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 $p_1 * v_1^n= p_2 * v_2 * [...]$?
