1.3 State variables and equations of state

State Variables and Properties

State variables in thermodynamic systems

A state variable is any property that describes a thermodynamic system at equilibrium. The defining feature: state variables depend only on the current condition of the system, not on how it got there. Temperature, pressure, volume, and mass are the most common examples.

Think of it like elevation on a mountain. Your altitude at the summit is the same whether you hiked the north trail or the south trail. The path doesn't matter; only where you are now matters. That's exactly how state variables work.

Because state variables are path-independent, you can use them to calculate other important thermodynamic quantities:

• Internal energy ($U$)
• Enthalpy ($H$)
• Entropy ($S$)

These derived quantities are themselves state functions, meaning they too depend only on the current state of the system.

Intensive vs. extensive properties

Thermodynamic properties fall into two categories based on whether they scale with system size.

Intensive properties don't depend on how much matter is present. If you split a system in half, each half retains the same value for these properties:

• Temperature
• Pressure
• Density

Extensive properties do depend on the amount of matter. They're additive, so combining two subsystems means summing their values:

• Volume
• Mass
• Total energy

A useful trick for remembering the difference: if you pour half a glass of water into another container, the temperature (intensive) stays the same, but the volume (extensive) in each container is now half the original.

You can often convert an extensive property into an intensive one by dividing by the amount of substance. For example, dividing volume (extensive) by the number of moles gives you molar volume (intensive).

Ideal Gas Law and Equations of State

Applications of the ideal gas law

An equation of state is a mathematical relationship connecting the state variables of a substance. The simplest and most widely used is the ideal gas law:

$PV = nRT$

where:

• $P$ = pressure of the gas
• $V$ = volume of the gas
• $n$ = number of moles
• $R$ = universal gas constant ($8.314 \, \frac{J}{mol \cdot K}$)
• $T$ = absolute temperature (in Kelvin)

An "ideal gas" is a hypothetical gas whose molecules have no intermolecular forces and occupy zero volume. No real gas behaves this way perfectly, but many gases approximate ideal behavior well at moderate temperatures and low pressures.

When you know any three of the four variables ($P$, $V$, $n$, $T$), you can solve for the fourth. For example, to find the volume of 2 mol of gas at 300 K and 1 atm (101,325 Pa):

$V = \frac{nRT}{P} = \frac{(2)(8.314)(300)}{101325} \approx 0.0492 \, m^3$

Equations of state for real substances

Real gases deviate from ideal behavior, especially at high pressures (molecules are forced close together) and low temperatures (molecules move slowly enough for attractive forces to matter). To handle this, more sophisticated equations of state account for intermolecular forces and the finite volume molecules actually occupy.

Two common examples:

1. Van der Waals equation:

$\left(P + \frac{a}{V_m^2}\right)(V_m - b) = RT$

The constant $a$ corrects for intermolecular attractions (which reduce pressure below the ideal prediction), and $b$ corrects for the actual volume occupied by the molecules. Both are substance-specific and looked up in tables.

1. Redlich-Kwong equation:

$P = \frac{RT}{V_m - b} - \frac{a}{\sqrt{T} \, V_m(V_m + b)}$

This provides better accuracy than Van der Waals for many gases, particularly at higher pressures, while still using only two substance-specific constants.

The key takeaway: the ideal gas law is your starting point and works well in many situations. When conditions push a gas far from ideal behavior, these more complex equations give significantly better predictions.