Thermodynamic Processes

Thermodynamic Processes

• For any point during the process $\frac{P}{T}=Constant$ $dW=Pdv=0\to W=0$ $\Delta U=\frac{3}{2}nR\Delta T=\frac{3}{2}Nk_B\Delta T$ for monatomic gases $\Delta U=Q-W\to\Delta U=Q=nC_v\Delta T$ where $C_v$ is molar specific heat at constant volume
• $C_v$ is equal to $\frac{3}{2} R$, $\frac{5}{2} R$ and $\frac{7}{2} R$ for monatomic, diatomic and multi-atomic gases respectively.

• For any point during the process $\frac{V}{T}=Constant$ $W=\int_{ }^{ }dW=\int_{ }^{ }PdV=P\Delta V=nR\Delta T$ $Q=nC_p \Delta T$ where $C_p$ is molar specific heat at Constant Pressure and is equal to $C_v+R$
• The change in the internal energy is always equal to $nC_v \Delta T$ independent of the type of the process.

• For any point during this process $PV=Constant$ $\begin{aligned} & \Delta U=nC_v \Delta T=0\to Q=W\\ & W=\int pdV=\int \dfrac{nRT}{V} dV=nRT\int_{V_i}^{V_f} \dfrac{dV}{V}=nRT \ln⁡\Big(\dfrac{V_f}{V_i}\Big) \end{aligned}$
• If compression or expansion happens very slowly, the gas can exchange heat with the environment to keep its temperature constant.

• $Q=0\to\Delta U=-W=-nC_V \Delta T$ For any point during an adiabatic process $PV^\gamma=Constant$ In addition, for any point $TV^{\gamma-1}=Constant'$ where $\gamma=\frac{C_p}{C_v}$ ($\gamma=\frac{5}{3}$ for monatomic gases )
• If compression or expansion happens quickly, the gas does not have time to transfer heat to the environment so, $Q=0$ and the process is adiabatic.

Key Terms

System Vs Surroundings

• open system → cup of coffee
• closed system → bottle of pop
• isolated system → perfect thermos