# How to derive pv^gamma=constant

• milan talaviya
In summary, pressure and volume of an ideal gas are expressed as pV^gamma=constant in a reversible adiabatic condition where gamma=Cp,m/Cv,m.
milan talaviya

## Homework Statement

The relationship between pressure and volume of an ideal gas is expressed as pv=constant in a reversable isothermal condition. Show that the relationship between pressure and volume of the same gas is expressed as pV^gamma=constant in a reversible adiabatic condition where gamma=Cp,m/Cv,m.

## Homework Equations

gamma=Cp,m/Cv,m.
pv=constant, rev. ideal

## The Attempt at a Solution

p1v1^(Cpm/Cvm)=p2v2^(Cpm/Cvm) take ln and mult both sides by the Cvm/Cpm
ln p1v1 = ln p2v2 e to both sides
p1v1=p2v2
p1v1/p2v2 = 1

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please answer it if you have better than these.
your's thankfully milan talaviya

Hello Milan, welcome to PF.

You have to show that pVγ= constant assuming a reversible adiabatic process on an ideal gas.

What is an adiabatic process? How does the internal energy change in an adiabatic process? What are Cp and Cv?

ehild

milan talaviya said:

## The Attempt at a Solution

p1v1^(Cpm/Cvm)=p2v2^(Cpm/Cvm) take ln and mult both sides by the Cvm/Cpm
ln p1v1 = ln p2v2 e to both sides
p1v1=p2v2
p1v1/p2v2 = 1
You haven't proved anything here. For one, you've made an algebra mistake. The more serious error is that you started with what you're supposed to be proving. That sort of argument isn't valid logically.

milan talaviya said:

## Homework Statement

The relationship between pressure and volume of an ideal gas is expressed as pv=constant in a reversable isothermal condition. Show that the relationship between pressure and volume of the same gas is expressed as pV^gamma=constant in a reversible adiabatic condition where gamma=Cp,m/Cv,m.

## Homework Equations

gamma=Cp,m/Cv,m.
pv=constant, rev. ideal

## The Attempt at a Solution

p1v1^(Cpm/Cvm)=p2v2^(Cpm/Cvm) take ln and mult both sides by the Cvm/Cpm
ln p1v1 = ln p2v2 e to both sides
p1v1=p2v2
p1v1/p2v2 = 1
The adiabatic condition that you have to prove is not:

$(PV)^{\frac{C_p}{C_v}} = \text{constant}$

Rather, it is :

$$PV^{\frac{C_p}{C_v}} = \text{constant}$$

which means:

$$\frac{P_1}{P_2} = (\frac{V_2}{V_1})^{\frac{C_p}{C_v}}$$

1. Start with the first law and find an expression for dU in terms of PdV (hint: what is dQ if it is adiabatic?).

2. Then express dU in terms of dT and substitute your answer in 1. for dU.

3. Finally express dT in terms of d(PV). (hint: use R = Cp-Cv).

AM

## 1. How is the equation pv^gamma=constant derived?

The equation pv^gamma=constant is derived through the use of the ideal gas law, which states that the product of pressure (p) and volume (v) of an ideal gas at a constant temperature is equal to a constant value. The exponent gamma (γ) is also known as the adiabatic index, which is a measure of a gas's ability to expand or compress without exchanging heat. When solving for this constant value, we use the principles of thermodynamics and equations of state to arrive at the final equation.

## 2. What is the significance of the equation pv^gamma=constant?

The equation pv^gamma=constant is significant because it describes the relationship between pressure and volume for an ideal gas. It is used in various fields of science, such as physics and chemistry, to calculate the behavior of gases under different conditions. It also helps to understand the changes in pressure and volume that occur during adiabatic processes, where no heat is exchanged.

## 3. Can the equation pv^gamma=constant be applied to real gases?

While the equation pv^gamma=constant is derived for ideal gases, it can still be applied to real gases under certain conditions. Real gases behave similarly to ideal gases at low pressures and high temperatures, where the particles are far apart and have high kinetic energies. However, at high pressures and low temperatures, real gases deviate from ideal gas behavior due to intermolecular forces between particles.

## 4. How does the value of gamma affect the behavior of an ideal gas?

The value of gamma (γ) has a significant impact on the behavior of an ideal gas. It is a measure of the gas's ability to expand or compress without exchanging heat. A higher value of gamma indicates that the gas is less compressible and has a steeper slope on a pressure-volume graph. This means that the gas will experience larger changes in pressure for a given change in volume and vice versa.

## 5. What are some practical applications of the equation pv^gamma=constant?

The equation pv^gamma=constant has various practical applications in different fields of science and engineering. It is used in the design of internal combustion engines, gas turbines, and other heat engines to predict the behavior of gases during the compression and expansion processes. It is also used in the study of atmospheric science to understand the behavior of air masses and the adiabatic processes that occur in the Earth's atmosphere. Additionally, it is used in the analysis of sound waves and in the design of acoustic systems.

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