The energy density `u/V` of an ideal gas is related to its pressure P as

Options

A. `(U)/(V)=3p`
B. `(U)/(V)=(3)/(2)p`
C. `(U)/(V)=(p)/(3)`
D. `(U)/(V)=(5)/(2)p`

Answer

To find the relationship between the energy density \( \frac{U}{V} \) of an ideal gas and its pressure \( P \), we can follow these steps:

### Step 1: Understand the Ideal Gas Law
The ideal gas law is given by the equation:
\[ PV = nRT \]
where:
- \( P \) is the pressure,
- \( V \) is the volume,
- \( n \) is the number of moles,
- \( R \) is the ideal gas constant,
- \( T \) is the temperature in Kelvin.

### Step 2: Determine the Internal Energy of an Ideal Gas
For an ideal gas, the internal energy \( U \) can be expressed as:
\[ U = \frac{3}{2} nRT \]
This equation indicates that the internal energy is directly proportional to the temperature of the gas.

### Step 3: Substitute \( nRT \) from the Ideal Gas Law
From the ideal gas law, we can express \( nRT \) as:
\[ nRT = PV \]
Substituting this into the internal energy equation gives:
\[ U = \frac{3}{2} PV \]

### Step 4: Calculate the Energy Density \( \frac{U}{V} \)
To find the energy density \( \frac{U}{V} \), we divide the internal energy \( U \) by the volume \( V \):
\[ \frac{U}{V} = \frac{\frac{3}{2} PV}{V} \]
This simplifies to:
\[ \frac{U}{V} = \frac{3}{2} P \]

### Conclusion
Thus, the relationship between the energy density \( \frac{U}{V} \) and the pressure \( P \) of an ideal gas is:
\[ \frac{U}{V} = \frac{3}{2} P \]
This corresponds to option B.