On the notion of pressure in the canonical ensemble

In summary, The pressure in a canonical ensemble is calculated by taking the derivative of the internal energy with respect to the volume, even though the volume is fixed. This is done by considering infinitesimal changes in volume and energy, resulting in the ratio being equal to the pressure.
  • #1
Derivator
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Hi folks,

since the volume V is fixed in a canonical ensemble I'm a bit confused about the fact, that the pressure is calculated as the derivation of the internal energy U with respect to the volume V.

Sure, P = dU/dV comes from dU = dQ + dW = tdS - pdV + ... But what does it mean to derivate with respect to the volume, when the volume of the system can't be changed, since it is fixed. (by the way: when the volume is fixed, pdV= p*0 =0, so p is "undefined", since it can take any value.)

--
derivator
 
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  • #2
The volume is not free to change by any significant amount, but we are considering the case where it is changed by an infinitesimal amount. In response, the system energy changes by an infinitesimal amount. The ratio is the pressure.
 

1. What is the canonical ensemble?

The canonical ensemble is a statistical mechanical model used to describe the behavior of a system in thermal equilibrium at a fixed temperature, volume, and number of particles. It is often used to study systems such as gases, liquids, and solids.

2. How is pressure related to the canonical ensemble?

In the canonical ensemble, pressure is a measure of the force exerted by the particles of the system on its boundaries. It is related to the average kinetic energy of the particles and the volume of the system. As the temperature or number of particles changes, the pressure of the system may also change.

3. What is the significance of pressure in the canonical ensemble?

Pressure plays a crucial role in the canonical ensemble as it affects the thermodynamic properties of the system, such as its energy and entropy. It is also used to determine the equilibrium state of the system and can provide information about the interactions between particles.

4. How is pressure calculated in the canonical ensemble?

Pressure can be calculated in the canonical ensemble using the ideal gas law, which relates pressure to the number of particles, temperature, and volume of the system. It can also be calculated using the virial expansion, which takes into account the interactions between particles.

5. What are the assumptions made in the canonical ensemble regarding pressure?

The canonical ensemble assumes that the system is in thermal equilibrium and that the particles are non-interacting. It also assumes that the volume and number of particles are constant, and that the system can be described by a probability distribution function.

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