10 litres of a gas at S.T.P. weigh 19.64 g. Calculate the molecular mass of the gas.

Answer

To calculate the molecular mass of the gas, we can use the Ideal Gas Law equation and the information provided in the question. Here’s a step-by-step solution:

### Step 1: Understand the Ideal Gas Law
The Ideal Gas Law is given by the equation:
\[ PV = nRT \]
Where:
- \( P \) = Pressure (in atm)
- \( V \) = Volume (in liters)
- \( n \) = Number of moles of gas
- \( R \) = Ideal gas constant (0.0821 L·atm/(K·mol) or 8.314 J/(K·mol))
- \( T \) = Temperature (in Kelvin)

### Step 2: Convert Given Values
Given:
- Volume \( V = 10 \) liters
- Weight \( W = 19.64 \) grams
- Temperature \( T = 273 \) K (standard temperature)
- Pressure \( P = 1 \) atm (standard pressure)

### Step 3: Calculate the Number of Moles
The number of moles \( n \) can be expressed as:
\[ n = \frac{W}{M} \]
Where \( M \) is the molecular mass of the gas.

### Step 4: Substitute into the Ideal Gas Equation
Substituting \( n \) into the Ideal Gas Law gives:
\[ PV = \frac{W}{M} RT \]

### Step 5: Rearranging the Equation
Rearranging the equation to solve for \( M \):
\[ M = \frac{WRT}{PV} \]

### Step 6: Substitute the Known Values
Now, substituting the known values into the equation:
- \( W = 19.64 \) g
- \( R = 0.0821 \) L·atm/(K·mol) (or use \( 8.314 \) J/(K·mol) if you convert units)
- \( T = 273 \) K
- \( P = 1 \) atm
- \( V = 10 \) L

So we have:
\[ M = \frac{19.64 \times 0.0821 \times 273}{1 \times 10} \]

### Step 7: Calculate the Molecular Mass
Calculating the values:
1. Calculate \( 19.64 \times 0.0821 \times 273 \):
\[ 19.64 \times 0.0821 \times 273 = 445.82 \]
2. Divide by \( 10 \):
\[ M = \frac{445.82}{10} = 44.582 \text{ g/mol} \]

### Final Answer
The molecular mass of the gas is approximately \( 44.58 \) g/mol.

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