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in this exercise, I have hydrogen and it follows this thermodynamics cycle,

1-2 } --> polytropic process with exponent $n = 1.1$,

2-3 } --> adiabatic process,

3-1 } --> isobaric process,

my question is only about polytropic process, $$p2 * v_2^n = p1 * v_1^n$$

I know that hydrogen in 2nd phase has pressure $p2 = 7 * 10^5 Pa$, and $v_2 = 25\space kg/m^3$ [yes, I've used specific volume]

now I can rewrite the general equation this way, $$ p1 * v1^n = C $$ where $C = p2 * v2^n$ [I've calculate C]. now, I have to find both p1 and v1, knowing that $T_1 = 238.15 K$, and $T_2 = 253.15 K$, meaning I know temperatures in the 1st phase and in the 2nd phase.

my question is: why do I have to use this equation? $$v_1 = v_2 * (\frac{T_2}{T_1})*exp{\frac{1}{1.1-1}}$$

[note: I've used exp{ [...]}, but it's not equal to $e^x$, I've just raised to a power this way because it's easier to read] why is volume (or specific volume) related to temperature in a polytropic process with $n = 1.1$?

where does this fraction $\frac{T_2}{T_1}$ come from?

another question, why this equation ignores the pressures? $$v_1 = v_2 * (\frac{T_2}{T_1})*exp{\frac{1}{1.1-1}}$$

shouldn't it be something like $p_1 * v_1^n= p_2 * v_2 * [...]$?

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  • $\begingroup$ Have you tried substituting the ideal gas law into the starting equation?? $\endgroup$
    – Chet Miller
    Mar 23 at 10:55
  • $\begingroup$ @ChetMiller if i substitute p with mRT/v both sides, i've got: $mRT = mRT$ $\endgroup$
    – Gabriel Burzacchini
    Mar 23 at 11:00

1 Answer 1

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$$PV^n=C$$$$\left(\frac{mRT}{V}\right)V^n=C$$$$TV^{n-1}=\frac{C}{mR}=C'$$

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