Question

At an initial pressure of 100 kPa, 1 kg air is compressed reversibly from 15 litres to 1 litre at final pressure of 2000 kPa. Neglecting other losses, the Polytropic Exponent (n) is ________. (Rounded off to three decimal places)

Hint:

Use P1 V1^n equals P2 V2^n and solve for n from the pressure and volume ratios.

Answer 1

Correct Answer: 1.106

Step 1: Set up the polytropic law in log form.
$P_1 V_1^n = P_2 V_2^n$ rearranges to $(V_1/V_2)^n = P_2/P_1$.
Taking $\log_{10}$ of both sides gives $n \log_{10}(V_1/V_2) = \log_{10}(P_2/P_1)$.

Step 2: Plug in numbers using base-10 logs.
$V_1/V_2 = 15/1 = 15$ and $P_2/P_1 = 2000/100 = 20$.
$\log_{10}(15) = 1.1761$ and $\log_{10}(20) = 1.3010$.
$n = \dfrac{1.3010}{1.1761}$

Step 3: Divide to get n.
$n \approx 1.106$
This matches the natural-log route exactly, since the log base cancels out in the ratio.

Final Answer:
The polytropic exponent works out to 1.106, comfortably inside the 1.100 to 1.111 window. \[ \boxed{n \approx 1.106} \]