Help with Ideal gas equation of state

[jaredogden]

I am reading through my thermodynamics book. Going over a air-standard otto cycle example problem. For reference the example problem is as follows:

The temperature at the beginning of the compression process of an air-standard Otto cycle with a compression
ratio of 8 is 5408R, the pressure is 1 atm, and the cylinder volume is 0.02 ft3. The maximum temperature during
the cycle is 36008R. Determine (a) the temperature and pressure at the end of each process of the cycle, (b) the
thermal efficiency, and (c) the mean effective pressure, in atm.


After finding the values for u₁ and v_r1 from a table and using the air-standard relation of v_r2=(V₂/V₁)(v_r1) which I followed. Then using that to interpolate a value for T₂ and u₂, the solution then states that since process 2-3 occurs at constant volume we can find p₂ by the following equation:

p₂ = p₁((T₂/T₁)(V₁/V₂))

I'm trying to understand where this is coming from because I don't see this on any ideal gas tables. I'm sure I'm missing some simple relation but if anyone can help explain where this relation has come from and why we can use it I would appreciate it.

 

[jack action]

$P_2 V_2 = m_2 RT_2$ and $P_1 V_1 = m_1 RT_1$

We know that:

$m_2 = m_1$

So:

$\frac{P_2 V_2}{T_2} = m_2 R$ and $\frac{P_1 V_1}{T_1} = m_1 R$

Or:

$\frac{P_2 V_2}{T_2} = \frac{P_1 V_1}{T_1}$

$P_2 = P_1 \frac{ T_2}{T_1}\frac{V_1}{V_2}$

 

[jaredogden]

Wow that easy huh?.. I feel stupid now. I guess I didn't fully understand how to use the ideal gas law all these years. Thanks a ton