# State Variables and Equation of State

State variables are physical quantities that describe the state of a thermodynamic system. They are called "state" variables because they define the current state of the system and are independent of how the system reached that state.

The most common state variables are temperature T, pressure P, volume V, and the number of particles n in the system. Other examples of state variables include internal energy U and entropy S.

The values of state variables determine the behavior of the system. For example, the temperature and pressure of a gas in a container determine the gas's volume and how it will interact with the walls of the container. The state variables also allow us to calculate other thermodynamic properties, such as heat capacity, thermal conductivity, and coefficient of thermal expansion.

All physical quantities are not state variables. For example, work and heat are not state variables, since their values depend on how the system reached a particular state.

An equation of state relates the state variables with each other. This relationship allows us to calculate one state variable from the others. The ideal gas law is an example of an equation of state that relates the pressure, volume, and temperature of an ideal gas: \begin{align} PV = nRT \end{align} where P is the pressure of the gas, V is its volume, n is the number of moles of gas, R is the gas constant, and T is the temperature of the gas in Kelvin.

Other equations of state include the van der Waals equation for real gases \begin{align} \left(P+a\frac{n^2}{V^2}\right)(V-nb)=nRT \end{align} where $a$ and $b$ are constants.

## Problems from IIT JEE

Problem (IIT JEE 1997): The equation of state of a real gas is given by \begin{align} \left( p+\frac{a}{V^2}\right) (V-b)=RT, \end{align} where $p$, $V$ and $T$ are pressure, volume and temperature respectively and $R$ is the universal gas constant. The dimensions of the constant $a$ in this equation is________

Solution: The dimensions of $\frac{a}{V^2}$ and $p$ are same. Thus, $[a]=[pV^2]=[\mathrm{ML^5T^{-2}}]$.

Problem (JEE Mains 2021): In thermodynamics, heat and work are

1. point functions
2. path functions
3. intensive thermodynamic state variables
4. extensive thermodynamic state variables

Problem (IIT JEE 2008): An ideal gas is expanding such that $PT^2=\text{constant}$. The coefficient of volume expansion of the gas is

1. $1/T$
2. $2/T$
3. $3/T$
4. $4/T$

Solution: The coefficient of volume expansion is defined as \begin{align} & \gamma=\frac{1}{V}\frac{\mathrm{d}V}{\mathrm{d}T}. \end{align} The ideal gas equation, $PV=nRT$, gives $P=nRT/V$. Substitute $P$ in $PT^2=\text{constant}$ to get \begin{align} & V=kT^3, \end{align} for some constant $k$. Differentiate above equation with respect to $T$ and substitute in the first equation to get $\gamma=3/T$.