127 mL of a gas at 136°C and 758 mm pressure weigh 0.4524 g. If 1 mL of hydrogen at S.T.P. weighs 0.00009 g, calculate the vapour density and molecular mass of the gas.

Answer

To solve the problem step by step, we will use the Ideal Gas Law and the definitions of vapor density and molecular mass.

### Step 1: Convert the given values to appropriate units
- **Volume (V)**: 127 mL = 127/1000 = 0.127 L
- **Temperature (T)**: 136°C = 136 + 273 = 409 K
- **Pressure (P)**: 758 mmHg = 758/760 = 0.995 atm

### Step 2: Use the Ideal Gas Law to find the molecular mass
The Ideal Gas Law is given by the equation:
\[ PV = nRT \]
Where:
- P = pressure in atm
- V = volume in liters
- n = number of moles
- R = ideal gas constant (0.0821 L·atm/(K·mol))
- T = temperature in Kelvin

We can rearrange this equation to solve for the number of moles (n):
\[ n = \frac{PV}{RT} \]

### Step 3: Substitute the values into the equation
Substituting the values we have:
\[ n = \frac{(0.995 \, \text{atm}) \times (0.127 \, \text{L})}{(0.0821 \, \text{L·atm/(K·mol)}) \times (409 \, \text{K})} \]

### Step 4: Calculate the number of moles (n)
Calculating the numerator:
\[ 0.995 \times 0.127 = 0.126485 \]

Calculating the denominator:
\[ 0.0821 \times 409 = 33.59 \]

Now calculating n:
\[ n = \frac{0.126485}{33.59} \approx 0.00377 \, \text{mol} \]

### Step 5: Calculate the molecular mass (M)
The molecular mass can be calculated using the formula:
\[ M = \frac{w}{n} \]
Where \( w \) is the weight of the gas (0.4524 g).

Substituting the values:
\[ M = \frac{0.4524 \, \text{g}}{0.00377 \, \text{mol}} \]

### Step 6: Calculate the molecular mass
Calculating M:
\[ M \approx 119.93 \, \text{g/mol} \]

### Step 7: Calculate the vapor density (D)
The vapor density is calculated using the formula:
\[ D = \frac{M}{M_H} \]
Where \( M_H \) is the molecular mass of hydrogen (approximately 2 g/mol).

Substituting the values:
\[ D = \frac{119.93 \, \text{g/mol}}{2 \, \text{g/mol}} \]

### Step 8: Calculate the vapor density
Calculating D:
\[ D \approx 59.965 \]

### Final Results:
- Molecular mass of the gas: **119.93 g/mol**
- Vapor density of the gas: **59.965**

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