The density of phosphine gas is `"1.27 g dm"^(-3)` at `50^(@)C` and `0.987xx10^(5)Pa.` Calculate its molar mass.

Explanation

To calculate the molar mass of phosphine gas using the given density, temperature, and pressure, we can follow these steps:

### Step 1: Convert Temperature to Kelvin
The temperature is given as \(50^\circ C\). To convert it to Kelvin, we use the formula:
\[ T(K) = T(°C) + 273.15 \]
Substituting the value:
\[ T = 50 + 273.15 = 323.15 \, K \]

### Step 2: Identify Given Values
- Density (\(D\)) = \(1.27 \, g/dm^3\)
- Pressure (\(P\)) = \(0.987 \times 10^5 \, Pa\)
- Temperature (\(T\)) = \(323.15 \, K\)

### Step 3: Use the Ideal Gas Equation
The ideal gas equation is given by:
\[ PV = nRT \]
Where:
- \(n\) = number of moles
- \(R\) = ideal gas constant

We can express the number of moles (\(n\)) in terms of mass (\(W\)) and molar mass (\(M\)):
\[ n = \frac{W}{M} \]
Substituting this into the ideal gas equation gives:
\[ PV = \frac{W}{M}RT \]
Rearranging this, we can express \(PM\) as:
\[ PM = \frac{W}{V}RT \]

### Step 4: Relate Density to Mass and Volume
Density (\(D\)) is defined as:
\[ D = \frac{W}{V} \]
Thus, we can substitute \(W/V\) with \(D\):
\[ PM = DRT \]

### Step 5: Solve for Molar Mass
Rearranging the equation to find the molar mass (\(M\)):
\[ M = \frac{DRT}{P} \]

### Step 6: Substitute the Values
Now we can substitute the values into the equation. First, we need to ensure that \(R\) is in the correct units. The value of \(R\) in \(Pa \cdot m^3/(mol \cdot K)\) is:
\[ R = 8.314 \, Pa \cdot m^3/(mol \cdot K) = 8.314 \times 10^3 \, Pa \cdot dm^3/(mol \cdot K) \]
Now substituting the values:
\[ M = \frac{(1.27 \, g/dm^3)(8.314 \times 10^3 \, Pa \cdot dm^3/(mol \cdot K))(323.15 \, K)}{0.987 \times 10^5 \, Pa} \]

### Step 7: Calculate the Molar Mass
Calculating the numerator:
\[ 1.27 \times 8.314 \times 10^3 \times 323.15 \approx 34,555.2 \, g \cdot Pa \cdot K/(mol \cdot dm^3) \]
Now, dividing by the pressure:
\[ M \approx \frac{34,555.2}{0.987 \times 10^5} \approx 34.55 \, g/mol \]

### Final Result
The molar mass of phosphine gas is approximately:
\[ \boxed{34.55 \, g/mol} \]