A sample of an ideal gas occupies a volume V at pressure P and absolute temperature T. The masss 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 given its volume \( V \), pressure \( P \), absolute temperature \( T \), and the mass of each molecule \( m \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the relationship between mass, volume, and density**:
The density \( \rho \) of a gas is defined as the mass \( M \) of the gas divided by its volume \( V \):
\[ \rho = \frac{M}{V} \]

2. **Express the total mass of the gas**:
If \( n \) is the number of molecules in the gas, and \( m \) is the mass of each molecule, then the total mass \( M \) of the gas can be expressed as:
\[ M = n \cdot m \]

3. **Use the ideal gas law**:
The ideal gas law states that:
\[ PV = nRT \]
where \( R \) is the ideal gas constant and \( T \) is the absolute temperature.

4. **Rearranging the ideal gas law**:
From the ideal gas law, we can express the number of moles \( n \) as:
\[ n = \frac{PV}{RT} \]

5. **Substituting for mass in the density formula**:
Now substitute \( n \) from the ideal gas law into the expression for total mass \( M \):
\[ M = n \cdot m = \left(\frac{PV}{RT}\right) \cdot m \]

6. **Substituting into the density formula**:
Now substitute \( M \) into the density formula:
\[ \rho = \frac{M}{V} = \frac{\left(\frac{PV}{RT}\right) \cdot m}{V} \]

7. **Simplifying the expression**:
The volume \( V \) cancels out:
\[ \rho = \frac{Pm}{RT} \]

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