Answer :
To find the molar mass of the gas, we can use the Ideal Gas Law equation:
[tex]\[ PV = nRT \][/tex]
Here, [tex]\( P \)[/tex] is the pressure, [tex]\( V \)[/tex] is the volume, [tex]\( n \)[/tex] is the number of moles, [tex]\( R \)[/tex] is the ideal gas constant, and [tex]\( T \)[/tex] is the temperature in Kelvin. Our goal is to determine [tex]\( n \)[/tex] first so we can find the molar mass.
### Step-by-step Solution:
1. Convert Mass:
- The mass of the gas is given as 38.8 mg. First, convert this to grams:
[tex]\[
\text{mass in grams} = \frac{38.8 \text{ mg}}{1000} = 0.0388 \text{ g}
\][/tex]
2. Convert Volume:
- The volume of the gas is given as 224 mL. Convert this to liters:
[tex]\[
\text{volume in liters} = \frac{224 \text{ mL}}{1000} = 0.224 \text{ L}
\][/tex]
3. Convert Temperature:
- The temperature is given as [tex]\(55^\circ C\)[/tex]. Convert this to Kelvin:
[tex]\[
\text{temperature in Kelvin} = 55 + 273.15 = 328.15 \text{ K}
\][/tex]
4. Convert Pressure:
- The pressure is given as 886 torr. Convert this to atmospheres (atm):
[tex]\[
\text{pressure in atm} = \frac{886 \text{ torr}}{760} \approx 1.166 \text{ atm}
\][/tex]
5. Calculate Moles (n):
- Using the Ideal Gas Law rearranged to solve for [tex]\( n \)[/tex]:
[tex]\[
n = \frac{PV}{RT}
\][/tex]
- [tex]\( P = 1.166 \text{ atm} \)[/tex]
- [tex]\( V = 0.224 \text{ L} \)[/tex]
- [tex]\( R = 0.0821 \text{ L atm/mol K} \)[/tex] is the ideal gas constant in these units.
- [tex]\( T = 328.15 \text{ K} \)[/tex]
[tex]\[
n \approx \frac{1.166 \times 0.224}{0.0821 \times 328.15} \approx 1.28 \times 10^{-5} \text{ moles}
\][/tex]
6. Calculate Molar Mass:
- The molar mass [tex]\( M \)[/tex] is calculated by dividing the mass by the number of moles:
[tex]\[
M = \frac{\text{mass in grams}}{n}
\][/tex]
[tex]\[
M = \frac{0.0388 \text{ g}}{1.28 \times 10^{-5} \text{ moles}} \approx 3040.66 \text{ g/mol}
\][/tex]
Thus, the molar mass of the gas is approximately 3040.66 g/mol.
[tex]\[ PV = nRT \][/tex]
Here, [tex]\( P \)[/tex] is the pressure, [tex]\( V \)[/tex] is the volume, [tex]\( n \)[/tex] is the number of moles, [tex]\( R \)[/tex] is the ideal gas constant, and [tex]\( T \)[/tex] is the temperature in Kelvin. Our goal is to determine [tex]\( n \)[/tex] first so we can find the molar mass.
### Step-by-step Solution:
1. Convert Mass:
- The mass of the gas is given as 38.8 mg. First, convert this to grams:
[tex]\[
\text{mass in grams} = \frac{38.8 \text{ mg}}{1000} = 0.0388 \text{ g}
\][/tex]
2. Convert Volume:
- The volume of the gas is given as 224 mL. Convert this to liters:
[tex]\[
\text{volume in liters} = \frac{224 \text{ mL}}{1000} = 0.224 \text{ L}
\][/tex]
3. Convert Temperature:
- The temperature is given as [tex]\(55^\circ C\)[/tex]. Convert this to Kelvin:
[tex]\[
\text{temperature in Kelvin} = 55 + 273.15 = 328.15 \text{ K}
\][/tex]
4. Convert Pressure:
- The pressure is given as 886 torr. Convert this to atmospheres (atm):
[tex]\[
\text{pressure in atm} = \frac{886 \text{ torr}}{760} \approx 1.166 \text{ atm}
\][/tex]
5. Calculate Moles (n):
- Using the Ideal Gas Law rearranged to solve for [tex]\( n \)[/tex]:
[tex]\[
n = \frac{PV}{RT}
\][/tex]
- [tex]\( P = 1.166 \text{ atm} \)[/tex]
- [tex]\( V = 0.224 \text{ L} \)[/tex]
- [tex]\( R = 0.0821 \text{ L atm/mol K} \)[/tex] is the ideal gas constant in these units.
- [tex]\( T = 328.15 \text{ K} \)[/tex]
[tex]\[
n \approx \frac{1.166 \times 0.224}{0.0821 \times 328.15} \approx 1.28 \times 10^{-5} \text{ moles}
\][/tex]
6. Calculate Molar Mass:
- The molar mass [tex]\( M \)[/tex] is calculated by dividing the mass by the number of moles:
[tex]\[
M = \frac{\text{mass in grams}}{n}
\][/tex]
[tex]\[
M = \frac{0.0388 \text{ g}}{1.28 \times 10^{-5} \text{ moles}} \approx 3040.66 \text{ g/mol}
\][/tex]
Thus, the molar mass of the gas is approximately 3040.66 g/mol.
