Equations used in an ideal gas system are listed here, but they might be not complete.

equation of state

Ideal gas obeys equation of state

$$\tag{1} PV = nRT, $$

known as the ideal gas law ¹, with $P$ is pressure, $V$ is volume, $n$ is number of moles of gas, $R$ is universal gas constant, and $T$ is temperature.

isobaric process

An isobaric process is a thermodynamic process in which the pressure remains constant ², where the state variables are

$$\tag{2a} P_j = P_i, $$

$$\tag{2b} V_j \ne V_i, $$

$$\tag{2c} T_j \ne T_i, $$

in this process from state $i$ to state $j$ and

$$\tag{2d} \frac{V_j}{T_j} = \frac{V_i}{T_i} $$

is relation between the two states.

isochoric process

An isochoric process is a thermodynamic process during which the volume of the closed system undergoing such a process remains constant, that is exemplified by the heating or the cooling of the contents of a sealed, inelastic, undeformable container ³. There state variables are

$$\tag{3a} P_j \ne P_i, $$

$$\tag{3b} V_j = V_i, $$

$$\tag{3c} T_j \ne T_i, $$

in this process from state $i$ to state $j$ and

$$\tag{3d} \frac{P_j}{T_j} = \frac{P_i}{T_i} $$

is relation between the two states.

isothermal process

An isothermal process is A thermodynamic process that occurs at constant temperature ⁴, where the state variables are

$$\tag{4a} P_j \ne P_i, $$

$$\tag{4b} V_j \ne V_i, $$

$$\tag{4c} T_j = T_i, $$

in this process from state $i$ to state $j$ and

$$\tag{4d} P_j V_j = P_i V_i $$

is relation between the two states.

adiabatic process

An adiabatic process is one in which no heat is gained or lost by the system ⁵, where the state variables

$$\tag{5a} P_j \ne P_i, $$

$$\tag{5b} V_j \ne V_i, $$

$$\tag{5c} T_j \ne T_i, $$

in this process from state $i$ to state $j$ and

$$\tag{5d} P_j V_j^\gamma = P_i V_i^\gamma $$

is relation between the two states.

specific heat ratio

It is an additional variable in ideal gas system ⁶

$$\tag{6} \gamma = \frac{C_P}{C_V}, $$

where at different temperature different gas has different value⁷. The $C_P$ and $C_V$ are molar heat capacity at constant pressure and volume, respetively.

work

From state $i$ to state $j$ work done by is defined as ⁸

$$\tag{7a} W_{i \rightarrow j} = \int_{V_i}^{V_j} p dV. $$

For isobaric process Eqn (7a) simply turns into

$$\tag{7b} W_{i \rightarrow j}^{\rm isobaric} = p \Delta V = p (V_j - V_i). $$

For isochoric process it becomes

$$\tag{7c} W_{i \rightarrow j}^{\rm isochoric} = 0, $$

since $V_j = V_i$. Then, for isothermal process $P = \frac{NRT}{V}$ that makes Eqn (7a)

$$\tag{7d} W_{i \rightarrow j}^{\rm isothermal} = \int_{V_i}^{V_j} \frac{nRT}{V} dV = nRT \ln \left( \frac{V_j}{V_i} \right). $$

Finally, for adiabatic process

$$\tag{7e} W_{i \rightarrow j} = \int_{V_i}^{V_j} \frac{c}{V^\gamma} dV = \frac{c}{1-\gamma} (V_j^{1-\gamma} - V_i^{1-\gamma}). $$

using $PV^\gamma = c$, where $c = nRTV^{\gamma - 1}$ is a constant. With help of Eqn (6) it can obtained that

$$\tag{7f} 1 - \gamma = 1 - \frac{C_P}{C_V} = \frac{C_V - C_P}{C_V} = - \frac{R}{C_V}. $$

Substitute back the result to Eqn (7e) will produce

$$\tag{7g} W_{i \rightarrow j} = \int_{V_i}^{V_j} \frac{c}{V^\gamma} dV = -\frac{c C_V}{R} (V_j^{1-\gamma} - V_i^{1-\gamma}). $$

Then $c = nRT_i V_i^{\gamma - 1}$ at state $i$ and $c = nRT_j V_j^{\gamma - 1}$ at state $j$. Substitute both to Eqn (7g) will give

$$\tag{7h} W_{i \rightarrow j}^{\rm adiabatic} = -( n C_V T_j - n C_V T_i) = - n C_V (T_j - T_i) = - n C_V \Delta T. $$

heat

For process from state $i$ to state $j$, heat can be obtained from

$$\tag{8a} Q = \int n C(T) dT, $$

where

$$\tag{8b} C(T) = C_P, $$

$$ Q = n C_P \Delta T $$

for isobaric process,

$$\tag{8c} C(T) = C_V, $$

$$ Q = n C_V \Delta T $$

for isochoric process,

$$\tag{8d} C(T) = 0, $$

$$ Q = 0 $$

for adiabatic process,

$$\tag{8e} C(T) = C_{\rm eff} = \frac{1}{T} \frac{T}{V} \frac{dV}{dT}, $$

$$ Q = n R T \ln \frac{V_j}{V_i} $$

for isothermal process, and

$$\tag{8f} C(T) = C_{\rm process} = \frac{dQ}{ndT} $$

for arbitrary process ⁹.

change in internal energy

For any process, the change of internal energy is simply

$$\tag{9} \Delta U = n C_V \Delta T, $$

where $\Delta T = T_j - T_i$ with initial state $i$ and final state $j$.

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