11.2.2 Non-flow processes

Ideal Gas Assumption

An ideal gas is one that follows the gas laws perfectly, where:

• There are no interactions between gas molecules other than perfectly elastic collisions.
• No intermolecular forces act between molecules, which means no potential energy. Thus, the gas's internal energy depends solely on the kinetic energy of its particles. The ideal gas equation for a non-flow process is:

$$pV = nRT$$

Where:

• $p$ = pressure,
• $V$ = volume,
• $n$ = number of moles of gas,
• $R$ = molar gas constant,
• $T$ = temperature in Kelvin. For a closed system, the amount of gas ($n$) remains constant, so $\frac{pV}{T} = \text{constant}$.

This equation can also be expressed as:

$$\frac{p_1 V_1}{T_1} = \frac{p_2 V_2}{T_2}$$

which is a useful form for comparisons at two states.

Types of Non-Flow Processes

1. Adiabatic Process

• An adiabatic process is where no heat enters or leaves the system, meaning $Q = 0$.
• For an adiabatic process, the internal energy change ( $$\Delta U) is equal to the work done by/on the system.
• Ideal gas behaviour implies that internal energy depends only on temperature, so:
• When a gas expands (does work), temperature decreases.
• When a gas compresses (work is done on it), temperature increases. For adiabatic processes, the product of pressure and volume raised to the adiabatic constant $\gamma$ is constant:

$$pV^{\gamma} = \text{constant}$$

This can also be expressed as:

$$p_1 V_1^{\gamma} = p_2 V_2^{\gamma}$$

where $\gamma$ depends on the type of gas.

2. Isothermal Process

• An isothermal process is where temperature remains constant, so $\Delta U = 0$.
• Here, all energy transfer is in the form of work done by/on the gas $($i.e.,$Q = W )$.
• The product of pressure and volume remains constant, following Boyle's Law:

$$pV = \text{constant}$$

This is expressed as:

$$p_1 V_1 = p_2 V_2$$

3. Constant Pressure Process

• In a constant pressure process, pressure remains constant as volume changes.
• The work done ($W$) can be calculated using:

$$W = p \Delta V$$

where $\Delta V$ is the change in volume.

Derivation:

Work done can be derived using the formula for force and area, assuming a piston system:

• $W = F \times d$
• Since $F = pA$ (pressure times area),
• And $d \times A = \Delta V$ (distance moved times area is volume change),
• This gives $W = p \Delta V$.

4. Constant Volume Process

• In a constant volume process, the volume remains constant. Hence, no work is done by or on the system $( W = 0)$.
• According to the first law of thermodynamics:

$$Q = \Delta U$$

This implies that all energy transfer results in a change in internal energy, affecting the temperature.

• If energy is added, temperature increases.
• If energy is removed, temperature decreases.

Applications of Non-Flow Processes

Understanding these processes is essential in thermodynamics, especially for systems where gas is heated or cooled in closed conditions (e.g., piston engines, closed cylinders).