Thermodynamics: two pistons; different pressures, volumes, and temperatures

In summary, when the valve is opened, the temperature and pressure in the two vessels become uniform.
  • #1
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Homework Statement

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Two thermally insulated vessels are connected by a narrow tube fitted with a valve that is initially closed as shown in the figure. One vessel of volume V1 = 15.2 L, contains oxygen at a temperature of T1 = 280 K and a pressure of P1 = 1.77 atm. The other vessel of volume V2 = 23.0 L contains oxygen at a temperature of T2 = 460 K and a pressure of P2 = 2.35 atm. When the valve is opened, the gases in the two vessels mix and the temperature and pressure become uniform throughout.

What is the final temperature?

What is the final pressure?

Homework Equations


I really do not know which are relevantPiVi/ Ti = PfVf/Tf

1 atm = 101325 pa

1L = 1000cm3ΔE internal = Q (isovolumetric process)

ΔE internal = W (adiabatic process)

ΔE internal = Q + W (first law of thermodynamics)

The Attempt at a Solution


I really do not know where to start or what formula to use so any hints would be helpful. We didn't have very much time to cover all this at all (very watered down) so if there is something I am supposed to give and don't give, then sorry.

The way I see it, I think you would solve for the tube as if it were another vessel / piston, but I am not sure how to even do that :(

P1 = 179345 pa
V1 = 152 m3
T1 = 280 kelvins

P2 = 238113 pa
V2 = 230 m3
T2 = 460 kelvins
 
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  • #2
as the system is isolated ,its internal energy will remain constant.
oxygen is diatomic, so internal energy of of n moles =2.5nRT=2.5PV
Ei=Ef
Ei=2.5P1V1+2.5P2V2
let final pressure be P,then
Ef=2.5P(v1+V2)
so,P=(P1V1+P2V2)/(V1+V2)

initial moles of oxygen=(P1V1)/RT1 +(P2V2)/RT2
so,P*(V1+V2)=(initial moles)*R*T (T is final temp.)
solve to get,T.
 
  • #3
Thank you so much. I got them both now even though you practically did the formula for me.

I knew that it was an isolated system too but didn't see how that would help either but...

pcm said:
oxygen is diatomic, so internal energy of of n moles =2.5nRT=2.5PV
Don't know where you got this from. And for temperature, I didn't realize you could use the ideal gas law like that (adding the pressure / volumes of the cylinders equal to find n).Thanks.
 
  • #4
oxygen is diatomic ,so it has 3 translational and 2 rotational degrees of freedom.
so,internal energy per mole of oxygen is (3+2)/2 *RT ...(equipartition of energy theorem)
 
  • #5


The first step in solving this problem would be to use the ideal gas law, PV = nRT, to calculate the number of moles of oxygen in each vessel. This can be done by rearranging the equation to solve for n and plugging in the given values for pressure, volume, and temperature. Once you have the number of moles, you can use the equation n1V1 = n2V2 to find the final volume of the combined gases.

Next, you can use the equation for the first law of thermodynamics, ΔE = Q + W, to calculate the change in internal energy of the system. Since the vessels are thermally insulated, there is no heat transfer (Q = 0), and the only work done is by the gases expanding to fill the larger combined volume. This can be calculated using the equation W = -PΔV.

Once you have the change in internal energy, you can use the equation ΔE = ncΔT, where c is the molar specific heat capacity of oxygen, to solve for the final temperature. This equation assumes that the gases undergo an isochoric (constant volume) process.

Finally, you can use the ideal gas law again to calculate the final pressure of the combined gases. Plug in the final volume and temperature, and solve for pressure using the equation P = nRT/V.

Remember to convert all units to the correct form (i.e. use joules for energy and kelvins for temperature) and to keep track of significant figures throughout your calculations.
 

Related to Thermodynamics: two pistons; different pressures, volumes, and temperatures

1. What is thermodynamics?

Thermodynamics is a branch of physics that deals with the relationships between heat, energy, and work. It studies how these factors affect the behavior of matter and how they can be used to understand and predict the physical properties of systems.

2. What are pistons in thermodynamics?

In thermodynamics, pistons are devices used to convert heat energy into mechanical energy. They consist of a cylindrical container with a movable disc (piston) that can be pushed or pulled by pressure differences within the system. They are commonly used in engines and refrigeration systems.

3. How do two pistons with different pressures, volumes, and temperatures behave?

When two pistons with different pressures, volumes, and temperatures are connected, they will reach an equilibrium state where the pressures and temperatures between the two pistons are equal. This is known as the Second Law of Thermodynamics, which states that heat will flow from a higher temperature to a lower temperature until thermal equilibrium is reached.

4. How does thermodynamics explain the relationship between pressure, volume, and temperature?

Thermodynamics explains the relationship between pressure, volume, and temperature through the Ideal Gas Law, which states that the product of pressure and volume is directly proportional to the absolute temperature of a gas. This means that as one of these variables changes, the others will also change in a predictable manner.

5. What are the practical applications of thermodynamics with two pistons?

The practical applications of thermodynamics with two pistons include the operation of engines, refrigeration systems, and heat pumps. It is also used in the design of power plants, turbines, and other energy conversion systems. Understanding thermodynamics is crucial for many industrial processes and advancements in technology.

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