A given sample of an ideal gas occupise a volume V at a pressure p and absolute temperature T.The mass of each molecule of the gas is m. Which of the following is the density of the gas ?

Options

A. `(P)/((kT))`
B. `(Pm)/((kT))`
C. `(P)/((kTV))`
D. `mkT`

Answer

To find the density of the gas given the conditions in the problem, we can follow these steps:

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

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

### Step 3: Substitute for Moles in the Ideal Gas Law
Substituting \( n \) into the ideal gas law gives:
\[ PV = \frac{m}{M} RT \]

### Step 4: Rearranging for Density
Density \( \rho \) is defined as mass per unit volume:
\[ \rho = \frac{m}{V} \]
From the equation \( PV = \frac{m}{M} RT \), we can rearrange it to express \( m \) in terms of \( P \), \( V \), and \( T \):
\[ m = \frac{PVM}{RT} \]

### Step 5: Substitute Mass into the Density Equation
Now substituting \( m \) back into the density equation:
\[ \rho = \frac{m}{V} = \frac{PVM}{RTV} \]
This simplifies to:
\[ \rho = \frac{PM}{RT} \]

### Step 6: Relate Molar Mass to Molecular Mass
Since \( M \) (molar mass) 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 gives:
\[ \rho = \frac{P(m \cdot N_A)}{RT} \]

### Step 7: Use Boltzmann's Constant
We know that the universal gas constant \( R \) can be related to Boltzmann's constant \( k_B \):
\[ R = N_A k_B \]
Substituting \( R \) into the density equation:
\[ \rho = \frac{P(m \cdot N_A)}{N_A k_B T} \]
This simplifies to:
\[ \rho = \frac{Pm}{k_B T} \]

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