Expansion of a Gas into a Vacuum

[kurebo]

Homework Statement

A stop-cock which connects an evacuated flask with the atmosphere is opened and
air at 1 atm and 17°C (290 K) enters. (a) What will be the temperature of the air in the flask
before any heat has been transferred to the walls of the flask? Assume air to be a perfect
gas and its molal heat capacity at constant pressure to be 7.00 calories per degree.
Answer=132 °C (405 K)

2. The attempt at a solution

I started working on this under the assumption that it was a throttling question with constant enthalpy:
H2-H1=0
but I'm stuck at the point where you define enthalpy for each state. You can define the work done by the gas as:
w=int(Cv dT,T,290,T2) where Cv=Cp-R (Cp=7 cal/K;Cv=5.013 cal/K)

Any ideas on where to go?

 

[stewartcs]

Sounds like a free expansion of an ideal gas problem so the temperature change will be zero, i.e. the final temp is the same as the initial temp.

CS

 

[piercas]

First, let's define the initial state of the gas in the flask as state 1, with pressure P1 = 0 (vacuum) and temperature T1 = 17°C = 290 K. The final state, after the gas has expanded into the flask, can be defined as state 2, with pressure P2 = 1 atm and an unknown temperature T2.

Next, we can use the ideal gas law to relate the initial and final states:
P1V1 = nRT1 and P2V2 = nRT2
where V1 and V2 are the volumes of the flask and the gas, n is the number of moles of gas, and R is the ideal gas constant.

Since the flask is initially evacuated, we can assume that the volume of the gas is equal to the volume of the flask, so V1 = V2. This allows us to eliminate the volume terms from the above equations and solve for the ratio of temperatures:
T2/T1 = P2/P1 = 1/0 = undefined

This means that the final temperature T2 is undefined and cannot be determined from the given information. We need to make some additional assumptions or use more information to solve for T2.

One option is to assume that the expansion of the gas is reversible and adiabatic, meaning that no heat is transferred to the walls of the flask and the process occurs slowly enough that the gas can be considered to be in thermal equilibrium at all times. In this case, we can use the adiabatic expansion equation:
T2/T1 = (P2/P1)^((γ-1)/γ)
where γ is the ratio of specific heats (Cp/Cv) for the gas. Since we are given Cp = 7 cal/K and Cv = 5.013 cal/K, we can calculate γ = Cp/Cv = 7/5.013 = 1.396.

Plugging in the values for P2/P1 and γ, we get:
T2/T1 = (1/0)^((1.396-1)/1.396) = undefined
This again leads to an undefined final temperature T2.

Another option is to assume that the expansion is reversible and isothermal, meaning that the temperature of the gas remains constant throughout the process. In this case, the final temperature T2 would be equal to the initial temperature T1 = 290 K.

Without