The volume of a gas at pressure `1.2 xx 10^(7) Nm^(-2)` and temperature `127^(@)C` is `2.0` litre. Find the number of molecules in the gas.

Answer

To solve the problem of finding the number of molecules in a gas given its pressure, volume, and temperature, we can follow these steps:

### Step 1: Write down the given data
- Pressure (P) = \(1.2 \times 10^7 \, \text{N/m}^2\)
- Temperature (T) = \(127^\circ C\)
- Volume (V) = \(2.0 \, \text{liters}\)

### Step 2: Convert the temperature to Kelvin
To convert Celsius to Kelvin, we use the formula:
\[ T(K) = T(°C) + 273 \]
So,
\[ T = 127 + 273 = 400 \, K \]

### Step 3: Convert the volume from liters to cubic meters
1 liter is equal to \(10^{-3}\) cubic meters. Therefore,
\[ V = 2.0 \, \text{liters} = 2.0 \times 10^{-3} \, \text{m}^3 \]

### Step 4: Use the Ideal Gas Equation
The Ideal Gas Equation is given by:
\[ PV = nRT \]
Where:
- \(P\) = pressure
- \(V\) = volume
- \(n\) = number of moles
- \(R\) = gas constant (\(8.314 \, \text{J/(mol K)}\))
- \(T\) = temperature in Kelvin

### Step 5: Rearrange the equation to solve for \(n\)
\[ n = \frac{PV}{RT} \]

### Step 6: Substitute the values into the equation
Substituting the known values:
\[ n = \frac{(1.2 \times 10^7 \, \text{N/m}^2)(2.0 \times 10^{-3} \, \text{m}^3)}{(8.314 \, \text{J/(mol K)})(400 \, K)} \]

### Step 7: Calculate \(n\)
Calculating the numerator:
\[ 1.2 \times 10^7 \times 2.0 \times 10^{-3} = 2.4 \times 10^4 \]
Calculating the denominator:
\[ 8.314 \times 400 = 3325.6 \]
Now substituting back:
\[ n = \frac{2.4 \times 10^4}{3325.6} \approx 7.22 \, \text{moles} \]

### Step 8: Calculate the number of molecules
To find the number of molecules, we use Avogadro's number, which is \(6.022 \times 10^{23} \, \text{molecules/mole}\):
\[ \text{Number of molecules} = n \times N_A \]
\[ = 7.22 \times 6.022 \times 10^{23} \]

### Step 9: Perform the final calculation
Calculating the number of molecules:
\[ = 7.22 \times 6.022 \approx 43.46 \times 10^{23} \approx 4.346 \times 10^{24} \, \text{molecules} \]

### Final Answer
The number of molecules in the gas is approximately \(4.346 \times 10^{24}\).

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