Take the mass of carbon dioxide
remaining in the bottle after
leak as a system
The Process on the T-s Diagram
Molar Mass, Gas Constant for Various Common Gases
Specific Heats of Some
Common Ideal Gases
Take the bottle as the system. Denote the state before leak as state 1 and the state after leak as state 2.
From the assumptions made, carbon dioxide is modeled as an idea
gas. With the ideal-gas equation of state, the mass remains in the bottle after leak is
m2 = P2V2/RT2
where P2 and V2 are known. If the temperature
T2 is determined, the mass remaining m2 can be
obtained. The gas constant R of carbon dioxide is 188.9 J(kg-K).
The leaking process is adiabatic and reversible, hence it is an isentropic
process. The second relation of isentropic process for ideal gases is:
Assume the final temperature is 350 K and use the average value
of the initial temperature and the final temperature to determine the value of k.
Tav = (500 + 350 )/2 = 425 K
From the table, k equals to 1.246 at 425 K.
With all the data known, T2 can be obtained by substituting
the values of T1,
P1, and P2 into the above equation.
T2 = 500(1/5)0.246/1.246 = 363.9 K
The calculated final temperature is close to 350 K (the assumed
temperature), hence no iteration is needed.
Inserting T2, P2, and V2 into the
expression for m2 gives,
m2 = (101,325)(200/1,000)/((188.9)(363.9))
= 295 g