The total pressure exerted by the mixture of equal moles of two gases is `5xx10^(3)NM^(-2)` in a container of volume 2 litres at 273K. Calculate the number of moles of the gases mixed.

Explanation

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

\[ PV = nRT \]

Where:
- \( P \) = total pressure (in Pascals)
- \( V \) = volume (in cubic meters)
- \( n \) = total number of moles of gas
- \( R \) = universal gas constant (8.314 J/(mol·K))
- \( T \) = temperature (in Kelvin)

### Step 1: Identify the given values
- Total pressure \( P = 5 \times 10^3 \, \text{N/m}^2 \)
- Volume \( V = 2 \, \text{litres} = 2 \times 10^{-3} \, \text{m}^3 \) (since 1 litre = \( 10^{-3} \, \text{m}^3 \))
- Temperature \( T = 273 \, \text{K} \)
- Gas constant \( R = 8.314 \, \text{J/(mol·K)} \)

### Step 2: Set up the equation
Since the mixture consists of equal moles of two gases, we can denote the number of moles of each gas as \( n \). Therefore, the total number of moles \( n_{\text{total}} \) is:

\[ n_{\text{total}} = n + n = 2n \]

### Step 3: Substitute the values into the Ideal Gas Law
Now we can substitute the known values into the Ideal Gas Law equation:

\[ P V = (2n) R T \]

Substituting the values:

\[ 5 \times 10^3 \times 2 \times 10^{-3} = (2n) \times 8.314 \times 273 \]

### Step 4: Simplify the equation
Calculating the left side:

\[ 5 \times 10^3 \times 2 \times 10^{-3} = 10 \, \text{N} \]

Now, we can rewrite the equation:

\[ 10 = (2n) \times 8.314 \times 273 \]

### Step 5: Calculate \( 8.314 \times 273 \)
Calculating the right side:

\[ 8.314 \times 273 \approx 2270.562 \]

Now we have:

\[ 10 = (2n) \times 2270.562 \]

### Step 6: Solve for \( n \)
Rearranging the equation to solve for \( n \):

\[ 2n = \frac{10}{2270.562} \]

Calculating \( n \):

\[ n = \frac{10}{2 \times 2270.562} \approx \frac{10}{4541.124} \approx 0.0022 \, \text{moles} \]

### Step 7: Find the number of moles of each gas
Since the number of moles of each gas is \( n \):

- Moles of each gas = \( n \approx 0.0022 \, \text{moles} \)

### Final Answer
The total number of moles of the gases mixed is \( 2n \approx 0.0044 \, \text{moles} \).

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