Helium gas occupies a volume of 0.04m^3 at a pressure of 2*10^5 N/m^2 and a temperature of 300K,calculate:
1.) the mass of helium
2.)the root mean square speed of its molecule
3.)the root mean square speed at 432K when the gas is heated at constant pressure to this temperature
The answer for the question
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To calculate the given quantities for helium gas, we can use the ideal gas law and the kinetic theory of gases.
1.) The mass of helium:
To calculate the mass of helium, we need to know the number of moles of helium gas first. We can use the ideal gas law equation to find this:
PV = nRT
Where:
P = pressure of the gas (2 * 10^5 N/m^2)
V = volume of the gas (0.04 m^3)
n = number of moles of the gas (unknown)
R = ideal gas constant (8.314 J/(mol*K))
T = temperature of the gas (300 K)
Rearranging the equation to solve for n:
n = PV / RT
Plugging in the values:
n = (2 * 10^5 N/m^2) * (0.04 m^3) / (8.314 J/(mol*K) * 300 K)
Simplifying:
n ≈ 0.03214 moles
The molar mass of helium is approximately 4 g/mol. So, to calculate the mass of helium, we can multiply the number of moles by the molar mass:
Mass = n * molar mass
Mass = 0.03214 moles * 4 g/mol
Mass ≈ 0.12856 g
Therefore, the mass of helium is approximately 0.12856 g.
2.) The root mean square speed of helium molecules:
The root mean square (rms) speed of gas molecules can be calculated using the kinetic theory of gases with the formula:
v(rms) = √(3RT / M)
Where:
v(rms) = root mean square speed
R = ideal gas constant (8.314 J/(mol*K))
T = temperature of the gas (300 K)
M = molar mass of helium (4 g/mol)
Plugging in the values:
v(rms) = √(3 * 8.314 J/(mol*K) * 300 K) / 4 g/mol
Simplifying:
v(rms) ≈ √6231.6 J/mol / 4 g/mol
v(rms) ≈ √(1557.9 m^2/s^2) / (10^3 m^2/kg)
v(rms) ≈ 39.5 m/s
Therefore, the root mean square speed of helium molecules is approximately 39.5 m/s.
3.) The root mean square speed at 432K:
To calculate the root mean square speed at a different temperature, we can use the same formula as before:
v(rms) = √(3RT / M)
Where:
v(rms) = root mean square speed
R = ideal gas constant (8.314 J/(mol*K))
T = temperature of the gas (432 K)
M = molar mass of helium (4 g/mol)
Plugging in the values:
v(rms) = √(3 * 8.314 J/(mol*K) * 432 K) / 4 g/mol
Simplifying:
v(rms) ≈ √10309.944 J/mol / 4 g/mol
v(rms) ≈ √(2577.486 m^2/s^2) / (10^3 m^2/kg)
v(rms) ≈ 50.8 m/s
Therefore, the root mean square speed of helium molecules at 432K is approximately 50.8 m/s.