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

[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.

Thanks in advance

 

[Chestermiller]

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)?

 

[Master1022]

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

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

 

[Chestermiller]

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

 

[Master1022]

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?

 

[Chestermiller]

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.

 

[Master1022]

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.