# Help with Ideal gas equation of state

• jaredogden
In summary, the conversation revolved around solving an air-standard Otto cycle example problem, where the temperature and pressure at the end of each process, the thermal efficiency, and the mean effective pressure needed to be determined. The solution involved finding values for u1 and vr1, using the air-standard relation of vr2=(V2/V1)(vr1), and using the ideal gas law to find p2. However, the person in the conversation was struggling to understand where this relation came from and how to use it, but eventually came to understand it with the help of others.
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 vr1 from a table and using the air-standard relation of vr2=(V2/V1)(vr1) 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.

$P_2 V_2 = m_2 RT_2$ 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:

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

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

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

## 1. What is the ideal gas equation of state?

The ideal gas equation of state, also known as the ideal gas law, is a mathematical relationship that describes the behavior of ideal gases. It relates the pressure, volume, temperature, and number of moles of a gas through the equation PV = nRT, where P is pressure, V is volume, n is the number of moles, R is the gas constant, and T is temperature.

## 2. How is the ideal gas equation of state derived?

The ideal gas equation of state is derived from the kinetic theory of gases, which states that gas particles are in constant motion and do not interact with each other. This theory is combined with the gas laws that describe the relationship between pressure, volume, and temperature, and the Avogadro's law which states that equal volumes of gases at the same temperature and pressure contain the same number of molecules.

## 3. What are the units of the ideal gas constant (R)?

The units of the ideal gas constant depend on the units used for pressure, volume, temperature, and the number of moles in the equation. In SI units, the ideal gas constant has a value of 8.314 J/mol·K, where J represents joules, mol represents moles, and K represents kelvin.

## 4. Can the ideal gas equation of state be used for real gases?

The ideal gas equation of state is an approximation that is most accurate for gases at low pressures and high temperatures. It does not take into account the intermolecular forces and volume of gas particles. For real gases, the Van der Waals equation of state is used, which includes correction factors for these factors.

## 5. How is the ideal gas equation of state used in practical applications?

The ideal gas equation of state is used in various practical applications, such as in the design of gas storage tanks and in the calculation of gas densities. It is also used in the ideal gas law calculator to solve for unknown variables, and in the gas laws experiments to demonstrate the relationship between pressure, volume, and temperature of gases.

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