A gas has volume `V` and pressure `p`. The total translational kinetic energy of all the molecules of the gas is

Options

A. 3/2 pV only if th gas in monoatomic
B. 3/2 pV only if the gas is diatomic
C. 3/2 pV in all cases
D. None of above

Explanation

To find the total translational kinetic energy of all the molecules of a gas with volume \( V \) and pressure \( p \), we can follow these steps:

### Step 1: Understand the relationship between kinetic energy and temperature
The average translational kinetic energy \( E \) of a single molecule of an ideal gas is given by the formula:
\[ E = \frac{3}{2} k T \]
where \( k \) is the Boltzmann constant and \( T \) is the absolute temperature in Kelvin.

### Step 2: Relate pressure, volume, and temperature
For an ideal gas, the equation of state is given by:
\[ PV = nRT \]
where \( P \) is the pressure, \( V \) is the volume, \( n \) is the number of moles, \( R \) is the universal gas constant, and \( T \) is the temperature.

### Step 3: Express \( kT \) in terms of \( PV \)
We can express the temperature \( T \) in terms of \( P \), \( V \), and \( n \):
\[ T = \frac{PV}{nR} \]
Substituting this into the expression for average kinetic energy gives:
\[ E = \frac{3}{2} k \left(\frac{PV}{nR}\right) \]

### Step 4: Calculate the total kinetic energy for all molecules
The total number of molecules \( N \) in the gas can be expressed as:
\[ N = n N_A \]
where \( N_A \) is Avogadro's number. The total translational kinetic energy \( E_{total} \) for all molecules is then:
\[ E_{total} = N \cdot E = N \cdot \frac{3}{2} k T \]
Substituting for \( N \) and \( T \):
\[ E_{total} = n N_A \cdot \frac{3}{2} k \left(\frac{PV}{nR}\right) \]

### Step 5: Simplify the expression
Using the relation \( k = \frac{R}{N_A} \), we can simplify:
\[ E_{total} = n N_A \cdot \frac{3}{2} \left(\frac{R}{N_A}\right) \left(\frac{PV}{nR}\right) \]
This simplifies to:
\[ E_{total} = \frac{3}{2} PV \]

### Final Result
Thus, the total translational kinetic energy of all the molecules of the gas is:
\[ E_{total} = \frac{3}{2} pV \]