(a) Derive the following formula for a quasi-static adiabatic process for the ideal gas, assuming \(\gamma\) to be constant: $$ \frac{T}{P(\gamma-1) / \gamma}=\text { const. } $$ (b) Helium \(\left(\gamma=\frac{5}{3}\right)\) at \(300 \mathrm{~K}\) and \(1 \mathrm{~atm}\) pressure is compressed quasi-statically and adiabatically to a pressure of 5 atm. Assuming that the helium behaves like the ideal gas, calculate the final temperature.

The Ideal Gas Law is a fundamental principle used to relate the pressure, volume, and temperature of an ideal gas. It is expressed as \[ PV = nRT \] where

• P is the pressure
• V is the volume
• n is the number of moles
• R is the universal gas constant
• T is the temperature.

This formula is used as a foundational part of deriving other relationships in thermodynamics. It is crucial to understand that this law assumes ideal gas behavior where interactions between molecules are negligible, and the gas molecules occupy no volume themselves. For an adiabatic process, we often combine this law with other equations.

The First Law of Thermodynamics states that energy cannot be created or destroyed, only transformed. Formally, it is written as \[ \text{ΔU} = Q - W \] where

• ΔU is the change in internal energy
• Q is the heat added to the system
• W is the work done by the system.

For an adiabatic process, no heat is exchanged (Q = 0), simplifying the equation to \[ \text{ΔU} = -W \] This means that any work done by or on the gas directly translates to a change in its internal energy. This simplification is essential when delving into adiabatic processes in ideal gases.

An adiabatic process in thermodynamics is one in which no heat is transferred to or from the system. For this specific process in an ideal gas, we use the following key equation: \[ PV^{\theta} = \text{const} \] where

• P is the pressure
• V is the volume
• θ is a constant that depends on the specific heat capacities of the gas

In our case of ideal gases, the exponent θ can be expressed as \[ \theta = \frac{\text{\textgamma} - 1}{\text{\textgamma}} \] where γ is the heat capacity ratio (γ = C_p / C_v), the ratio of the specific heat at constant pressure to the specific heat at constant volume. This relationship allows us to derive further equations that connect temperature and pressure without involving volume directly, providing a more straightforward analysis for adiabatic processes.

One of the significant results derived from combining the Ideal Gas Law and the adiabatic process equation is the relationship between temperature and pressure. For an adiabatic process, the derived formula \[ \frac{T}{P^{(\text{\textgamma} - 1)/ \text{\textgamma}}} = \text{const} \] shows that the ratio of the temperature to the pressure raised to the adiabatic exponent remains constant. This formula helps in solving practical problems like the one provided in the initial exercise, a critical insight for processes where heat exchange is not involved. When applying this formula, we can calculate interchanging states of pressure and temperature, as shown in the exercise example where helium gas is compressed adiabatically. Knowing initial temperature and pressure allows us to find the final state parameters, vital for various scientific and engineering applications.