Work done on container receiving gas from high pressure container

  • #1
zenterix
621
81
Homework Statement
A thin-walled metal container of volume ##V## contains a gas at high pressure. COnnected to the container is a capillary tube and stopcock. When the stopcock is opened slightly, the gas leaks slowly into a cylinder equipped with a nonleaking, frictionless piston, where the pressure remains constant at the atmospheric value ##P_0##.

(a) Show that, after as much gas as possible has leaked out, an amount of work

$$W=-P_0(V_0-V)$$

has been done, where ##V## is the volume of the gas at atmospheric pressure and temperature.

(b) How much work would be done if the gas leaked directly into the atmosphere?
Relevant Equations
$$W=-\int_{V_i}^{V_f} PdV$$
1696832815801.png


It seems to me that we can already answer b): if gas leaks from the high pressure container to the atmosphere, there is no expansion work. The container loses gas, which means in ##PV=nRT## we have ##n## going down and ##P## going down, and perhaps ##T## going down.

But no work.

As for a), I assume we have an approximately quasi-static process happening: the flow of gas is extremely slow such that we have approximately uniform temperature in each container individually throughout the process.

Specifically for the container with the piston, the pressure is always approximately the atmospheric pressure ##P_0##. What is really happening is that in many infinitesimally small occurrences, the pressure is ##P_0+dP##, the container expands a little bit, and the pressure goes down to ##P_0## again.

We can use equations of state for the gas in each container at every point in the process.

For the high pressure container,

$$W=-\int_V^V PdV=0$$

For the container with the piston,

$$W=-\int_{V_i}^{V_f} P_0 dV=$$

Now, ##V_0=V+V_f## then ##V_f=V_0-V##, and so if ##V_i=0##, then we get the desired result:

$$W=-P_0(V_f-V_i)=-P_0(V_0-V-0)=-P_0(V_0-V)$$

Seems to be correct, now that I wrote it all out here.
 

Attachments

  • 1696832796947.png
    1696832796947.png
    14.7 KB · Views: 45
Last edited:
Physics news on Phys.org
  • #2
It seems to me that this is not done correctly. Is V the volume of the high pressure container or the volume of gas at atmospheric temperature and pressure? What it Vo supposed to be?
 

Related to Work done on container receiving gas from high pressure container

What is the definition of work done on a container receiving gas from a high-pressure container?

Work done on a container receiving gas from a high-pressure container refers to the energy transferred to the gas as it moves from a region of high pressure to a region of lower pressure. This process typically involves changes in volume and pressure, which can be quantified using thermodynamic principles.

How is the work done calculated in this process?

The work done on the container can be calculated using the formula W = PΔV, where W is the work done, P is the pressure of the gas, and ΔV is the change in volume of the gas. For more complex scenarios involving non-constant pressure, the work done can be calculated using the integral W = ∫ P dV.

What factors affect the amount of work done on the container?

The amount of work done on the container is affected by several factors, including the initial and final pressures of the gas, the volume change of the container, the temperature of the gas, and the properties of the gas itself (e.g., whether it behaves ideally or non-ideally).

Is the process of gas transfer always adiabatic?

No, the process of gas transfer is not always adiabatic. An adiabatic process is one in which no heat is exchanged with the surroundings. Depending on the specific conditions and setup, the gas transfer can be isothermal (constant temperature), isobaric (constant pressure), or follow other thermodynamic paths.

What are some practical applications of understanding the work done on a container receiving gas from a high-pressure container?

Understanding the work done on a container receiving gas from a high-pressure container has practical applications in various fields, including engineering (e.g., designing pressure vessels and gas pipelines), refrigeration and air conditioning systems, industrial gas storage and transport, and even in scientific research involving gas dynamics and thermodynamics.

Similar threads

  • Introductory Physics Homework Help
Replies
3
Views
1K
  • Introductory Physics Homework Help
Replies
4
Views
856
  • Introductory Physics Homework Help
Replies
12
Views
914
  • Introductory Physics Homework Help
Replies
10
Views
1K
  • Introductory Physics Homework Help
Replies
27
Views
5K
  • Introductory Physics Homework Help
Replies
8
Views
1K
  • Introductory Physics Homework Help
Replies
11
Views
1K
Replies
5
Views
706
  • Introductory Physics Homework Help
Replies
1
Views
1K
  • Introductory Physics Homework Help
Replies
2
Views
1K
Back
Top