Answer :
The molar mass of the unknown gas is approximately 98.5 g/mol, and the correct option is A. 98.5.
To calculate the molar mass, we can use the combined gas law equation, which combines the relationships between pressure, volume, and temperature for gases. The combined gas law is given by:
(P1 × V1) / (T1 × n1) = (P2 × V2) / (T2 × n2)
Where P1, V1, and T1 are the initial pressure, volume, and temperature, respectively, and P2, V2, and T2 are the final pressure, volume, and temperature, respectively. In this case, we are given the densities in grams per liter (g/L) and the molar volume of any gas at STP (standard temperature and pressure) is 22.4 L.
First, we need to find the number of moles (n) of the unknown gas. We know the density (d) is equal to mass (m) divided by volume (V), so:
d = m / V
m = d × V
Now, we can find the number of moles using the molar mass (M):
n = m / M
Substitute the value of the molar volume at STP (V = 22.4 L) and the given density (1.62 g/L) into the equation:
n = (1.62 g/L × 22.4 L) / M
Next, we need to use the ideal gas law equation to find the molar mass. The ideal gas law is given by:
PV = nRT
where R is the ideal gas constant (0.0821 L.atm/mol.K), and T is the temperature in Kelvin. Convert the given temperature (22.5 degrees Celsius) to Kelvin:
T = 22.5 + 273.15 = 295.65 K
Now, we can solve for M:
M = (P × V) / (n × R × T)
Substitute the values:
M = (2.14 atm × 22.4 L) / (n × 0.0821 L.atm/mol.K × 295.65 K)
M = 47.8568 / n
Now, equate the two expressions for M:
M = 47.8568 / [(1.62 g/L × 22.4 L) / M]
Solve for M:
M^2 = 47.8568 / (1.62 × 22.4)
M^2 = 1.45
M = √1.45
M ≈ 1.20 g/mol
Since the molar mass cannot be negative, we discard the negative root.
Therefore, the molar mass of the unknown gas is approximately 98.5 g/mol, and the correct option is A. 98.5.
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