# When to use the adiabatic index in the polytropic process formulae?

• Master1022
In summary, the equation states that the final volume is the same as the initial volume if the pressure and temperature remain the same. However, if the pressure or temperature is changed, then the final volume will be different than the initial volume.
Master1022
Homework Statement
Air undergoes a polytropic process from 1.2 bar and 300 K to 4 bar and 500 K. Find the polytropic exponent, n (there is more that follows on from this, but I am not interested in those bits).
Relevant Equations
$p v^n = constant$
Hi,

I was doing this question and I was slightly confused as to whether I ought to just substitute $n = \gamma$ (the adiabatic constant) into the equation? The answers don't do this, but I was wondering why it was wrong for me to do so? This was only a small fraction of the question (which was worth very few marks), so I thought it would be an appropriate substitution to make given that we often assume air is a perfect gas.

I cannot really understand the reason not to use $n = \gamma$, apart from it yielding the wrong answer (correct answer is ~1.74 and I just let n = 1.4 from our textbook data table).

The answer scheme seems to suggest using: $$p_{1} v_1 ^ n = p_{2} v_2 ^ n$$
$$n = \frac{\log(\frac{p_2}{p_1})}{\log(\frac{v_1}{v_2})}$$

and we can calculate v1 and v2 from the conditions given.

Suppose you had m moles of gas. From the ideal gas law, at the initial pressure of 1.2 bars and initial temperature of 300 K, in terms of m, what is the initial volume (in m^3)? From the ideal gas law, at the final pressure of 4 bars and the final temperature of 500 K, in terms of m, what is the final volume (in m^3)?

Chestermiller said:
Suppose you had m moles of gas. From the ideal gas law, at the initial pressure of 1.2 bars and initial temperature of 300 K, in terms of m, what is the initial volume (in m^3)?

Is it: $V_{i} = \frac{m * R_0 * (300)}{1.2 * 10^5}$

Chestermiller said:
From the ideal gas law, at the final pressure of 4 bars and the final temperature of 500 K, in terms of m, what is the final volume (in m^3)?

Is it: $V_{f} = \frac{m * R_0 * (500)}{4* 10^5}$

OK. Now, if you substitute this into the equation ##P_iV_i^n=P_fV_f^n## and simplify, what do you get?

Chestermiller said:
OK. Now, if you substitute this into the equation ##P_iV_i^n=P_fV_f^n## and simplify, what do you get?
Thank you for your response. I understood how they arrived at the calculated n value. I was wondering why it wasn't the case that $n = \gamma$ here (beyond calculation purposes)? Is there any other indication that would suggest that we cannot make that assumption here?

Master1022 said:
Thank you for your response. I understood how they arrived at the calculated n value. I was wondering why it wasn't the case that $n = \gamma$ here (beyond calculation purposes)? Is there any other indication that would suggest that we cannot make that assumption here?
They didn't say that the process is adiabatic and reversible, which would then be consistent with ##n=\gamma## and the specific temperature change produced by an adiabatic reversible process. It this case, the temperature change was not one consistent with an adiabatic reversible process.

Chestermiller said:
They didn't say that the process is adiabatic and reversible, which would then be consistent with ##n=\gamma## and the specific temperature change produced by an adiabatic reversible process. It this case, the temperature change was not one consistent with an adiabatic reversible process.
This makes sense. Perfect.

Thank you very much.

## 1. What is the adiabatic index in the polytropic process formulae?

The adiabatic index, also known as the ratio of specific heats, is a thermodynamic parameter that describes the relationship between the pressure and volume of a gas during a reversible and adiabatic process. It is denoted by the symbol gamma (γ) and is commonly used in polytropic process formulae.

## 2. When should I use the adiabatic index in the polytropic process formulae?

The adiabatic index should be used when describing a process where there is no heat transfer between the system and its surroundings. This means that the process is adiabatic, and the change in internal energy is solely due to work done on or by the system.

## 3. How is the adiabatic index related to the specific heat of a gas?

The adiabatic index is related to the specific heat of a gas through the following equation: γ = Cp/Cv, where Cp is the specific heat at constant pressure and Cv is the specific heat at constant volume. This ratio is a constant for a particular gas and is dependent on the molecular structure of the gas.

## 4. What is the significance of the adiabatic index in the polytropic process formulae?

The adiabatic index is significant in the polytropic process formulae as it helps to describe the relationship between pressure and volume during an adiabatic process. It is also used in various thermodynamic calculations and is a crucial parameter in the study of thermodynamics and fluid mechanics.

## 5. Can the adiabatic index be different for different gases?

Yes, the adiabatic index can vary for different gases. This is because the value of the adiabatic index is dependent on the molecular structure and properties of the gas. For example, monatomic gases have a higher adiabatic index compared to diatomic gases due to the difference in their molecular structures.

Replies
3
Views
548
Replies
2
Views
2K
Replies
9
Views
865
Replies
6
Views
1K
Replies
4
Views
1K
Replies
1
Views
966
Replies
47
Views
4K
Replies
9
Views
355
Replies
9
Views
2K
Replies
11
Views
533