Under the identical conditions of temperature, the density of a gas X is two times to that of gas Y while molecular mass of gas Y is three times that of X. Calculate the ratio of pressure of X and Y.

Explanation

To solve the problem, we will use the Ideal Gas Law and the relationships between density, molecular mass, and pressure. Let's break it down step by step.

### Step 1: Understand the Given Information
- The density of gas X (denoted as \( \rho_X \)) is twice that of gas Y (denoted as \( \rho_Y \)):
\[ \rho_X = 2 \rho_Y \]
- The molecular mass of gas Y (denoted as \( M_Y \)) is three times that of gas X (denoted as \( M_X \)):
\[ M_Y = 3 M_X \]

### Step 2: Apply the Ideal Gas Law
The Ideal Gas Law is given by:
\[ PV = nRT \]
Where:
- \( P \) = Pressure
- \( V \) = Volume
- \( n \) = Number of moles
- \( R \) = Universal gas constant
- \( T \) = Temperature

Since the temperature and the universal gas constant are constant for both gases, we can express the pressure in terms of density and molecular mass.

### Step 3: Relate Pressure, Density, and Molecular Mass
From the Ideal Gas Law, we can express the number of moles \( n \) in terms of density and molecular mass:
\[ n = \frac{m}{M} \]
Where \( m \) is the mass of the gas.

The volume \( V \) can also be expressed as:
\[ V = \frac{m}{\rho} \]
Substituting \( n \) and \( V \) into the Ideal Gas Law gives:
\[ P \left(\frac{m}{\rho}\right) = \frac{m}{M} RT \]
Rearranging gives:
\[ P = \frac{\rho RT}{M} \]

### Step 4: Calculate the Ratio of Pressures
Now, we can find the ratio of pressures for gases X and Y:
\[ \frac{P_X}{P_Y} = \frac{\rho_X \cdot R \cdot T / M_X}{\rho_Y \cdot R \cdot T / M_Y} = \frac{\rho_X}{\rho_Y} \cdot \frac{M_Y}{M_X} \]

### Step 5: Substitute the Known Values
From the information given:
- \( \frac{\rho_X}{\rho_Y} = 2 \)
- \( \frac{M_Y}{M_X} = 3 \)

Substituting these values into the ratio of pressures:
\[ \frac{P_X}{P_Y} = 2 \cdot 3 = 6 \]

### Final Answer
Thus, the ratio of the pressures of gas X to gas Y is:
\[ \frac{P_X}{P_Y} = 6 \]

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