Kinetic energy per unit volume of a gas is

Options

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

Answer

To find the kinetic energy per unit volume of a gas, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Kinetic Energy of a Gas:**
The kinetic energy (KE) of a gas can be derived from the law of equipartition of energy. According to this law, the total energy is equally divided among the degrees of freedom (F) of the gas molecules.

2. **Kinetic Energy per Degree of Freedom:**
For each degree of freedom, the kinetic energy is given by:
\[ KE_{\text{per degree}} = \frac{1}{2} kT \]
where \( k \) is the Boltzmann constant and \( T \) is the temperature in Kelvin.

3. **Total Kinetic Energy for F Degrees of Freedom:**
If the gas has \( F \) degrees of freedom, the total kinetic energy for one molecule is:
\[ KE_{\text{total}} = \frac{F}{2} kT \]

4. **Kinetic Energy for One Mole of Gas:**
To find the kinetic energy for one mole of gas, we multiply the kinetic energy of one molecule by Avogadro's number \( N \):
\[ KE_{\text{one mole}} = \frac{F}{2} kT \cdot N \]

5. **Substituting the Value of \( k \):**
The Boltzmann constant \( k \) can be expressed in terms of the gas constant \( R \) as:
\[ k = \frac{R}{N} \]
Substituting this into the equation gives:
\[ KE_{\text{one mole}} = \frac{F}{2} \left(\frac{R}{N}\right) T \cdot N = \frac{F}{2} RT \]

6. **Kinetic Energy for N Moles:**
For \( N \) moles of gas, the kinetic energy becomes:
\[ KE_{\text{N moles}} = N \cdot \frac{F}{2} RT = \frac{F}{2} NRT \]

7. **Using the Ideal Gas Law:**
From the ideal gas law, we know that:
\[ PV = NRT \]
Therefore, we can substitute \( NRT \) with \( PV \):
\[ KE_{\text{N moles}} = \frac{F}{2} PV \]

8. **Kinetic Energy for Monoatomic Gas:**
For a monoatomic gas, the degrees of freedom \( F \) is 3. Thus, the kinetic energy becomes:
\[ KE = \frac{3}{2} PV \]

9. **Kinetic Energy per Unit Volume:**
To find the kinetic energy per unit volume, we divide the total kinetic energy by the volume \( V \):
\[ KE_{\text{per unit volume}} = \frac{KE}{V} = \frac{\frac{3}{2} PV}{V} = \frac{3}{2} P \]

### Final Result:
The kinetic energy per unit volume of a gas is given by:
\[ KE_{\text{per unit volume}} = \frac{3}{2} P \]