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

Options

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

Answer

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

### Step 1: Start with the Ideal Gas Law
The ideal gas law states:
\[ PV = nRT \]
For one mole of gas, this simplifies to:
\[ PV = RT \]

### Step 2: Define Internal Energy
The internal energy (E) of an ideal gas can be expressed as:
\[ E = \frac{3}{2} nRT \]
For one mole of gas, this becomes:
\[ E = \frac{3}{2} RT \]

### Step 3: Substitute for RT
From the ideal gas law, we can express RT in terms of pressure and volume:
\[ RT = PV \]
Substituting this into the internal energy equation gives:
\[ E = \frac{3}{2} PV \]

### Step 4: Express Energy Density
Energy density (E_d) is defined as the internal energy per unit volume:
\[ E_d = \frac{E}{V} \]
Substituting the expression for E:
\[ E_d = \frac{\frac{3}{2} PV}{V} = \frac{3}{2} P \]

### Step 5: Relate Pressure to Energy Density
From the energy density equation, we can express pressure in terms of energy density:
\[ P = \frac{2}{3} E_d \]

### Final Relation
Thus, the relation between pressure (P) and energy (E) of the gas can be summarized as:
\[ P = \frac{2}{3} E_d \]
Where \( E_d \) is the energy density. If we consider \( E \) as the total internal energy, we can write:
\[ P = \frac{2}{3} E \]

### Conclusion
The final answer is:
\[ P = \frac{2}{3} E \]

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