A sample of an ideal gas occupies a volume V at pressure P and absolute temperature T. The mass of each molecule is m, then the density of the gas is

Options

A. mKT
B. `(Pm)/(KT)`
C. `(P)/(km)`
D. `(P)/(KT)`

Explanation

To find the density of an ideal gas, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Given Information**:
- We have a sample of an ideal gas occupying a volume \( V \).
- The pressure of the gas is \( P \).
- The absolute temperature of the gas is \( T \).
- The mass of each molecule of the gas is \( m \).

2. **Recall the Ideal Gas Law**:
- The ideal gas law is given by the equation:
\[ PV = nRT \]
- Here, \( n \) is the number of moles, \( R \) is the universal gas constant, and \( T \) is the absolute temperature.

3. **Relate Moles to Mass**:
- The number of moles \( n \) can be expressed in terms of the mass of the gas. If \( M \) is the total mass of the gas, then:
\[ n = \frac{M}{M_m} \]
- Where \( M_m \) is the molar mass of the gas. In terms of the mass of each molecule \( m \), the total mass \( M \) can be expressed as:
\[ M = N \cdot m \]
- Here, \( N \) is the total number of molecules.

4. **Substituting into the Ideal Gas Law**:
- Substitute \( n \) into the ideal gas equation:
\[ PV = \frac{M}{M_m}RT \]
- Rearranging gives:
\[ M = \frac{PV \cdot M_m}{RT} \]

5. **Calculate Density**:
- The density \( \rho \) of the gas is defined as mass per unit volume:
\[ \rho = \frac{M}{V} \]
- Substitute the expression for \( M \):
\[ \rho = \frac{PV \cdot M_m}{RT \cdot V} \]
- Simplifying this gives:
\[ \rho = \frac{PM_m}{RT} \]

6. **Expressing Molar Mass**:
- The molar mass \( M_m \) can also be expressed in terms of the mass of a single molecule \( m \) and Avogadro's number \( N_A \):
\[ M_m = N_A \cdot m \]
- Substitute this back into the density equation:
\[ \rho = \frac{P(N_A \cdot m)}{RT} \]

7. **Final Density Expression**:
- Thus, the density of the ideal gas can be expressed as:
\[ \rho = \frac{Pm}{kT} \]
- Where \( k \) is the Boltzmann constant, which relates the temperature to the energy of the gas molecules.

### Final Answer:
The density of the ideal gas is given by:
\[ \rho = \frac{Pm}{kT} \]