Why does an ideal gas satisfy ##(\partial U/\partial P)_T=0##?

  • #1
zenterix
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Why does an ideal gas satisfy ##\left (\frac{\partial U}{\partial P}\right )_T = 0##?
The book I am reading says that by definition, the ideal gas satisfies the equations

$$PV=nRT\tag{1}$$

$$\left (\frac{\partial U}{\partial P}\right )_T = 0\tag{2}$$

where does (2) come from? In other words, what justifies this equation in the definition above?
 
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  • #2
The internal energy of an ideal (monoatomic) gas is ##3RnT/2##. Differentiating with respect to ##P## with ##T## constant is clearly zero.
 
  • #3
Orodruin said:
The internal energy of an ideal (monoatomic) gas is ##3RnT/2##. Differentiating with respect to ##P## with ##T## constant is clearly zero.
The thing is, I believe that equation comes from the kinetic theory of the ideal gas right.

The chapter of the book that I am on is a few chapters before talking about that theory. The only reason I know about that equation is from looking ahead.

I am wondering about some other justification not based on that theory.
 

1. Why does an ideal gas satisfy ##(\partial U/\partial P)_T=0##?

An ideal gas satisfies ##(\partial U/\partial P)_T=0## because it does not exhibit any intermolecular forces or interactions between its particles. In an ideal gas, the internal energy only depends on the temperature, not on the pressure. Therefore, the change in internal energy with respect to pressure at constant temperature is zero.

2. How does the lack of intermolecular forces in an ideal gas lead to ##(\partial U/\partial P)_T=0##?

The lack of intermolecular forces in an ideal gas means that the internal energy of the gas is solely determined by the kinetic energy of its particles, which is directly related to the temperature. Since there are no interactions between particles affecting the internal energy, the change in internal energy with respect to pressure at constant temperature is zero.

3. Can a real gas satisfy ##(\partial U/\partial P)_T=0## under certain conditions?

Under certain conditions, a real gas can approximate the behavior of an ideal gas and satisfy ##(\partial U/\partial P)_T=0##. This typically occurs at low pressures and high temperatures, where the effects of intermolecular forces become negligible compared to the kinetic energy of the gas molecules.

4. How does the equation ##(\partial U/\partial P)_T=0## relate to the first law of thermodynamics?

The equation ##(\partial U/\partial P)_T=0## is a consequence of 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 the case of an ideal gas, where the internal energy is independent of pressure at constant temperature, the work done by the system is solely due to changes in volume, not pressure.

5. What are the implications of ##(\partial U/\partial P)_T=0## for the behavior of an ideal gas in different processes?

The fact that ##(\partial U/\partial P)_T=0## for an ideal gas means that its internal energy remains constant with respect to pressure at constant temperature. This has implications for various processes involving ideal gases, such as isothermal processes where the temperature is held constant. In such processes, the internal energy of the ideal gas does not change with pressure variations, leading to predictable behavior in terms of work and heat exchange.

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