A gas bulb of `1 L` capacity contains `2.0xx10^(11)` molecules of nitrogen exerting a pressure of `7.57xx10^(3)Nm^(-2)`. Calculate the root mean square (rms) speed and the temperature of the gas molecules. If the ratio of the most probable speed to the root mean square is `0.82`, calculate the most probable speed for these molecules at this temperature.

Options

A. 232.88 m/sec
B. 809.12 m/sec
C. 444.18 m/sec
D. 494.22 m/sec

Answer

To solve the problem step by step, we will follow the outlined procedure based on the information given in the question.

### Step 1: Convert Volume to Cubic Meters
Given that the volume of the gas bulb is 1 liter, we need to convert this to cubic meters.

\[ 1 \text{ L} = 1000 \text{ cm}^3 = 10^{-3} \text{ m}^3 \]

### Step 2: Calculate the Number of Moles
We are given the number of molecules of nitrogen gas, which is \(2.0 \times 10^{11}\). To find the number of moles, we will use Avogadro's number (\(N_A = 6.022 \times 10^{23} \text{ molecules/mol}\)).

\[ n = \frac{N}{N_A} = \frac{2.0 \times 10^{11}}{6.022 \times 10^{23}} \approx 3.32 \times 10^{-13} \text{ moles} \]

### Step 3: Use the Ideal Gas Law to Find Temperature
The ideal gas equation is given by:

\[ PV = nRT \]

We can rearrange this to solve for temperature \(T\):

\[ T = \frac{PV}{nR} \]

Substituting the known values:
- \(P = 7.57 \times 10^{3} \text{ N/m}^2\)
- \(V = 10^{-3} \text{ m}^3\)
- \(n \approx 3.32 \times 10^{-13} \text{ moles}\)
- \(R = 8.314 \text{ J/(mol K)}\)

\[ T = \frac{(7.57 \times 10^{3}) \times (10^{-3})}{(3.32 \times 10^{-13}) \times (8.314)} \]

Calculating this gives:

\[ T \approx 274.2 \text{ K} \]

### Step 4: Calculate the Root Mean Square Speed
The formula for the root mean square speed (\(v_{rms}\)) is:

\[ v_{rms} = \sqrt{\frac{3RT}{M}} \]

Where \(M\) is the molar mass of nitrogen gas, which is approximately \(28 \times 10^{-3} \text{ kg/mol}\).

Substituting the values:

\[ v_{rms} = \sqrt{\frac{3 \times 8.314 \times 274.2}{28 \times 10^{-3}}} \]

Calculating this gives:

\[ v_{rms} \approx 494.22 \text{ m/s} \]

### Step 5: Calculate the Most Probable Speed
We are given the ratio of the most probable speed (\(v_{mp}\)) to the root mean square speed:

\[ \frac{v_{mp}}{v_{rms}} = 0.82 \]

Thus, we can find \(v_{mp}\):

\[ v_{mp} = 0.82 \times v_{rms} = 0.82 \times 494.22 \approx 405.26 \text{ m/s} \]

### Final Results
- **Root Mean Square Speed**: \(v_{rms} \approx 494.22 \text{ m/s}\)
- **Temperature**: \(T \approx 274.2 \text{ K}\)
- **Most Probable Speed**: \(v_{mp} \approx 405.26 \text{ m/s}\)