Air fills a rooms in winter at `7^@ C` and in summer at `37^@ C`. If the pressure is the same in winter and summer, the ratio of the weight of the air filled in winter and that in summer is.

Options

A. `2.2`
B. `1.75`
C. `1.1`
D. `3.3`

Explanation

To solve the problem, we will use the Ideal Gas Law and the relationship between weight and temperature. Here's a step-by-step solution:

### Step 1: Convert temperatures from Celsius to Kelvin
- For winter: \( T_w = 7^\circ C = 7 + 273 = 280 \, K \)
- For summer: \( T_s = 37^\circ C = 37 + 273 = 310 \, K \)

### Step 2: Write 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 in Kelvin

### Step 3: Relate weight to number of moles
The weight \( W \) of the air can be expressed in terms of the number of moles:
\[ W = n \cdot M \]
Where \( M \) is the molar mass of air. Thus, we can rewrite the Ideal Gas Law as:
\[ PV = \frac{W}{M} RT \]
From this, we can isolate \( W \):
\[ W = \frac{PVM}{RT} \]

### Step 4: Establish the relationship between weight and temperature
Since the pressure \( P \) and volume \( V \) are constant for both winter and summer, we can say:
\[ W \propto \frac{1}{T} \]
This means that the weight of air is inversely proportional to the temperature.

### Step 5: Calculate the ratio of weights
We want to find the ratio of the weight of air in winter to the weight of air in summer:
\[ \frac{W_w}{W_s} = \frac{T_s}{T_w} \]
Substituting the values we found:
\[ \frac{W_w}{W_s} = \frac{T_s}{T_w} = \frac{310 \, K}{280 \, K} \]

### Step 6: Simplify the ratio
Calculating the ratio:
\[ \frac{W_w}{W_s} = \frac{310}{280} = \frac{31}{28} \approx 1.107 \]

### Conclusion
Thus, the ratio of the weight of the air filled in winter to that in summer is approximately \( 1.1 \).