Can you check my answer? Thank you
What is the molar mass of a gas that behaves ideally (i.e. obeys the perfect gas law pV=nRT) if it has a density of 1.92 g dm-3 at 150 kPa and 25 C?
I got;
M=p(density)RT/(P)=31.7 kg mol^-1
31.7 is ok but are sure that's kg/mol. The density is given in g/L. Besides, a molar mass of 31,700 for a gas seems unreasonably high to me.
To calculate the molar mass of the gas, we can use the ideal gas law equation:
PV = nRT
Where:
P is the pressure in Pascal (Pa),
V is the volume in cubic meters (m^3),
n is the number of moles,
R is the ideal gas constant (8.314 J/(mol·K)),
T is the temperature in Kelvin (K).
First, let's convert the given values to the appropriate units:
- Pressure: 150 kPa = 150,000 Pa
- Density: 1.92 g/dm^3 = 1.92 g/L = 1920 g/m^3 (since 1 dm^3 = 1 L)
- Temperature: 25 degrees Celsius = 25 + 273.15 = 298.15 K
Next, we need to find the number of moles (n) using the ideal gas law. We can rearrange the equation to solve for n:
n = PV / RT
Substituting the given values:
n = (150,000 Pa) * (V) / ((8.314 J/(mol·K)) * (298.15 K))
Now, let's convert the given density to molar mass:
- Density: 1920 g/m^3
- Volume (V) can be calculated by dividing the mass (m) by density (d):
V = m / d = (1920 g) / (1920 g/m^3) = 1 m^3
Note: The calculation assumes a volume of 1 m^3.
Substituting the values into the equation for n, we get:
n = (150,000 Pa) * (1 m^3) / ((8.314 J/(mol·K)) * (298.15 K))
Simplifying the equation:
n ≈ 63.77 mol
Finally, to find the molar mass (M), we can rearrange the density equation:
density = mass / volume
mass = density * volume
mass = 1.92 g/dm^3 * 1 dm^3
mass = 1.92 g
Substituting the values into the formula:
M = mass / n
M = 1.92 g / 63.77 mol
Calculating the molar mass:
M ≈ 0.0301 g/mol
Therefore, the molar mass of the gas is approximately 0.0301 g/mol.
To calculate the molar mass of the gas using the ideal gas law, you need to rearrange the equation to solve for the molar mass (M). Here's how you can do it step by step:
1. Start with the ideal gas law equation:
PV = nRT
2. Identify the given values:
- Density (ρ) = 1.92 g dm^(-3)
- Pressure (P) = 150 kPa
- R is the ideal gas constant (8.314 J K^(-1) mol^(-1))
- Temperature (T) = 25 degrees Celsius = 298 K
3. Convert density to mass:
Density (ρ) = Mass (m) / Volume (V)
Rearranging the equation, we get: Mass (m) = Density (ρ) * Volume (V)
Since the units given for density are g dm^(-3), we need to convert it to kg m^(-3) to match the units of the ideal gas constant (R) in the equation.
1 g dm^(-3) = 1 kg m^(-3) / 1000
Therefore, the converted density is: 1.92 g dm^(-3) * 1 kg m^(-3) / 1000 = 0.00192 kg m^(-3)
4. Substitute the known values into the rearranged equation for mass:
Mass (m) = 0.00192 kg m^(-3) * V
5. Solve for the volume (V):
Volume (V) is given in dm^3 (decimeter cubed), which is equivalent to 0.001 m^3.
We can express V as:
V = (10^-3 m^3 / 1 dm^3) * V(dm^3)
Therefore, V = (10^-3) * V(dm^3)
6. Substitute the values of mass (m) and volume (V) into the ideal gas law equation:
PV = nRT
(P * V) = (m / M) * R * T
Rearrange the equation to solve for molar mass (M):
M = (m * R * T) / (P * V)
Now substitute the known values:
M = (0.00192 kg m^(-3) * V(dm^3) * R * 298 K) / (150 kPa * (10^-3) * V(dm^3))
Simplifying further, M = (0.00192 * R * 298) / 150
7. Calculate the molar mass:
Now, calculate M:
M = (0.00192 * 8.314 * 298) / 150
M ≈ 0.0317 kg mol^(-1)
So, after calculating everything, the molar mass of the gas is approximately 0.0317 kg mol^(-1).