100 g of an ideal gas is kept in a cylinder of 416 L volume at `27^@`C under 1.5 bar pressure. The molar mass of the gas is _______ g `mol^(-1)`. (Nearest integer)
(Given : R=0.083 L bar `K^(-1)mol^(-1)`)

Explanation

To find the molar mass of the ideal gas, we can use the Ideal Gas Law, which is given by the equation:

\[ PV = nRT \]

Where:
- \( P \) = pressure (in bar)
- \( V \) = volume (in liters)
- \( n \) = number of moles
- \( R \) = ideal gas constant (in L bar K\(^{-1}\) mol\(^{-1}\))
- \( T \) = temperature (in Kelvin)

### Step 1: Convert the temperature to Kelvin
The temperature given is \( 27^\circ C \). To convert this to Kelvin, we use the formula:

\[ T(K) = T(°C) + 273.15 \]

So,

\[ T = 27 + 273.15 = 300.15 \, K \]

### Step 2: Use the Ideal Gas Law to find the number of moles (n)
We can rearrange the Ideal Gas Law to solve for \( n \):

\[ n = \frac{PV}{RT} \]

Substituting the given values:
- \( P = 1.5 \, \text{bar} \)
- \( V = 416 \, \text{L} \)
- \( R = 0.083 \, \text{L bar K}^{-1} \text{mol}^{-1} \)
- \( T = 300.15 \, K \)

Now, substituting these values into the equation:

\[ n = \frac{(1.5 \, \text{bar})(416 \, \text{L})}{(0.083 \, \text{L bar K}^{-1} \text{mol}^{-1})(300.15 \, K)} \]

### Step 3: Calculate the number of moles (n)
Calculating the numerator:

\[ 1.5 \times 416 = 624 \, \text{L bar} \]

Calculating the denominator:

\[ 0.083 \times 300.15 \approx 24.9 \, \text{L bar K}^{-1} \text{mol}^{-1} \]

Now substituting these values back into the equation for \( n \):

\[ n = \frac{624}{24.9} \approx 25.04 \, \text{mol} \]

### Step 4: Calculate the molar mass (M)
The molar mass \( M \) can be calculated using the formula:

\[ M = \frac{\text{mass}}{n} \]

Where the mass of the gas is given as 100 g. Therefore:

\[ M = \frac{100 \, g}{25.04 \, mol} \approx 3.99 \, g/mol \]

### Step 5: Round to the nearest integer
The nearest integer to 3.99 is 4.

### Final Answer
The molar mass of the gas is approximately **4 g/mol**.

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