Proving the Ideal Gas Equation of State

In summary, to show the equation of state of an ideal gas with N molecules in a container of volume V at temperature T is P=NkT, we use the result A=-NkTlnZ and take the derivative with respect to V while keeping T constant. By using the ideal gas law and expressing Z in terms of V, we can derive the desired result.
  • #1
yellowstone33
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Homework Statement



Show that the equation of state of an ideal gas with N molecules in a container of volume V at temperature T is P=NkT. Use the result derived earlier.

The result is A=-NkTlnZ, where Z is the partition function and A is the Helmholtz free energy.

Homework Equations



The Attempt at a Solution



P=-([tex]\delta[/tex]A)/([tex]\delta[/tex]V)T

P=NkT([tex]\delta[/tex]lnZ)/([tex]\delta[/tex]V)T

Here, I'm stuck. Surely, ([tex]\delta[/tex]lnZ)/([tex]\delta[/tex]V)T=0?

It seems so trivial. I don't understand what I'm missing. Thank you in advance.
 
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  • #2


Hello,

You are on the right track! The key here is to use the fact that the partition function Z is related to the volume V through the ideal gas law, which states that PV = NkT. So, we can express Z as Z = (N/V)^N. Now, we can take the derivative of lnZ with respect to V, keeping T constant:

(dlnZ)/dV = (d/dV)(ln((N/V)^N)) = (d/dV)(Nln(N) - Nln(V)) = -N/V = -NkT/V

Substituting this into your equation, we get:

P = NkT(dlnZ)/dV = NkT(-NkT/V) = NkT^2/V

But we know that PV = NkT, so we can substitute this in and get:

P = NkT^2/V = (PV)/V = NkT

Which is the desired result! I hope this helps. Keep up the good work!
 

1. What is the Ideal Gas Equation of State?

The Ideal Gas Equation of State, also known as the Ideal Gas Law, is a mathematical relationship between the pressure, volume, temperature, and amount of gas in a system. It is represented by the equation PV = nRT, where P is the pressure, V is the volume, n is the number of moles of gas, R is the ideal gas constant, and T is the temperature.

2. How can the Ideal Gas Equation of State be proven?

The Ideal Gas Equation of State can be proven through various experimental methods, such as measuring the pressure and volume of a gas at different temperatures and using the data to calculate the value of the gas constant, R. It can also be derived from the kinetic theory of gases, which explains the behavior of gases in terms of the motion and collisions of individual particles.

3. What is the significance of the Ideal Gas Equation of State?

The Ideal Gas Equation of State is a fundamental law in chemistry and physics, as it describes the behavior of ideal gases and provides a basis for understanding the properties and interactions of real gases. It is also used in various practical applications, such as in the design of gas storage tanks and in the analysis of gas mixtures in industrial processes.

4. What are the assumptions of the Ideal Gas Equation of State?

The Ideal Gas Equation of State makes several assumptions about the behavior of gases, including that the gas particles have negligible volume and do not interact with each other, and that the gas behaves in accordance with the kinetic theory of gases. It also assumes that the gas is at a constant temperature and that the gas constant, R, is constant regardless of the type of gas being studied.

5. Are there any limitations to the Ideal Gas Equation of State?

While the Ideal Gas Equation of State is a useful tool in many situations, it is not always accurate in describing the behavior of real gases. It does not take into account intermolecular forces, which can affect the volume and pressure of a gas, and it does not apply at very high pressures or low temperatures. Additionally, it only applies to ideal gases, which do not exist in the real world.

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