find the energy per unit volume, we need to divide the total energy \( E \) by the volume \( V \):
\[ e = \frac{E}{V} = \frac{\frac{3}{2} PV}{V} = \frac{3}{2} P \]
where \( e \) is the energy per unit volume.

### Step 5: Rearranging for pressure
Now, we can express pressure \( P \) in terms of energy per unit volume \( e \):
\[ P = \frac{2}{3} e \]

### Conclusion
Thus, the relationship between pressure (P) and energy (E) of a gas can be summarized as:
\[ P = \frac{2}{3} E \]
where \( E \) is the total energy of the gas.