The temperature of an open room of volume `30 m^(3)` increases from `17^(@)C to 27^(@)C` due to sunshine. The atmospheric pressure in the room remains `1 xx 10^(5) Pa`. If `n_(i) and n_(f)` are the number of molecules in the room before and after heating then `n_(f)` and `n_(i)` will be

Options

A. `2.5 xx 10^(25)`
B. `-2.5 xx 10^(25)`
C. `-1.61 xx 10^(23)`
D. `1.38 xx 10^(23)`

Explanation

To solve the problem, we will use the Ideal Gas Law, which is given by the equation:

\[ PV = nRT \]

Where:
- \( P \) = pressure (in Pascals)
- \( V \) = volume (in cubic meters)
- \( n \) = number of moles
- \( R \) = universal gas constant (\( 8.31 \, \text{J/(mol·K)} \))
- \( T \) = temperature (in Kelvin)

### Step 1: Convert the temperatures from Celsius to Kelvin
The initial temperature \( T_i \) and final temperature \( T_f \) are given in Celsius. We need to convert them to Kelvin:

\[ T_i = 17^\circ C = 17 + 273.15 = 290.15 \, K \approx 290 \, K \]
\[ T_f = 27^\circ C = 27 + 273.15 = 300.15 \, K \approx 300 \, K \]

### Step 2: Identify the constants
We are given:
- Volume \( V = 30 \, m^3 \)
- Pressure \( P = 1 \times 10^5 \, Pa \)
- Universal gas constant \( R = 8.31 \, J/(mol·K) \)

### Step 3: Calculate the initial number of moles \( n_i \)
Using the Ideal Gas Law for the initial state:

\[ n_i = \frac{PV}{RT_i} \]

Substituting the values:

\[ n_i = \frac{(1 \times 10^5 \, Pa)(30 \, m^3)}{(8.31 \, J/(mol·K))(290 \, K)} \]

Calculating \( n_i \):

\[ n_i = \frac{3 \times 10^6}{2403.9} \approx 1240.2 \, mol \]

### Step 4: Calculate the final number of moles \( n_f \)
Using the Ideal Gas Law for the final state:

\[ n_f = \frac{PV}{RT_f} \]

Substituting the values:

\[ n_f = \frac{(1 \times 10^5 \, Pa)(30 \, m^3)}{(8.31 \, J/(mol·K))(300 \, K)} \]

Calculating \( n_f \):

\[ n_f = \frac{3 \times 10^6}{2493} \approx 1203.5 \, mol \]

### Step 5: Calculate the change in number of moles
The change in the number of moles is given by:

\[ \Delta n = n_f - n_i \]

Substituting the values:

\[ \Delta n = 1203.5 - 1240.2 = -36.7 \, mol \]

### Step 6: Convert moles to number of molecules
To find the number of molecules, we use Avogadro's number \( N_a = 6.022 \times 10^{23} \, molecules/mol \):

\[ N_f = n_f \times N_a \]
\[ N_i = n_i \times N_a \]

Calculating \( N_i \) and \( N_f \):

\[ N_i = 1240.2 \times 6.022 \times 10^{23} \approx 7.46 \times 10^{26} \, molecules \]
\[ N_f = 1203.5 \times 6.022 \times 10^{23} \approx 7.25 \times 10^{26} \, molecules \]

### Final Answer
- \( n_i \approx 7.46 \times 10^{26} \, molecules \)
- \( n_f \approx 7.25 \times 10^{26} \, molecules \)