Procedure:
(a) To calculate the pressure of the molecular cloud, we can use the ideal gas law equation: PV = nRT, where P is the pressure, V is the volume, n is the number of moles, R is the ideal gas constant, and T is the temperature.
First, we need to calculate the number of moles of H2 in the molecular cloud. We are given that there are 106 molecules per cubic centimeter. To convert this to moles, we divide by Avogadro's number (6.022 × 10^23 molecules/mol):
Number of moles = (106 molecules/cm^3) / (6.022 × 10^23 molecules/mol)
Next, we need to convert the volume from cubic centimeters to cubic meters. Since 1 cm^3 = 1 × 10^-6 m^3:
Volume = (1 × 10^-6 m^3)
Now, we can substitute the values into the ideal gas law equation and solve for pressure:
P * (1 × 10^-6 m^3) = [(106 molecules/cm^3) / (6.022 × 10^23 molecules/mol)] * (8.314 J/(mol·K)) * (20 K)
Solving for P, we get:
P = [(106 molecules/cm^3) / (6.022 × 10^23 molecules/mol)] * (8.314 J/(mol·K)) * (20 K) / (1 × 10^-6 m^3)
(b) To calculate the root mean square speed of the gas molecules, we can use the equation:
v = √(3RT/M), where v is the root mean square speed, R is the ideal gas constant, T is the temperature, and M is the molar mass.
In this case, the molar mass of H2 is given as 2.01589 g/mol. We need to convert this to kilograms:
Molar mass = 2.01589 g/mol * (1 kg / 1000 g)
Now, we can substitute the values into the equation and solve for v:
v = √(3 * (8.314 J/(mol·K)) * (20 K) / (2.01589 kg/mol))
(c) To calculate the translational kinetic energy per molecule, we can use the equation:
KE = (3/2) * k * T, where KE is the kinetic energy, k is the Boltzmann constant, and T is the temperature.
Substituting the values, we get:
KE = (3/2) * (1.381 × 10^-23 J/K) * (20 K)
To calculate the energy for all molecules of the gas in a m^3, we multiply the kinetic energy per molecule by the number of molecules:
Energy = [(106 molecules/cm^3) / (6.022 × 10^23 molecules/mol)] * (8.314 J/(mol·K)) * (20 K) * (1 × 10^-6 m^3)
(d) To calculate the new pressure and temperature if the gas expands adiabatically to double its volume, we can use the adiabatic expansion equation:
P1 * V1^γ = P2 * V2^γ, where P1 and V1 are the initial pressure and volume, P2 and V2 are the final pressure and volume, and γ is the adiabatic index (for diatomic gases, γ = 7/5).
In this case, the initial volume is 1 × 10^-6 m^3. To double the volume, the final volume will be 2 × (1 × 10^-6 m^3).
Substituting the values into the equation, we get:
P1 * (1 × 10^-6 m^3)^γ = P2 * (2 × 1 × 10^-6 m^3)^γ
To calculate the new pressure, we need to solve for P2:
P2 = P1 * (1 × 10^-6 m^3)^γ / (2 × 1 × 10^-6 m^3)^γ
To calculate the new temperature, we can use the equation:
T2 = T1 * (V1 / V2)^(γ-1)
Substituting the values, we get:
T2 = 20 K * (1 × 10^-6 m^3 / (2 × 1 × 10^-6 m^3))^(7/5 - 1)