A 3 mg sample of an unknown ideal gas is subjected to an adiabatic and
mechanically reversible expansion from an initial pressure of 2 atmospheres to a final pressure
of 300 torr. Given that gamma
for the gas is 1.532 and that the temperature drop achieved in the
expansion is 231 K, calculate both the initial and final temperatures. Go on to determine
delta H and the molecular weight of the gaseous species if the energy lost in the expansion is
0.126 J. empirical relation
To solve this problem, we can use the ideal gas law and the adiabatic expansion equation. Here are the steps to find the initial and final temperatures, as well as delta H and the molecular weight of the gas:
Step 1: Calculate the initial temperature (T1):
- We know the initial pressure (P1) is 2 atmospheres and the sample size (n) is 3 mg.
- We'll use the ideal gas law: 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.
- Rearranging the equation, T1 = PV / (nR)
- Plug in the values: T1 = (2 atm) * V / (3 mg * R)
Step 2: Calculate the final temperature (T2):
- We know the final pressure (P2) is 300 torr.
- We'll use the adiabatic expansion equation: P1 * V1^(gamma) = P2 * V2^(gamma), where gamma is the heat capacity ratio for the gas, P is the pressure, and V is the volume.
- Rearranging the equation, V2 = (P1 / P2)^(1/gamma) * V1
- Plug in the values: V2 = (2 atm / 300 torr)^(1/1.532) * V1
- The temperature drop achieved in the expansion is given as 231 K, so T2 = T1 - 231 K.
Step 3: Calculate delta H:
- Delta H is the energy lost in the expansion, given as 0.126 J.
- Delta H = n * Cp * delta T, where n is the number of moles, Cp is the molar heat capacity at constant pressure, and delta T is the temperature change.
- Rearranging the equation, n = delta H / (Cp * delta T)
- Plug in the values: n = 0.126 J / (Cp * 231 K)
Step 4: Determine the molecular weight (MW) of the gas species:
- MW = (mass / moles), where the mass is given as 3 mg.
- Plug in the values: MW = (3 mg / n)
By following these steps and knowing the values of the given parameters, you can calculate the initial and final temperatures, delta H, and the molecular weight of the gas.