# Adiabatic Process: C_v ≠ 0 Despite dQ = 0

• alejandrito29
In summary: If the force acting on a mass is zero, then the mass can be anything, including zero. But it doesn't have to be zero. Likewise, if the heat and work are both zero, the change in internal energy can be anything, including zero. But it doesn't have to be zero.

#### alejandrito29

alejandrito29 said:
The low is:

$$dQ=dU+p dV$$

but the specific heat to volumen (in a perfect gas) cte is:

$$C_v = \frac{dQ}{dT} = \frac{dU}{dT}$$

why if in adiabatic proces $$dQ=0$$, then $$C_v \neq 0$$ ??

Cv is a property of the gas.

From Newton's second law, m = F/a. Does the mass go to zero if there is no force acting on it?

AM

I think you're tricking yourself with some algebraic manipulations instead of thinking about the actual physics.

Here's what I think you're saying:
"We have Cv= dU/dT = (dQ-pdV)/dT|V = dQ/dT. Now if dQ=0 for an adiabatic process, then dQ/dT=0."

I think in this pseudo-algebra, the problem is that you can't possibly compute dQ/dT because dT is also zero if you insist there is no heat AND no work. If we have dU=dQ-pdV and both dQ and dV are zero, then dU is zero. Your equation basically changes into to 0=0 and it's not surprising you run into trouble when trying to take derivatives of this.

Let's look at it from another angle. The equation of state for the ideal gas is PV=NkT. Now in your example we have a closed system so N is fixed (dN = 0), and if we are using the specific heat at constant volume, then we are assuming dV=0. So we have V dP=Nk dT. Without doing any work (which would require nonzero dV), the only possible way to change the temperature is by increasing the system's pressure. The only way we could increase the system's pressure is by adding heat, but for an adiabatic process this is zero. so we have Nk dT = V dP = 0. So when you write dQ/dT = 0/dT = 0/0. So the pseudo-algebra has a divide by zero error.

Edit: Andrew Mason gave a nice analogy that is probably a better explanation than mine. But his rhetorical question "Does the mass go to zero if there is no force acting on it?" also can be interpreted in terms of the m=F/a=0/0 fallacy.

Last edited:

## What is an adiabatic process?

An adiabatic process is a thermodynamic process in which there is no transfer of heat or matter between a system and its surroundings. This means that the system is insulated and does not exchange energy with its surroundings.

## What is C_v?

C_v, also known as the heat capacity at constant volume, is a measure of the amount of heat required to raise the temperature of a substance by one degree while keeping its volume constant.

## Why is C_v not equal to 0 in an adiabatic process with dQ = 0?

C_v is not equal to 0 in an adiabatic process because it is a measure of the amount of heat required for a change in temperature at constant volume, not the amount of heat actually exchanged in the process. Even though there is no transfer of heat in an adiabatic process, the temperature can still change due to work being done on or by the system.

## What does dQ = 0 mean in an adiabatic process?

In an adiabatic process, dQ = 0 means that there is no transfer of heat between the system and its surroundings. This can occur when the system is well-insulated or when the process happens very quickly, giving no time for heat to be exchanged.

## Can an adiabatic process with dQ = 0 still have a change in temperature?

Yes, an adiabatic process with dQ = 0 can still have a change in temperature due to work being done on or by the system. This is because the change in temperature is determined by the amount of heat needed to change the temperature at a constant volume, not the actual exchange of heat during the process.