The number of molecules per unit volume (n) of a gas is given by

Options

A. `(P)/(kT)`
B. `(kT)/(P)`
C. `(P)/(kT)`
D. `(RT)/(P)`

Answer

To find the number of molecules per unit volume (n) of a gas, we can use the ideal gas law and some fundamental constants. Here’s a step-by-step solution:

### Step-by-Step Solution:

1. **Start with the Ideal Gas Law**:
The ideal gas law is given by the equation:
\[ PV = nRT \]
where:
- \( P \) = pressure of the gas
- \( V \) = volume of the gas
- \( n \) = number of moles of the gas
- \( R \) = universal gas constant
- \( T \) = absolute temperature of the gas

2. **Rearrange the Ideal Gas Law**:
To find the number of moles per unit volume, we can rearrange the equation:
\[ n = \frac{PV}{RT} \]

3. **Convert Moles to Molecules**:
The number of molecules \( N \) can be expressed in terms of moles using Avogadro's number \( N_A \):
\[ N = n \times N_A \]
Substituting the expression for \( n \) from the previous step:
\[ N = \left(\frac{PV}{RT}\right) \times N_A \]

4. **Express Volume per Unit Volume**:
Since we are interested in the number of molecules per unit volume, we can express the volume \( V \) as 1 (for unit volume):
\[ N = \frac{P \times 1}{RT} \times N_A = \frac{P}{RT} \times N_A \]

5. **Using Boltzmann's Constant**:
We know that the universal gas constant \( R \) can be expressed in terms of Boltzmann's constant \( k \):
\[ R = N_A \times k \]
Substituting this into our equation gives:
\[ N = \frac{P}{kT} \]

6. **Final Result**:
Therefore, the number of molecules per unit volume \( n \) is given by:
\[ n = \frac{P}{kT} \]

### Final Answer:
The number of molecules per unit volume \( n \) of a gas is:
\[ n = \frac{P}{kT} \]