Densities of two gases are in the ratio `1: 2` and their temperatures are in the ratio `2:1`, then the ratio of their respective pressure is

Options

A. `1:1`
B. `1:2`
C. `2:1`
D. `4:1`

Explanation

To solve the problem, we need to use the relationship between density, pressure, and temperature of gases. The formula we will use is derived from the ideal gas law, which states:

\[ \text{Density} (\rho) = \frac{P \cdot M}{R \cdot T} \]

Where:
- \( \rho \) is the density of the gas,
- \( P \) is the pressure,
- \( M \) is the molar mass,
- \( R \) is the universal gas constant,
- \( T \) is the temperature in Kelvin.

Given:
1. The densities of two gases are in the ratio \( 1:2 \).
2. The temperatures of the two gases are in the ratio \( 2:1 \).

Let’s denote:
- \( \rho_1 \) and \( \rho_2 \) as the densities of gas 1 and gas 2 respectively,
- \( T_1 \) and \( T_2 \) as the temperatures of gas 1 and gas 2 respectively,
- \( P_1 \) and \( P_2 \) as the pressures of gas 1 and gas 2 respectively.

### Step 1: Write the density ratio
From the problem, we have:
\[ \frac{\rho_1}{\rho_2} = \frac{1}{2} \]

### Step 2: Write the temperature ratio
From the problem, we have:
\[ \frac{T_1}{T_2} = \frac{2}{1} \]

### Step 3: Express densities in terms of pressures and temperatures
Using the density formula:
\[ \rho_1 = \frac{P_1 \cdot M}{R \cdot T_1} \quad \text{and} \quad \rho_2 = \frac{P_2 \cdot M}{R \cdot T_2} \]

### Step 4: Set up the ratio of densities
Substituting the expressions for density into the ratio:
\[ \frac{\rho_1}{\rho_2} = \frac{\frac{P_1 \cdot M}{R \cdot T_1}}{\frac{P_2 \cdot M}{R \cdot T_2}} = \frac{P_1 \cdot T_2}{P_2 \cdot T_1} \]

### Step 5: Substitute the known ratios
From the density ratio:
\[ \frac{1}{2} = \frac{P_1 \cdot T_2}{P_2 \cdot T_1} \]

From the temperature ratio:
\[ T_1 = 2T_2 \quad \text{(from } \frac{T_1}{T_2} = 2\text{)} \]

### Step 6: Substitute \( T_1 \) in the density ratio equation
Substituting \( T_1 \) into the density ratio equation:
\[ \frac{1}{2} = \frac{P_1 \cdot T_2}{P_2 \cdot (2T_2)} \]

### Step 7: Simplify the equation
This simplifies to:
\[ \frac{1}{2} = \frac{P_1}{2P_2} \]

### Step 8: Cross-multiply to find the pressure ratio
Cross-multiplying gives:
\[ P_1 = P_2 \]

### Conclusion
Thus, the ratio of the pressures \( P_1 : P_2 \) is:
\[ \boxed{1 : 1} \]