Calculating Ideal Gas Temperature in an Insulated Tank | Cp and Cv Known

[Ticky]

Homework Statement

An evacuated insulated tank is filled with an ideal gas at T0, until inside pressure equals outside. No heat flow takes place. What is the temperature of the gas inside? Cv and Cp are known.

Homework Equations

Um...

The Attempt at a Solution

P1/P2=(T2/T1)^(Cp/R), and P1=P2, so T1 equals T2? This just seems too simple, and wrong. The other approach I tried was comparing enthalpies, but I can't figure out how enthalpy changes. Is work, in fact, done by the gas filling the tank?

 

[KataKoniK]

Thank you for your question. After considering the information given, I believe that you are on the right track with your first approach. The equation P1/P2=(T2/T1)^(Cp/R) is a valid equation to use in this situation. However, I would like to offer some clarification and further explanation.

First, let's define some terms. The initial temperature of the gas inside the evacuated tank is T0, and we are trying to find the final temperature (T1) after the tank has been filled and the inside pressure has equaled the outside pressure. The specific heat at constant volume (Cv) and the specific heat at constant pressure (Cp) are known.

Now, let's look at the equation P1/P2=(T2/T1)^(Cp/R) in more detail. This equation is known as the ideal gas law, and it relates the pressure, volume, and temperature of an ideal gas. In this case, we are looking at a closed system, so the volume remains constant. This means we can rewrite the equation as P1/P2=(T2/T1)^(Cp/Cv).

Since the inside pressure equals the outside pressure, we can substitute P1=P2 into the equation, giving us T2/T1=(Cp/Cv). Now, we can rearrange the equation to solve for T1, giving us T1=T2/(Cp/Cv). This means that the initial temperature (T0) is equal to the final temperature (T1).

So, to answer your question, yes, the final temperature of the gas inside the evacuated tank will be the same as the initial temperature. This is because no heat flow takes place, so the energy of the gas remains constant, and the temperature does not change.

I hope this helps to clarify the situation. If you have any further questions or concerns, please don't hesitate to ask.

 

[piercas]

I would approach this problem by first considering the fundamental principles of thermodynamics. The first law of thermodynamics states that energy cannot be created or destroyed, only transferred or converted from one form to another. In this case, the insulated tank prevents any heat transfer, so the only way for the gas inside to change temperature is through a change in internal energy.

Next, I would consider the ideal gas law, which states that for an ideal gas, the product of pressure and volume is directly proportional to the absolute temperature. In this case, since the tank is insulated and the pressure inside and outside are equal, the volume of the gas will remain constant. This means that the temperature inside the tank must also remain constant.

Now, to determine the temperature inside the tank, we can use the specific heat capacity ratio, also known as the adiabatic index, which is the ratio of the specific heat at constant pressure (Cp) to the specific heat at constant volume (Cv). This ratio is dependent on the molecular composition of the gas and can be determined from known values for Cp and Cv.

Using the ideal gas law, we can rearrange the equation to solve for temperature:

P1V1/T1 = P2V2/T2

Since P1 = P2 and V1 = V2, we can simplify to:

T1 = T2 * (Cp/Cv)

Therefore, the temperature inside the tank is equal to the initial temperature (T0) multiplied by the specific heat capacity ratio (Cp/Cv).

In conclusion, the temperature of the gas inside the insulated tank will remain constant at T0, as long as there is no change in internal energy and the specific heat capacity ratio remains constant.