# In a polytropic process, how can I prove the validity of this equation: $v_1 = v_2 (\frac{T_2}{T_1})\exp{\frac{1}{1.1-1}}$?

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

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

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

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

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