Helium is monatomic with a relative atomic mass of 4. calculate (i) the density of helium gas at a temperature of 0 degrees Celsius and a pressure of 101kPa and (ii) the rms speed of the helium atoms under these conditions
http://en.wikipedia.org/wiki/Root_mean_square_speed
i)0.18
ii)1300
To calculate the density of helium gas at a temperature of 0 degrees Celsius and a pressure of 101 kPa, we can use the ideal gas law. The ideal gas law states that the product of pressure (P), volume (V), and the number of moles of gas (n) is directly proportional to the product of the gas constant (R) and the temperature (T) in Kelvin.
The ideal gas law equation is expressed as:
PV = nRT
To solve for density, we need to rearrange the equation to solve for n/V (number of moles per unit volume) and then divide it by the molar mass of helium.
Let's calculate:
(i) Density of Helium gas:
First, let's convert 0 degrees Celsius to Kelvin.
T(K) = T(°C) + 273.15 = 0 + 273.15 = 273.15 K
The molar mass of helium is 4 g/mol.
The gas constant R is 8.314 J/(mol·K).
The equation becomes:
PV = nRT
n/V = P/RT
Substituting the values:
n/V = (101kPa) / (8.314 J/(mol·K) × 273.15 K)
n/V = (101 × 10^3 Pa) / (8.314 J/(mol·K) × 273.15 K)
n/V = 16.3897 mol/m^3
Now, we divide by the molar mass of helium (4 g/mol):
Density = (16.3897 mol/m^3) / (4 g/mol)
Density = 4.0974 g/m^3
Therefore, the density of helium gas at a temperature of 0 degrees Celsius and a pressure of 101 kPa is approximately 4.0974 g/m^3.
(ii) RMS (Root Mean Square) speed of Helium atoms:
The RMS speed of gas molecules is given by the equation:
v_rms = √((3RT)/M)
Where:
v_rms is the RMS speed
R is the gas constant (8.314 J/(mol·K))
T is the temperature in Kelvin (273.15 K)
M is the molar mass of helium (4 g/mol)
Substituting the values:
v_rms = √((3 × 8.314 J/(mol·K) × 273.15 K)/(4 g/mol))
v_rms = √(6182 J/(mol·g))
v_rms ≈ 1572.25 m/s
Therefore, the RMS speed of helium atoms at a temperature of 0 degrees Celsius and a pressure of 101 kPa is approximately 1572.25 m/s.