How Does the Van der Waals Equation Define Adiabatic Processes for Real Gases?

In summary: I hope that helps!In summary, an equation for an adiabatic process of a gas obeying the van der Waals equation is derived using the equations for entropy and the gas's equation of state. The assumption is made that the heat capacity is nearly constant over the range of temperatures for the process. This is justified by the fact that for a van der Waals gas, the heat capacity is a function of temperature only and is identical to the ideal gas heat capacity. The resulting equation is given by ##T(v-b)^{R/c_v}=constant##, where ##c_v## is the heat capacity and ##R## is the ideal gas constant.
  • #1
arpon
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Homework Statement


Show that for a gas obeying the van der Waals equation ##\left(P+\frac{a}{v^2}\right)(v-b)=RT##, with ##c_v## a function of ##T## only, an equation for an adiabatic process is $$T(v-b)^{R/c_v}=constant$$

Homework Equations


##TdS=c_vdT+T\left(\frac{\partial P}{\partial T}\right)_v dv##

The Attempt at a Solution


For reversible adiabatic process, ##dS=0##.
So, from the third ##TdS## equation,
$$c_vdT+T\left(\frac{\partial P}{\partial T}\right)_v dv=0$$
$$c_vdT+T\left(\frac{R}{v-b}\right)dv=0~~~$$ [Using equation of state]
$$c_v\frac{dT}{T}=-\frac{RdV}{v-b}$$
If ##c_v## is a constant, integrating both sides, we have:
$$T(v-b)^{R/c_v}=constant$$
But, in this case, ##c_v## is a function of ##T##.
Any help would be appreciated.
 
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  • #2
I guess they must be assuming that, over the range of temperatures for the process, the heat capacity is nearly constant. We certainly often do this for an ideal gas, and, since the heat capacity for a van der waals gas is a function only of temperature, it must be identical to the ideal gas heat capacity.
 
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  • #3
Chestermiller said:
since the heat capacity for a van der waals gas is a function only of temperature, it must be identical to the ideal gas heat capacity
Could you please explain this part? The heat capacity for ideal gas is a constant while for van der waals gas, it is a function of temperature. What did you actually mean by they are identical?
 
  • #4
arpon said:
Could you please explain this part? The heat capacity for ideal gas is a constant while for van der waals gas, it is a function of temperature. What did you actually mean by they are identical?
The heat capacity of a real gas, in the limit of very low pressures, is a function of temperature. We engineers take this into account in our definition of an ideal gas (by regarding an ideal gas as having a temperature-dependent heat capacity), but physicists have idealized it further (by regarding an ideal gas as having a constant heat capacity). Now, if the heat capacity of a van der Waals gas is a function of temperature only (and not volume and pressure), it must be the same temperature-dependent function as we engineers refer to for an ideal gas.
 
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Related to How Does the Van der Waals Equation Define Adiabatic Processes for Real Gases?

1. What is an adiabatic process?

An adiabatic process is a thermodynamic process in which there is no exchange of heat between the system and its surroundings. This means that the temperature of the system remains constant throughout the process.

2. How does the adiabatic process of real gas differ from ideal gas?

In ideal gas, the particles do not interact with each other and follow the ideal gas law. However, in real gas, there is some interaction between the particles, and the gas may deviate from the ideal gas law. This results in a slight difference in the behavior of real gas in an adiabatic process compared to ideal gas.

3. What is the equation for the adiabatic process of real gas?

The equation for the adiabatic process of real gas is PVγ = constant, where P is the pressure, V is the volume, and γ is the specific heat ratio of the gas.

4. What is the significance of the adiabatic process in real-life applications?

The adiabatic process is commonly seen in real-life applications, such as in the compression and expansion of gases in engines. It is also used in the study of weather patterns and atmospheric processes.

5. What is the role of specific heat ratio (γ) in the adiabatic process of real gas?

The specific heat ratio (γ) is a measure of the heat capacity of a gas at constant pressure and volume. It determines the rate at which the temperature of the gas changes during an adiabatic process. A higher specific heat ratio results in a slower temperature change, while a lower ratio leads to a faster temperature change.

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