To find the molar mass of the unknown gas, we can follow these steps:
1. Convert the given volume to liters:
The volume provided is 25.0 mL. Converting mL to L:
[tex]\[
\text{Volume} = \frac{25.0}{1000} = 0.0250 \text{ L}
\][/tex]
2. Convert the given temperature to Kelvin:
The temperature provided is 25°C. Converting Celsius to Kelvin:
[tex]\[
\text{Temperature in Kelvin} = 25 + 273.15 = 298.15 \text{ K}
\][/tex]
3. Use the Ideal Gas Law to find the number of moles (n) of the gas:
The ideal gas law is:
[tex]\[
PV = nRT
\][/tex]
Rearranging to solve for [tex]\( n \)[/tex]:
[tex]\[
n = \frac{PV}{RT}
\][/tex]
Where:
- [tex]\( P = 0.995 \text{ atm} \)[/tex]
- [tex]\( V = 0.0250 \text{ L} \)[/tex]
- [tex]\( R = 0.0821 \text{ L·atm/(mol·K)} \)[/tex]
- [tex]\( T = 298.15 \text{ K} \)[/tex]
Plugging in the values:
[tex]\[
n = \frac{(0.995 \text{ atm}) \times (0.0250 \text{ L})}{(0.0821 \text{ L·atm/(mol·K)}) \times (298.15 \text{ K})}
\][/tex]
Calculating the number of moles [tex]\( n \)[/tex]:
[tex]\[
n \approx 0.0010162138710435834 \text{ moles}
\][/tex]
4. Calculate the molar mass:
The molar mass (M) can be determined using the formula:
[tex]\[
M = \frac{\text{mass}}{n}
\][/tex]
Where:
- The mass of the gas is 0.0200 grams.
- The number of moles [tex]\( n \)[/tex] is approximately 0.0010162138710435834 moles.
Plugging in the values:
[tex]\[
M = \frac{0.0200 \text{ grams}}{0.0010162138710435834 \text{ moles}}
\][/tex]
Calculating the molar mass:
[tex]\[
M \approx 19.68089648241206
\][/tex]
Considering significant figures, the mass given (0.0200 grams) has four significant figures, the pressure (0.995 atm) has three significant figures, the volume (25.0 mL converted to 0.0250 L) has three significant figures, and the gas constant [tex]\( R \)[/tex] typically contributes three or more significant figures. Thus, the final result should be capped at three significant figures according to the least count in the measurements.
Therefore, the molar mass of the unknown gas is:
[tex]\[
\boxed{19.7}
\][/tex]
