# Adiabatic equation in dependence on volume and pressure

• steroidjunkie
In summary, the adiabatic equation in dependence on volume and pressure is a thermodynamic equation derived from the first law of thermodynamics. It describes the relationship between volume and pressure during an adiabatic process and is used in various applications such as heat engine efficiency and weather forecasting. However, it has limitations in that it assumes a reversible process and ideal gas behavior, and does not account for changes in temperature.
steroidjunkie

## Homework Statement

Find: (a) Equation of state $$f (p, V, T)$$ and (b) Adiabatic equation in dependence on volume and pressure. Internal energy $$U(V, S)=\frac{1}{aV} ln(\frac{S}{\gamma})$$ where a and ##\gamma## are positive constants.

## Homework Equations

(a) ##dU=TdS-pdV \rightarrow##
##T=(\frac{dU}{dS})_V##
##p=-(\frac{dU}{dV})_{S}##

##T=(\frac{d}{dS})_V \cdot \frac{1}{aV} ln(\frac{S}{\gamma})=\frac{1}{aV} \cdot \frac{\gamma}{S} \cdot \frac{1}{\gamma}=\frac{1}{a \gamma V} ##
##p=-(\frac{d}{dV})_S \frac{1}{aV} ln(\frac{S}{\gamma})=- \frac{1}{a} \cdot (- \frac{1}{V^2}) \cdot ln(\frac{S}{\gamma})=\frac{1}{aV^2} ln(\frac{S}{\gamma}) ##

(b) ##pV^{\gamma}=NkT##

##\gamma=?##
##\gamma=\frac{C_p}{C_V}##
##C_p=(\frac{dU}{dT})_p##
##C_V=(\frac{dU}{dT})_V##

## The Attempt at a Solution

(b) ##C_p=(\frac{d}{dT})_p \frac{1}{aV} ln(\frac{S}{\gamma})##

##T=\frac{1}{a \gamma V} \rightarrow T \gamma=\frac{1}{a V}##
##C_p=(\frac{d}{dT})_p T \gamma ln(\frac{S}{\gamma})##
##p=\frac{1}{aV^2} ln(\frac{S}{\gamma}) \rightarrow paV^2=ln(\frac{S}{\gamma})##
##C_p=(\frac{d}{dT})_p T \gamma paV^2=\gamma paV^2##

##C_V=(\frac{d}{dT})_V \frac{1}{aV} ln(\frac{S}{\gamma})##
##C_V=(\frac{d}{dT})_V T \gamma ln(\frac{S}{\gamma})##
##C_V=(\frac{d}{dT})_V T \gamma paV^2=\gamma paV^2##

##\gamma=\frac{C_p}{C_V}= \frac{\gamma paV^2}{\gamma paV^2}=1##

I need help with (b) part of the problem. I know this is not correct and I assume I did something wrong while substituting, but I have no idea what. If you know something please post.
Thanks.

You need help on part a also. Your equation for T is incorrect. After you get that corrected, you will need to elimate S between the equations for S and T.

In part b, your equation for Cp is incorrect.

Chet

With regard to part b: how does S vary along an adiabatic reversible path?

Regarding a mistake with (a) ##T=\frac{1}{aV} \cdot \frac{\gamma}{S} \cdot \frac{1}{\gamma}=\frac{1}{aSV}##
Thank you for noticing.

(b) The right formula: ##C_V=(\frac{dH}{dT})_p##

##H=U+pV=U+U=2U##
##C_p=(\frac{d}{dT})_p (2U)=2 \cdot C_V=2SpaV^2##

##\gamma=\frac{C_p}{C_V}=\frac{2SpaV^2}{SpaV^2}=2##

For part a,

$$S=γe^{aV^2p}=\frac{1}{aVT}$$

This is the p-V-T relationship they were looking for.

For part b,
From your final equation for p, you could have immediately seen that pV2 is constant if S is constant along a reversible adiabat. I'm not so sure about the NkT part, however. I don't think that that is correct.

Chet

I see. I could have stated that: ##pV^2=const##
and then from the pressure equation: ## const=\frac{1}{a}ln(\frac{S}{\gamma})##
Then I substitute variables in constant in order to get a dependence on T:
##pV^2=\frac{1}{a}ln(\frac{S}{\gamma}) \rightarrow apV^2=ln(\frac{S}{\gamma}) \rightarrow e^{apV^{\gamma}}=\frac{S}{\gamma} ##
## \rightarrow \gamma e^{ap V^{\gamma}}=S ##

and: ##T=\frac{1}{aSV} \rightarrow S=\frac{1}{aTV} ##

which leads to: ##\gamma e^{apV^{\gamma}}= \frac{1}{aTV} \rightarrow e^{apV^{\gamma}}= \frac{1}{a \gamma TV} \rightarrow apV^\gamma ##
## = ln(\frac{1}{a \gamma TV}) \rightarrow pV^{\gamma}= \frac{1}{a}ln(\frac{1}{a \gamma TV}) ##

Solution: ## pV^2= \frac{1}{a}ln(\frac{1}{2aTV}) ##

Thank you one more time.

Last edited:

## 1. What is the adiabatic equation in dependence on volume and pressure?

The adiabatic equation in dependence on volume and pressure is a thermodynamic equation that describes the relationship between the volume and pressure of a gas during an adiabatic process, meaning a process that occurs without the exchange of heat with the surroundings.

## 2. How is the adiabatic equation derived?

The adiabatic equation is derived from 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. For an adiabatic process, there is no heat added or removed, so the first law simplifies to the equation dU = -pdV.

## 3. What is the difference between the adiabatic equation and the ideal gas law?

The ideal gas law, PV = nRT, describes the relationship between pressure, volume, temperature, and the number of moles of an ideal gas. The adiabatic equation, on the other hand, only describes the relationship between volume and pressure during an adiabatic process, and does not take into account the temperature or number of moles of the gas.

## 4. How is the adiabatic equation used in real-world applications?

The adiabatic equation is used in many applications, such as determining the efficiency of a heat engine or predicting the behavior of gases in various industrial processes. It is also used in weather forecasting to understand the changes in temperature and pressure in the atmosphere.

## 5. What are the limitations of the adiabatic equation?

The adiabatic equation assumes that the process is reversible and that the gas follows the ideal gas law. In reality, there may be energy losses due to friction or non-ideal behavior of the gas, which can affect the accuracy of the equation. It also does not take into account any changes in temperature, which may be significant in certain situations.

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