Write mathematical equations of the first law of thermodynamics for the following processes:
(a) Adiabatic process
(b) Isochoric process

Hint: Depending on the value of the variables the thermodynamic process can be of various types, like, isothermal, isobaric, isochoric, adiabatic, etc. The \[{1^{st}}\] law of thermodynamic states that energy cannot be destroyed or created but it can be changed from one to another. So, the first law of thermodynamics gives information about the energy transfer process.

Complete step by step answer:
According to the First Law of Thermodynamics, the mathematical equation for calculating the change in internal energy is given by:
 \[\Delta U = q + W\]
W is the reversible work done in a thermodynamic process. To put it in simpler terms, reversible work is the amount of work we would need to do to achieve the initial conditions of the thermodynamic reaction. Since reversible work depends only on the initial and final conditions of the thermodynamic process, it is a state function.
q is the reversible heat in a thermodynamic process. To put it in simpler terms, it is the heat that is generated by the reversing of the thermodynamic process. It results in the initial state of q once the process is completely reversed. q is a path-dependent function.
of the examples of state function is the change in internal energy. This quantity is dependent only on the initial and final states of the reaction. Internal energy is represented by \[\Delta U\].
Now, at the adiabatic process, the heat transfer between the system and surroundings is not possible. It can also be said that an adiabatic process can be defined as a thermodynamic process wherein there is no exchange of heat from the system to its surroundings neither during compression nor expansion.
So, the equation of \[{1^{st}}\] law of thermodynamics would be,
 \[
  \Delta U = q + W \\
\Rightarrow \Delta U = 0 + W \\
\Rightarrow \Delta U = W \\
 \]
And at the isochoric process, the change of volume will be zero, therefore the work done would be zero, as the work done is equal to, \[ - P\Delta V\].
So, the equation of 1st law of thermodynamics would be,
 \[
  \Delta U = q + W \\
\Rightarrow \Delta U = q - P\Delta V \\
\Rightarrow \Delta U = q - P \times 0 \\
\Rightarrow \Delta U = q \\
 \]
Note: The second law of thermodynamic states that, for any natural process, the entropy will increase for a spontaneous reaction.
This concept of entropy is used to measure the spontaneity of any process or change of the system. According to the second law of thermodynamics for an isolated system (no exchange of heat or mass with the surroundings) the reversible process always proceeds to increase the entropy.
The mathematical representation of the second law of thermodynamics is \[\Delta {S_{universe}} > 0\].