A ideal gas has pressure 'p', volume 'V' and absolute temperature 'T'. It 'm' is the mass of each molecules and 'K' is the Boltzmann constant , the density of the gas is

Options

A. `("pm")/(MT)`
B. `(KT)/("pm")`
C. `(Km)/("pT")`
D. `(pK)/("Tm")`

Explanation

To find the density of an ideal gas given its pressure \( p \), volume \( V \), absolute temperature \( T \), mass of each molecule \( m \), and Boltzmann constant \( K \), 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 universal gas constant, and \( T \) is the absolute temperature.

### Step 2: Relate Moles to Mass
The number of moles \( n \) can be expressed in terms of the mass of the gas. If \( m \) is the mass of one molecule and \( M \) is the molar mass (mass of one mole of gas), then:
\[ n = \frac{m_{\text{total}}}{M} \]
where \( m_{\text{total}} \) is the total mass of the gas.

### Step 3: Express Total Mass in Terms of Density
The density \( \rho \) of the gas is defined as:
\[ \rho = \frac{m_{\text{total}}}{V} \]
From this, we can express the total mass as:
\[ m_{\text{total}} = \rho V \]

### Step 4: Substitute for Moles in the Ideal Gas Law
Substituting \( n \) in the ideal gas law:
\[ PV = \left(\frac{m_{\text{total}}}{M}\right)RT \]
Substituting \( m_{\text{total}} = \rho V \):
\[ PV = \left(\frac{\rho V}{M}\right)RT \]

### Step 5: Simplify the Equation
We can cancel \( V \) from both sides (assuming \( V \neq 0 \)):
\[ P = \frac{\rho RT}{M} \]

### Step 6: Solve for Density
Rearranging the equation to solve for density \( \rho \):
\[ \rho = \frac{PM}{RT} \]

### Step 7: Relate Molar Mass to Mass of Molecules
The molar mass \( M \) can be expressed in terms of the mass of a single molecule \( m \) and Avogadro's number \( N_A \):
\[ M = m \cdot N_A \]
Substituting this into the density equation:
\[ \rho = \frac{P(m \cdot N_A)}{RT} \]

### Step 8: Substitute \( R \) with Boltzmann's Constant
Since \( R = k \cdot N_A \) (where \( k \) is the Boltzmann constant), we can substitute \( R \):
\[ \rho = \frac{P(m \cdot N_A)}{k \cdot N_A \cdot T} \]
The \( N_A \) cancels out:
\[ \rho = \frac{Pm}{kT} \]

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