# First Law of Thermodynamics and its Applications

Internal energy (U) of a system is the sum of molecular kinetic and potential energies in the centre of mass reference frame. It is a state variable (p, V and T are other state variables). The change in internal energy in a cyclic process is zero. Heat (Q) is energy in transit.

The first law of thermodynamics is a statement of conservation of energy. According to the first law, the heat $\Delta Q$ supplied to a system is sum of the increase in internal energy $\Delta U$ of the system and work done by the system i.e., \begin{align} \Delta Q=\Delta U+\Delta W \end{align} The first law of thermodynamics for a cyclic process reduces to $\Delta Q=\Delta W$ (because $\Delta U=0$ in a cyclic process).

## Solved Problems from IIT JEE

### Problem from IIT JEE 2014

A thermodynamic system is taken from an initial state i with internal energy $U_\text{i}=100\;\mathrm{J}$ to the final state f along two different paths iaf and ibf, as schematically shown in the figure. The work done by the system along the path af, ib and bf are $W_\text{af}={200}\;\mathrm{J}$, $W_\text{ib}={50}\;\mathrm{J}$ and $W_\text{bf}={100}\;{J}$ respectively. The heat supplied to the system along the path iaf, ib and bf are $Q_\text{iaf}$, $Q_\text{ib}$ and $Q_\text{bf}$ respectively. If the internal energy of the system in the state b is $U_\text{b}={200}\;\mathrm{J}$ and $Q_\text{iaf}={500}\;\mathrm{J}$, the ratio $Q_\text{bf}/Q_\text{ib}$ is,

Solution: In a thermodynamics process, the heat supplied to the system, the increase in internal energy of the system, and the work done by the system are related by the first law of thermodynamics,

\begin{align} \Delta Q=\Delta U+\Delta W. \end{align}

The first law of thermodynamics for the process iaf gives, \begin{align} \label{hxb:eqn:1} Q_\text{iaf}=U_\text{iaf}+W_\text{iaf}=(U_\text{f}-U_\text{i})+(W_\text{ia}+ W_\text{af}). \end{align} Substitute $Q_\text{iaf}={500}\mathrm{J}$, $U_\text{i}={100}\;\mathrm{J}$, $W_\text{ia}=0$ (constant volume), and $W_\text{af}={200}\mathrm{J}$ to get $U_\text{f}={400}\;\mathrm{J}$.

In the process ib, \begin{align} \label{hxb:eqn:2} Q_\text{ib}= U_\text{ib}+ W_\text{ib}=(U_\text{b}-U_\text{i})+ W_\text{ib}. \end{align} Substitute $U_\text{b}={200}\;\mathrm{J}$, $U_\text{i}={100}\;\mathrm{J}$, and $W_\text{ib}={50}\;\mathrm{J}$ to get $Q_\text{ib}={150}\;\mathrm{J}$.

In the process bf, \begin{align} \label{hxb:eqn:3} Q_\text{bf}=U_\text{bf}+ W_\text{bf}=(U_f-U_b)+ W_\text{bf}. \end{align} Substitute $U_\text{f}={400}\;\mathrm{J}$, $U_\text{b}={200}\;\mathrm{J}$ and $W_\text{bf}={100}\;\mathrm{J}$ to get $Q_\text{bf}={300}\;\mathrm{J}$. Thus, $Q_\text{bf}/Q_\text{ib}=300/150=2$.

### Problem from IIT JEE 2001

In a given process of an ideal gas, $\text{d}W=0$ and $\text{d}Q<0$. Then for the gas,

1. the temperature will decrease.
2. the volume will increase.
3. the pressure will remain constant.
4. the temperature will increase.

Solution: First law of thermodynamics, $\mathrm{d}Q=\mathrm{d}U+\mathrm{d}W$, gives $\mathrm{d}U<0$. For an ideal gas, internal energy decreases due to decrease in temperature.

## Questions on First Law of Thermodynamics

Question 1: One gram of water is vaporized at 100 deg C and atmospheric pressure. The volume water increases from 1 cm3 to 1671 cm3 in this process. If latent heat of vaporization is 2256 J/g then which of the following statement in wrong?

A. Heat supplied to the system is 2256 J
B. Work done by the system is 169.2 J
C. Increase in internal energy is 2086.8 J
D. Increase in internal energy of the system is 2425.2 J

Question 2: Which of the following statement is meaningful/correct?

A. Heat Q and work W are state variables.
B. Change in internal energy of a system may be non-zero in a cyclic process.
C. A gas in a given state can have 100 J of internal energy
D. A gas in a given state can have 100 J of heat

Question 3 (based on NCERT): Which of the following statement is not true?

A. Extensive variables indicates size of the system
B. Pressure and temperature are intensive variables
C. Internal energy and volume are extensive variables
D. The product pdV is an intensive variable