How can I solve this thermodynamics problem involving an adiabatic process?

In summary, the problem involves a cylinder with n moles of an ideal gas undergoing an adiabatic process. The work done in this process can be calculated using W=-\int Pdv and the condition PV^\gamma=constant. By substituting P=\frac{constant}{V^\gamma} into the integral and evaluating from Vi to Vf, the work done is given by W=(\frac{1}{\gamma - 1}(PfVf - PiVi), where Pf and Pi are the final and initial pressures respectively.
  • #1
thenewbosco
187
0
hello the problem is as stated:
a cylinder containing n moles of an ideal gas undergoes an adiabatic process. using [tex]W=-\int Pdv[/tex] and using the condition [tex]PV^\gamma=constant[/tex], show that the work done is:
[tex]W=(\frac{1}{\gamma - 1}(PfVf - PiVi)[/tex] where Pf is final pressure, Pi is initial pressure...
I tried substituting that [tex]P=\frac{constant}{V^\gamma}[/tex] into the integral, and evaluating from Vi to Vf, but this still leaves the gamma as an exponent. how can i go about solving this one?
thanks
 
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  • #2
thenewbosco said:
hello the problem is as stated:
a cylinder containing n moles of an ideal gas undergoes an adiabatic process. using [tex]W=-\int Pdv[/tex] and using the condition [tex]PV^\gamma=constant[/tex], show that the work done is:
[tex]W=(\frac{1}{\gamma - 1}(PfVf - PiVi)[/tex] where Pf is final pressure, Pi is initial pressure...
I tried substituting that [tex]P=\frac{constant}{V^\gamma}[/tex] into the integral, and evaluating from Vi to Vf, but this still leaves the gamma as an exponent. how can i go about solving this one?
thanks

Replace back [tex] V^{\gamma} [/tex] by constant/P.

ehild
 
  • #3


To solve this problem, you can use the first law of thermodynamics, which states that the change in internal energy of a system is equal to the heat added to the system minus the work done by the system. In an adiabatic process, there is no heat transfer, so the change in internal energy is equal to the work done by the system.

To start, you can write the first law of thermodynamics as:

dU = -PdV

Since the process is adiabatic, there is no heat transfer, so dU = dQ = 0. This means that:

0 = -PdV

Next, you can substitute the given condition PV^γ=constant into the equation, giving:

0 = -\frac{constant}{V^γ} dV

Integrating both sides from Vi to Vf, you can solve for the work done:

W = - \int_{Vi}^{Vf} \frac{constant}{V^γ} dV

Using integration by substitution, you can solve for the work done:

W = \frac{1}{γ - 1} (PfVf - PiVi)

This is the same result as the equation given in the problem. So, the work done in an adiabatic process can be calculated using this equation. Keep in mind that this equation applies to an ideal gas, so it may not be applicable to other systems. Also, make sure to double check your calculations and units to ensure accuracy.
 

1. What is thermodynamics?

Thermodynamics is a branch of physics that deals with the study of heat, energy, and their relationship to work. It explains how energy is transferred and transformed in different systems, and how this affects the behavior of matter.

2. What is the first law of thermodynamics?

The first law of thermodynamics, also known as the law of conservation of energy, states that energy cannot be created or destroyed, only transformed from one form to another. This means that the total amount of energy in a closed system remains constant.

3. What is the second law of thermodynamics?

The second law of thermodynamics states that in any thermodynamic process, the overall entropy (a measure of disorder) of a closed system will always increase. This law explains why heat always flows from hotter objects to colder objects, and why it is impossible to achieve a 100% efficient heat engine.

4. What is an example of a thermodynamics problem?

An example of a thermodynamics problem could be calculating the maximum efficiency of a heat engine, given the temperatures of the hot and cold reservoirs. This would involve applying the first and second laws of thermodynamics to determine the amount of work that can be produced from a given amount of heat.

5. How is thermodynamics used in real life?

Thermodynamics has many practical applications in everyday life. It is used in the design of engines, refrigerators, and air conditioners. It also plays a crucial role in understanding and predicting weather patterns, as well as in the study of chemical reactions and phase changes in materials.

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