[Physical Chemistry] Variables of State and Gas Dynamics

The concept of the physical state is an important part of understanding a substance. Essentially, it refers to the specific physical properties of a sample of a substance. Two samples are said to be in the same state if their physical properties are identical. The required variables to articulate the state of a system encompass the quantity of substance ($n$), the volume it engulfs ($V$), the exerted pressure ($p$), and the temperature ($T$).

Pressure

The force generated by a gas is a result of the unending bombardment of its molecules against the container walls. This incessant collision creates a uniform force, recognized as steady pressure. The standard international (SI) unit of pressure is the Pascal (Pa).

A gas’s pressure is a good indicator of whether a container holding the gas will achieve mechanical equilibrium with another gas if both are separated by a movable wall. This dynamic can be seen in two containers sharing a common movable wall. The gas with higher pressure will typically compress the gas with lower pressure, creating a shift until both pressures equalize, indicating mechanical equilibrium.

The measurement of atmospheric pressure is achieved using a barometer. The classic version of a barometer, invented by Galileo’s student Torricelli, features a sealed, inverted mercury tube. The pressure exerted by the atmosphere corresponds to the height of the mercury column when in mechanical equilibrium.

Inside a container, the pressure of a gas sample can be measured using a pressure gauge, a device that reacts to pressure. For example, a Bayard–Alpert pressure gauge operates based on gas molecule ionization, while a capacitance manometer works by monitoring diaphragm deflection. Moreover, certain semiconductors, which respond to pressure, function as transducers in solid-state pressure gauges.

Temperature

Temperature is another key variable of state. Historically, temperature measurement involved relating temperatures to the length of a liquid column. This approach gave birth to the Celsius scale, denoted as $\theta$ (theta) and expressed in degrees Celsius ($^\circ C$).

However, inconsistencies arose due to different liquid expansion rates and non-uniform expansion across a given range, resulting in different temperature readings between fixed points in thermometers made from different materials. The solution came in the form of the perfect-gas temperature scale, which relies on gas pressure and is independent of the gas type. This scale matches the thermodynamic temperature scale, and to avoid confusion, the term “thermodynamic temperature” is predominantly used.

On the thermodynamic temperature scale, temperatures are signified by $T$ and usually reported in kelvins ($K$). The relation between the Kelvin and Celsius scales is given by the formula:

\[T/K = \theta / ^\circ C + 273.15\]

This equation defines the Celsius scale in relation to the more fundamental Kelvin scale, indicating that a 1 $^\circ C$ temperature difference corresponds to a difference of 1K.

Equation of State

The state of a pure substance, theoretically, can be defined by four parameters: number of particles (n), volume (V), pressure (p), and temperature (T). However, experiments have consistently shown that specifying any three of these variables will invariably determine the fourth. Thus, it follows that each substance is characterized by its equation of state, an equation that correlates these four variables. This correlation can be mathematically presented as $p = f(T,V,n)$, highlighting that for any given substance, a specific combination of the variables n, T, and V will yield a particular pressure value. Although each substance possesses its distinct equation of state, in numerous instances, only a few explicit forms of the equations are known.

One of the essential exemplifications of the equation of state is demonstrated by the perfect gas or ideal gas. This particular form is described as $p = nRT/V$, where R stands for a constant, which is independent of the gas’s identity. The equation of state for a perfect gas was formulated by amalgamating a series of empirical laws, providing a foundational basis for understanding gases.

Historical experimental observations have contributed to the establishment of several individual gas laws, including Boyle’s Law and Charles’s Law. The former posits that pressure and volume are inversely proportional (pV = constant), while the latter states that volume is directly proportional to temperature ($V = constant × T$). These laws are examples of a limiting law, a concept that is strictly valid under specific conditions—in this case, as pressure approaches zero.

Despite the limitations, such laws provide a reasonably reliable approximation under normal pressures and are utilized across a wide range of chemical processes. For instance, when the temperature remains constant, changes in pressure result in inversely proportional changes in volume—an observation depicted graphically as hyperbolic isotherms. Similarly, under constant pressure conditions, volume varies linearly with temperature, generating isobars, while a constant volume scenario gives rise to isochores, representing the linear variation of pressure with temperature.

Individual Gas

The individual gas laws are coalesced into a single expression: $pV = constant × nT$, effectively amalgamating Boyle’s law, Charles’s law, and Avogadro’s principle. The resulting perfect gas law, $pV = nRT$, stands as the approximate equation of state for any gas and approaches absolute accuracy as the pressure of the gas nears zero. Gases that strictly adhere to this law under all conditions are deemed perfect or ideal gases.

It’s essential to note that real gases behave more like perfect gases at lower pressures and align precisely with the perfect gas law when pressure approaches zero. The gas constant R can be obtained by evaluating $R = pV/nT$ for a gas under zero pressure, thus ensuring that it’s behaving perfectly.

The molecular explanation of Boyle’s law posits that halving the volume of a gas sample effectively doubles the frequency of molecular collisions with the container walls, thereby doubling the pressure. In the context of Charles’s law, raising the gas temperature increases the average molecular speed, leading to more frequent and powerful collisions, and thus higher pressure.

The perfect gas law offers invaluable practical utility, aiding the calculation of gas properties under varying conditions. For instance, the molar volume of a perfect gas under standard ambient temperature and pressure (SATP) can be easily computed using the formula $V_m = RT/p$.

Gas Mixture

When dealing with mixtures of gases, it becomes crucial to assess the contribution of each constituent to the overall pressure. This can be achieved by determining the partial pressure, which can be defined using the mole fraction of each component.

Additionally, the total pressure of the mixture can be calculated as the sum of the partial pressures. This relationship, known as Dalton’s law, holds true for both real and perfect gases, and it states that the pressure exerted by a gas mixture equals the sum of the pressures that each gas would exert if it were alone in the container.

Reference

1. Atkins, P. W., Keeler, J. H. & De_Paula, J. Atkins’ physical chemistry. (Oxford university press, 2018).